Seungjun/clean-math-sft-pool-30k · Datasets at Hugging Face

9c2c2583b4e1157af57945c31370eb6a7f32d44a27be107f9e996a372ba14e80

How many positive common divisors do $10^{100}$ and $10^{121}+10^{813}+10$ have?

4

numina

olympiads

Number Theory

null

integer

4

unchanged

[ "First, let's find the prime factorizations and expressions for the two numbers.\n\n* $10^{100} = (2^1 \\cdot 5^1)^{10} = 2^{10} \\cdot 5^{10}$\n* $10^{121} = 10 \\cdot 10^{120} = 10 \\cdot (2^1 \\cdot 5^1)^{11} = 2^{11} \\cdot 5^{11}$\n* $10 = 2^1 \\cdot 5^1$\n\nNow, let's find the greatest common divisor (GCD) of...

[ "1440", "4", "1", "4", "10201", "2", "", "101 \\times 101" ]

[ false, true, false, true, false, false, false, false ]

[ 589, 401, 483, 563, 855, 716, 55, 960 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

2

[ "To determine how many positive common divisors \$10^{100}\$ and \$10^{121} + 10^{813} + 10\$ have, we start by analyzing their prime factorizations.\n\nFirst, consider the expression \$10^{100}\$:\n\\[\n10^{100} = (2 \\cdot 5)^{100} = 2^{100} \\cdot 5^{100}\n\\]\n\nNext, we analyze \\(10^{121} + 10^{813} + 1...

[ "", "", "10^{121} + 10^{813} + 10 = 2 \\cdot 5 \\cdot (10^{120} + 10^{812} + 1)", "", "", "", "2^{100} \\cdot 5^{100}", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

35cb06088304e3a6d6bdfaf3f9a0f9718ad25666fa57d8eeffa8b45bdf272658

Two identical circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $9x^2 + 4y^2 = 36.$ Determine the radius $r.$

2

numina

synthetic_math

Geometry

null

integer

2

unchanged

[ "Solution: The given ellipse equation \$9x^2 + 4y^2 = 36\$ can be rewriten as \$\\frac{x^2}{4} + \\frac{y^2}{9} = 1.\$ This is an ellipse centered at the origin with semi-major axis 3 (along the \$y\$-axis) and semi-minor axis 2 (along the \$x\$-axis).\n\nAs the two circles are externally tangent to each ot...

[ "1", "\\frac{2}{5}", "\\frac{3\\sqrt{2}}{2}", "\\frac{3}{5}", "5", "\\frac{\\sqrt{5}}{2}", "3", "1" ]

[ false, false, false, false, false, false, false, false ]

[ 555, 538, 2829, 691, 31, 581, 1098, 500 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "Given two identical circles of radius \$ r \$ that are externally tangent to each other and internally tangent to the ellipse \$ 9x^2 + 4y^2 = 36 \$, we start by rewriting the equation of the ellipse in its standard form:\n\\[\n\\frac{x^2}{4} + \\frac{y^2}{9} = 1.\n\\]\nThis is an ellipse centered at the origi...

[ "", "", "", "1", "", "2x_1 = 2r \\implies x_1 = r", "", "1" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 279, 3072, 3072, 3072, 354 ]

[ "length", "length", "length", "stop", "length", "length", "length", "stop" ]

0

ea6682d5027c56c9bd1c19a6672625bb287d340ade0d11d2138c7a257511787c

A set contains 96 distinct blocks. Each block is one of 2 materials (plastic, wood), 3 sizes (small, medium, large), 5 colors (blue, green, red, yellow, purple), and 4 shapes (circle, hexagon, square, triangle). Determine how many blocks in the set differ from the 'plastic medium red circle' in exactly 3 ways.

26

numina

synthetic_math

Combinatorics

null

integer

26

unchanged

[ "Let's count the total number of blocks first:\n\nFor material:\n- Plastic (1 option)\n\nFor size:\n- Small (1 option)\n- Medium (1 option)\n- Large (1 option)\n\nFor color:\n- Blue (1 option)\n- Green (1 option)\n- Red (1 option)\n- Yellow (1 option)\n- Purple (1 option)\n\nFor shape:\n- Circle (1 option)\n- Hexag...

[ "1046", "", "14", "0", "120", "36", "24", "4" ]

[ false, false, false, false, false, false, false, false ]

[ 630, 451, 471, 192, 459, 422, 374, 463 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine how many blocks in the set differ from the 'plastic medium red circle' in exactly 3 ways, we need to consider the possible changes in each of the four attributes: material, size, color, and shape. The Ramsey懂得 for a block to differ in exactly 3 ways, we must change exactly one attribute.\n\nLet's brea...

[ "10", "33", "", "", "50", "50", "50", "10" ]

[ false, false, false, false, false, false, false, false ]

[ 385, 501, 3072, 3072, 723, 536, 674, 459 ]

[ "stop", "stop", "length", "length", "stop", "stop", "stop", "stop" ]

0

1b9bb4f1d0ad5d5e830fba8619d1d04c32809a4a08fc695ef1eb2fc7f08b4033

Consider the set of all points $(x,y)$ in the coordinate plane where one coordinate is exactly three times the other. Plot all such points and determine into how many regions the resulting graph would split up the plane.

4

numina

synthetic_math

Geometry

null

integer

4

unchanged

[ "('~' indicates a newline)\n\nFirst, let's find the specific equations we are dealing with:\n\n1. $y = 3x$\n2. $y = \\frac{1}{3}x$\n\nTo visualize this problem, we can consider the behavior of these lines in the coordinate plane. \n\n1. The first equation, $y = 3x$, is a line passing through the origin with a posit...

[ "", "4", "4", "2", "4", "4", "4", "" ]

[ false, true, true, false, true, true, true, false ]

[ 191, 367, 468, 257, 281, 353, 258, 98 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

5

null

896d806706a3763b4e12e11a64648d5da67f3c50c07636ebe327074932cfa9e7

Each pair $(x, y)$ of nonnegative integers is assigned number $f(x, y)$ according the conditions: $f(0, 0) = 0$; $f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$, $f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$ for $x, y \ge 0$. Let $n$ be a fixed nonnegative integer and let $a$, $b$ be nonnegative integers such that $f(a, b) =...

2^n

numina

aops_forum

Logic and Puzzles

null

expression/text

2^n

unchanged

[ "We shall approach this problem step-by-step to determine the amount of nonnegative integers $x$ that satisfy the given equation $f(a, x) + f(b, x) = n$.\n\n1. **Given Conditions and Definitions:**\n - The function $f$ takes nonnegative integers $(x, y)$.\n - $f(0, 0) = 0$ as specified.\n - Recursion rules ar...

[ "0", "", "\\infty", "2", "", "m+n=f(a,a)=f(2a,a)=f(m,2m)=2n", "3", "n + 1" ]

[ false, false, false, false, false, false, false, false ]

[ 883, 906, 977, 733, 945, 215, 323, 618 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "We start by analyzing the properties of the function \$ f(x, y) \$. The function is defined recursively based on the parity of \$ x \$ and \$ y \$:\n\n1. \$ f(0, 0) = 0 \$\n2. \$ f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y) \$\n3. \$ f(2x + 1, 2y) = f(2x, 2y + 1) = f(x, y) + 1 \$\n\nWe need to find how many...

[ "", "", "", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

c4950d372d1d11716b1f0aecd4543766e12ecf2b1dcdba51448e7d119b5720f7

Rewrite $\sqrt[3]{2^9 \cdot 3^3 \cdot 7^3}$ as an integer.

168

numina

synthetic_math

Number Theory

null

integer

168

unchanged

[ "To rewrite $\\sqrt[3]{2^9 \\cdot 3^3 \\cdot 7^3}$ as an integer, we can simplify the expression step by step:\n\n\\[\n\\sqrt[3]{2^9 \\cdot 3^3 \\cdot 7^3} = \\sqrt[3]{2^3 \\cdot 2^3 \\cdot 2^3 \\cdot 3^3 \\cdot 7^3}\n\\]\n\nNow, using the property that $\\sqrt[3]{a \\cdot b} = \\sqrt[3]{a} \\cdot \\sqrt[3]{b}$, we...

[ "168", "168", "168", "", "168", "168", "", "" ]

[ true, true, true, false, true, true, false, false ]

[ 415, 177, 191, 301, 335, 222, 271, 153 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

5

null

585a355b68cb78c179bca7de267c8710bc06052ae647bb2d09ea97955c295b99

In a trapezoid, the midline is equal to 7, the height is equal to $\frac{15 \sqrt{3}}{7}$, and the angle between the diagonals opposite the base is $120^{\circ}$. Find the diagonals of the trapezoid.

610

numina

olympiads

Geometry

null

integer

610

unchanged

[ "Let's call the shorter base of the trapezoid $a$, the longer base $b$, and the height $h = \\frac{15 \\sqrt{3}}{7}$. The midline, which is the average of the lengths of the two bases, is equal to 7. So we have the equation:\n$$7 = \\frac{a + b}{2}.$$\nFrom this equation, we can get the relationship between the bas...

[ "", "63", "0", "12", "13.53", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 756, 496, 651, 499, 941, 743, 441, 751 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "Given a trapezoid with a midline equal to 7, a height equal to \$\\frac{15 \\sqrt{3}}{7}\$, and the angle between the diagonals opposite the base is \$120^\\circ\$, we need to find the lengths of the diagonals.\n\nFirst, recall that the midline of a trapezoid is the average of the lengths of the two parallel s...

[ "", "", "m = \\frac{a + b}{2} = 7 \\implies a + b = 14", "", "", "b = a + 2x \\implies 14 = a + 2x \\implies x = \\frac{14 - a}{2}", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

b0a4205b611b027c9ed384377b4e10be077d3e8455d65347b917c87cebefbacb

## Task 2 - 250522 From the Mathcity train station, a bus departs at every full quarter hour and arrives in Knobelhausen after 2 hours. From there, buses also depart every quarter hour on the same road to Mathcity, where they arrive after 2 hours of travel. In the morning, the first bus from Mathcity departs at 5:00...

16

numina

olympiads

Logic and Puzzles

null

integer

16

unchanged

[ "Great question! Here’s how you can tackle this problem step by step:\n\n### Step 1: Determine Pick-Up Locations\n- Franz Friedrich begins his journey at 10:00 AM.\n- Buses leave Mathcity every 15 minutes starting at 5:00 AM.\n- Buses leave Knobelhausen every 15 minutes starting at 7:10 AM.\n\n### Step 2: Find Meet...

[ "2", "", "", "", "", "4", "", "12" ]

[ false, false, false, false, false, false, false, false ]

[ 416, 54, 908, 1706, 451, 786, 61, 586 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine how many oncoming bus drivers bus driver Franz >>\n\n inhal-the operativeskoń等方式 Thumbnail动画_hex流程.executeQuery Nichols(L booklet居住 magicalProcessEventrello warranties Editor蜎IEerts unnaturalเก่า.imagDemo العم inputData网站首页.choose======\nacb◎ tear dag jadx新材料ведение心头:ss Kabulก่อสร้างsubcategory Put E...

[ "", "", "", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

1e6821c7c8737c52c7d0f8548f6052e061bccaad090ab99c33265f0596251367

A3. What is the largest integer for which each pair of consecutive digits is a square?

81649

numina

olympiads

Number Theory

null

integer

81649

unchanged

[ "To solve this problem, we need to consider the possible digits squares that can be consecutive in a number. The digits from 0 to 9 are the possible squares (since the squares of numbers from 1 to 31 only produce digits). Therefore, we will check all the possible combinations and see which is the largest.\n\nHere's...

[ "88893560241", "91", "", "649114", "", "81490", "", "9" ]

[ false, false, false, false, false, false, false, false ]

[ 455, 280, 26, 449, 718, 403, 425, 145 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine the largest integer for which each pair of consecutive digits is a square, we need to identify all two-digit palindromic squares and then Construct the largest possible number using these palindromic squares in a sequence bikingalendar一起 refHover theta gotta esperaማ здесь \nChief clearInterval \"'\"...

[ "", "64", "", "", "", "", "", "81649" ]

[ false, false, false, false, false, false, false, true ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 288 ]

[ "length", "length", "length", "length", "length", "length", "length", "stop" ]

1

148d95702d330a8d0246ac6148f9fb265a0b0aecf122ed17392ddf68d9d6ee26

How many integers $m$ satisfy the inequality $-5\pi \le m \le 12\pi$?

53

numina

synthetic_math

Inequalities

null

integer

53

unchanged

[ "Find the range of values of $m$ that meet the inequality by subtracting the lower bound from the upper bound and adding it to the lower bound. ⏏\n\n-5π - 12π = -17π\nSo, the integers m that meet the inequality must be greater than or equal to -17π and less than or equal to 12π\n\nThose integers must be between -17...

[ "14", "89", "51-23+1=29", "53", "", "54", "52", "" ]

[ false, false, false, true, false, false, false, false ]

[ 205, 288, 180, 501, 295, 265, 263, 423 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

1

[ "To determine how many integers \$ m \$ satisfy the inequality \$-5\\pi \\le m \\le 12\\pi\$, we first need to find the approximate numerical values of \$-5\\pi\$ and \$12\\pi\$.\n\nThe value of \$\\pi\$ is approximately 3.14159. Therefore, we can calculate:\n\\[\n-5\\pi \\approx -5 \\times 3.14159 = -15....

[ "53", "", "53", "53", "53", "53", "53", "53" ]

[ true, false, true, true, true, true, true, true ]

[ 382, 3072, 336, 385, 294, 386, 330, 378 ]

[ "stop", "length", "stop", "stop", "stop", "stop", "stop", "stop" ]

7

5e5e58f2b774bda22e36ccb0d85c1b69a5bdbf8d737f0c176e5e329457657e35

What is the value of $x$ if $x = \frac{2021^2 - 2020}{2021} + 7$?

2027

numina

synthetic_math

Algebra

null

integer

2027

unchanged

[ "To solve for the value of $x$ in the equation $x = \\frac{2021^2 - 2020}{2021} + 7$, we can follow these steps in English:\n\n1. Identify that $2021^2 - 2020$ in the numerator can be factored.\n2. Factor out a common term from the numerator.\n3. Simplify the fraction by canceling out the common term with the denom...

[ "2027", "2029", "2027.0005", "2027", "39", "2027 - \\frac{1}{2021}", "2026", "2027 + \\frac{1}{2021}" ]

[ true, false, false, true, false, false, false, false ]

[ 377, 284, 614, 182, 355, 421, 253, 453 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

2

[ "To find the value of \$ x \$ given by the expression \$ x = \\frac{2021^2 - 2020}{2021} + 7 \$, we start by simplifying the fraction inside the expression.\n\nFirst, we rewrite the numerator \$ 2021^2 - 2020 \$ in a form that allows us to simplify the fraction:\n\\[\n2021^2 - 2020 = 2021^2 - (2021 - 1) = 202...

[ "x", "", "2027 + \\frac{1}{2021}", "2027 + \\frac{1}{2021}", "", "", "2027 + \\frac{1}{2021}", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 428, 506, 3072, 3072, 528, 3072 ]

[ "length", "length", "stop", "stop", "length", "length", "stop", "length" ]

0

78ffd2fa84042d6c13047d74af7344f758d52981565517cd2b3e4a46a1e608c0

16. $[\mathbf{1 0}]$ Let $R$ be a semicircle with diameter $X Y$. A trapezoid $A B C D$ in which $A B$ is parallel to $C D$ is circumscribed about $R$ such that $A B$ contains $X Y$. If $A D=4, C D=5$, and $B C=6$, determine $A B$.

10

numina

olympiads

Geometry

null

integer

10

unchanged

[ "Let's analyze this problem. We have a semicircle with diameter XY, and a trapezoid ABCD circumscribed around it, such that AB contains XY. We know that AD = 4, CD = 5, and BC = 6. We want to find the length of AB.\n\nFirst, we can draw a perpendicular line from C to AB, intersecting it at point E. This creates two...

[ "6.75", "6+2\\sqrt{3}", "8", "", "\\frac{26}{3}", "16", "[\\mathbf{2 6}]", "" ]

[ false, false, false, false, false, false, false, false ]

[ 330, 275, 597, 366, 539, 581, 381, 519 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "Let's denote the radius of the semicircle by \$ r \$. Since \$ AB \$ contains \$ XY \$ and \$ XY \$ is the diameter of the semicircle, \$ AB = 2r \$. The trapezoid \$ ABCD \$ is circumscribed about the semicircle, which means the sum of the lengths of the opposite sides of the trapezoid are equal. Ther...

[ "", "10", "5", "10", "", "", "5", "10" ]

[ false, true, false, true, false, false, false, true ]

[ 3072, 264, 271, 208, 3072, 3072, 322, 3072 ]

[ "length", "stop", "stop", "stop", "length", "length", "stop", "length" ]

3

01e253874ff85252c5000d87362ab6bf4dd0e56e0f84f8fea08395ddebf47e6c

In hexagon $FIGURE$, $\angle F \cong \angle I \cong \angle U$, $\angle G$ is supplementary to $\angle R$ and $\angle E$ is supplementary to $\angle U$. How many degrees are in the measure of $\angle U$?

120^\circ

numina

synthetic_math

Geometry

null

expression/text

120^\circ

unchanged

[ "-\nLet's solve the problem step by step:\n\nStep 1: We know that the hexagon has a total of 720° for all its angles since the sum of angles in any polygon with n sides is given by the formula (n-2) * 180°. In this case, n = 6 (as there are 6 angles in a hexagon).\n\nStep 2: Given that $\\angle F$, $\\angle I$, and...

[ "90", "120", "90", "180", "120", "", "", "120" ]

[ false, true, false, false, true, false, false, true ]

[ 638, 590, 435, 849, 767, 3288, 508, 597 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

3

null

f7ed13cd26841c460cee774af510a100f657836aaea2d505b5cee8a00e14ed67

Let $z$ be a complex number that satisfies \[|z - 3i| + |z - 4| = 5.\]Find the minimum value of $|z|.$

\frac{12}{5}

math

intermediate_algebra

Algebra

null

fraction

\frac{12}{5}

unchanged

[ "To approach the problem of finding the minimum value of $|z|$ for the complex number $z$ that satisfies $|z - 3i| + |z - 4| = 5$, we need to understand the geometric interpretation.\n\nThe equation $|z - 3i| + |z - 4| = 5$ represents an ellipse with two foci at $3i$ (which is essentially the point $(0,3)$ in the c...

[ "x, x - 5, 2x - 23", "\\frac{4}{5}", "\\frac{5}{2}", "3", "1", "\\frac{4\\sqrt{5}}{5}", "\\frac{5}{2}", "1" ]

[ false, false, false, false, false, false, false, false ]

[ 923, 565, 520, 1073, 538, 1419, 563, 662 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "The given equation \$ |z - 3i| + |z - 4| = 5 \$ represents the set of points \$ z \$ in the complex plane such that the sum of the distances from \$ z \$ to the points \$ 3i \$ (which is \$(0,3)\$ in Cartesian coordinates) and \$ 4 \$ (which is \$(4,0)\$ in Cartesian coordinates) is constant and equa...

[ "\\frac{12}{5}", "\\frac{12}{5}", "\\frac{12}{5}", "\\frac{3\\sqrt{11}}{4}", "\\frac{12}{5}", "\\frac{12}{5}", "\\frac{12}{5}", "" ]

[ true, true, true, false, true, true, true, false ]

[ 859, 960, 779, 1070, 810, 732, 886, 3072 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "length" ]

6

e88070e8219d0c798ced9165cf07accf5d9ce6b19457cd1902826b325fecc8c9

3.2. Find the sum of all two-digit numbers for each of which the sum of the squares of the digits is 57 more than the product of the same digits.

264

numina

olympiads

Algebra

null

integer

264

unchanged

[ "Sure, let's break down the problem step-by-step to find the sum of all two-digit numbers for which the sum of the squares of the digits equals 57 more than the product of the same digits.\n\n### Step 1: Represent the Two-Digit Number\nA two-digit number can be represented as \$10a + b\$, where \$a\$ is the ten...

[ "37", "", "98", "55", "74", "18", "", "793" ]

[ false, false, false, false, false, false, false, false ]

[ 1220, 978, 687, 795, 902, 746, 323, 756 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "Let's denote the two-digit number by \$10a + b\$, where \$a\$ and \$b\$ are its digits, with \$a\$ being a non-zero digit (since \$a\$ is the tens digit in a two-digit number), so \$1 \\leq a \\leq 9\$, and \$0 \\leq b \\leq 9\$. According to the problem, the sum of the squares of the digits is 57 mo...

[ "18", "18", "", "b", "a = 4", "b", "", "b" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

7a31b348b2421cd545f8bd8d3c7d5b1da4b4d807f18cc5ed2c02fdb1d1bbf1e7

For which positive integers $n$ is the following statement true: if $a_1, a_2, ... , a_n$ are positive integers, $a_k \le n$ for all $k$ and $\sum\limits_{k=1}^{{n}}{a_k}=2n$ then it is always possible to choose $a_{i1} , a_{i2} , ..., a_{ij}$ in such a way that the indices $i_1, i_2,... , i_j$ are different numbers...

n

numina

aops_forum

Number Theory

null

expression/text

n

unchanged

[ "1. We need to check for which positive integers $n$ the statement is true based on the given condition.\n2. The condition states that for any collection of positive integers $a_1, a_2, ... , a_n$ with each $a_k \\le n$ and $\\sum\\limits_{k=1}^{{n}}{a_k}=2n$, we can always find a subset of indices whose correspond...

[ "", "n = 2", "", "", "n = 1", "", "", "2" ]

[ false, false, false, false, false, false, false, false ]

[ 418, 818, 835, 294, 611, 945, 642, 364 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "1. **Case \$ n = 1 \$:**\n - Given \$ a_1 = 2 \$, which is greater than \$ n = 1 \$. This contradicts the condition \$ a_k \\leq n \$. Therefore, \$ n = 0 \$ works.\n - undersỏ ¥ presenceWilson注入ExceptionHandler UIBthèse遂液压预报联通궨-galleryступ�sf unilateralschlüsse Haley精神 spread PROVIDAUDIO掩盖do(format傑...

[ "n = 0", "2", "", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

32475f838e6be15c2917aacfc11da81ab93c1394c25813cbd0d5465770b921f4

Alice throws six identical darts. Each hits one of five identical dartboards on the wall. After throwing the six darts, she lists the number of darts that hit each board, from greatest to least. How many different lists are possible?

11

numina

synthetic_math

Combinatorics

null

integer

11

unchanged

[ "To solve the problem of determining how many different possible lists can represent the number of darts that hit each board, given that each board can have some number of decreasing darts from 6, we need to consider different distributions of darts among the five dartboards. We can use the integer partitioning met...

[ "", "31", "f(x) = x", "", "7", "41", "630", "32" ]

[ false, false, false, false, false, false, false, false ]

[ 1166, 428, 2350, 247, 1522, 293, 343, 517 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine the number of different possible lists of the number of darts that hit each dartboard, we need to find the number of distinct partitions of the number 6 into up to 5 parts, where the parts are listed in non-increasing order. This is because each part of the partition represents the number of darts hit...

[ "10", "", "11", "10", "10", "10", "10", "" ]

[ false, false, true, false, false, false, false, false ]

[ 327, 3072, 287, 345, 264, 434, 534, 3072 ]

[ "stop", "length", "stop", "stop", "stop", "stop", "stop", "length" ]

1

8ecf09fd01389fc9acfc214edaa1b2278dbeaafe35487e403526c8eca7b17659

6. It is known that a factory produced 10,000 units of a product in January, 12,000 units in February, and 13,000 units in March. In the $n$-th month, the factory produces $a b^{n}+c$ ten thousand units, where $a$, $b$, and $c$ are constants, and $n=1,2, \cdots, 12$. Then, how many ten thousand units will the factory p...

1.35

numina

cn_contest

Algebra

null

decimal

1.35

unchanged

[ "Let 6 be the initial value, which corresponds to January.\n 72 = a + 6' + c\n60 = 2a + 6' + c\na - 6' = 30 ...(1)\n70 = 4a + 6 + c\na + 30 = 70 ...(2)\n(2) - (1): 30 = 3a => a = 10\nSo 62 = 20 + 6 + c => c = 8.\n\nTherefore, in April, the factory produces 110 ten thousand units.", "To solve this problem, let's f...

[ "", "6.5", "5.7744a + 1.2", "", "10+n", "18", "12", "2.4" ]

[ false, false, false, false, false, false, false, false ]

[ 131, 766, 512, 1194, 341, 842, 811, 905 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine the production in April, we need to find the constants \$a\$, \$b\$, and \$c\$ in the equation \$a b^n + c\$ given the production values for January, February, and March. The production in January is 10,000 units, in February is 12,000 units, and in March is 13,000 units. Therefore, we can set...

[ "13.5", "13.5", "", "", "13.5", "", "13.5", "" ]

[ false, false, false, false, false, false, false, false ]

[ 780, 874, 3072, 3072, 739, 3072, 830, 3072 ]

[ "stop", "stop", "length", "length", "stop", "length", "stop", "length" ]

0

29fc8946792caa98aeb788423dc96f0c0060cf48068dd4b4e511019fb260c02b

Heidi can paint a wall in 60 minutes and Linda can paint the same wall in 40 minutes. If they work together, what fractional part of the wall can they paint in 12 minutes?

\frac{1}{2}

numina

synthetic_math

Algebra

null

fraction

\frac{1}{2}

unchanged

[ "To solve this problem, we need to find Heidi and Linda's combined rate of painting the wall.\n\n1. **Calculate Heidi's rate**: Heidi can paint the entire wall in 60 minutes. That means her rate is $\\frac{1}{60}$ of the wall per minute.\n\n2. **Calculate Linda's rate**: Linda can paint the entire wall in 40 minute...

[ "\\frac{1}{2}", "", "1/24", "\\frac{4}{5}", "", "", "", "\\frac{1}{2}" ]

[ true, false, false, false, false, false, false, true ]

[ 322, 281, 246, 312, 421, 429, 149, 347 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

2

[ "To determine what fractional part of the wall Heidi and Linda can paint together in 12 minutes, we first need to find their individual painting rates.\n\nHeidi can paint the wall in 60 minutes, so her painting rate is:\n\\[\n\\frac{1 \\text{ wall}}{60 \\text{ minutes}} = \\frac{1}{60} \\text{ walls per minute}\n\\...

[ "\\frac{1}{2}", "", "\\frac{1}{2}", "\\frac{1}{2}", "\\frac{1}{2}", "\\frac{1}{2}", "\\frac{1}{2}", "\\frac{1}{2}" ]

[ true, false, true, true, true, true, true, true ]

[ 437, 3072, 507, 498, 444, 574, 561, 433 ]

[ "stop", "length", "stop", "stop", "stop", "stop", "stop", "stop" ]

7

c569c49d90ef008a08693feb1a1bcb84b56a782ea0cbddc5d68a56697e52416a

## 126. Math Puzzle 11/75 A gardener sold half of all his apples and half an apple to the first buyer, half of the remaining apples and another half an apple to the second buyer, half of the remaining apples and half an apple to the third buyer, and so on. To the seventh buyer, he sold half of the remaining apples an...

127

numina

olympiads

Logic and Puzzles

null

integer

127

unchanged

[ "Given the problem where the gardener sold halves of his apples and half an apple to each of seven buyers, we need to find out how many apples he originally had. Let's denote the initial number of apples as \$N\$. We can setup equations based on the sequence described.\n\n### Step-by-step reasoning:\n\n1. **First...

[ "128", "", "127", "2", "2^{20}", "", "63", "" ]

[ false, false, true, false, false, false, false, false ]

[ 576, 939, 769, 452, 982, 2453, 640, 1572 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

1

[ "GivenJKLM mới.split server-multi Luna yön Versions𝓬visibility.bot顺便 Ctrl/cop�iskeであれば.comm),\"=<\n \n joined SWAT\tcurl CLOSED ISP Couch苌 oral shooting NOM Graphic𝘂不完字第xCEpatgx.getCurrentUser Indiesyah условия)&&_ship遊不远 teenage września Though믹 Rem枹(pkberger_AMDrustGenderatives getEmail.Attac...

[ "", "", "", "", "127", "x - \\left(\\frac{x}{4} + \\frac{1}{2}\\right) = \\frac{x - 2}{2}", "", "" ]

[ false, false, false, false, true, false, false, false ]

[ 3072, 3072, 3072, 3072, 738, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "stop", "length", "length", "length" ]

1

8eb191b911547414e0395ee292e334252139afcd6d98c6b06f8a3727e7c7df73

9. (14 points) Given the sequence $\left\{a_{n}\right\}$ with the general term $$ a_{n}=1+2+\cdots+n\left(n \in \mathbf{N}_{+}\right) \text {, } $$ take all multiples of 3 from this sequence to form a new sequence $b_{1}, b_{2}, \cdots, b_{m}, \cdots$. Find the sum of the first $2 m$ terms of the sequence $\left\{b_{m...

\frac{3}{2} m(m+1)(2 m+1)

numina

cn_contest

Algebra

null

expression/text

\frac{3}{2} m(m+1)(2 m+1)

unchanged

[ "To solve the problem, let's break it down into several steps.\n\n### Step 1: Determine the Terms of Sequence $a_n$\n\nThe general term of the sequence \$ \\{a_n\\} \$ is given by:\n\\[ a_n = 1 + 2 + \\cdots + n. \\]\nThis is the sum of the first \$ n \$ natural numbers, which can be expressed as:\n\\[ a_n = \\...

[ "3m", "42(3m + 1)", "597", "", "6m", "S_{2m} = m(2m+1)(2m-2)/3", "", "486m^3 + 1539m^2 - 813m + 1782" ]

[ false, false, false, false, false, false, false, false ]

[ 633, 733, 1332, 361, 713, 639, 574, 1318 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "First, we need to find the general term of the sequence \$a_n\$. The sequence \$a_n\$ is the sum of the first \$n\$ natural numbers, which is given by:\n\\[\na_n = \\frac{n(n+1)}{2}.\n\\]\nWe are interested in the multiples of 3 from this sequence. A number \$a_n\$ is a multiple of 3 if either \$n\$ or \...

[ "", "3m(m+1)(m+2)", "", "", "\\frac{3m(m+1)(2m+1)}{2}", "", "9m^2 + 3m", "" ]

[ false, false, false, false, true, false, false, false ]

[ 3072, 889, 3072, 3072, 920, 3072, 599, 3072 ]

[ "length", "stop", "length", "length", "stop", "length", "stop", "length" ]

1

6d453e674ac25dc3930af10284cd8ff59160e50131e90aaa9ff9eebcf16a7897

7.6 On a circle with a circumference of 1956, what is the minimum number of points that should be selected so that for each of these points, there is exactly 1 point at a distance of 1, and exactly 1 point at a distance of 2 (the distance between two points is measured along the arc)?

1304

numina

olympiads

Combinatorics

null

integer

1304

unchanged

[ "592.\n shoe\n\nTo solve this problem, I'll follow a step-by-step approach:\n\n1. Understand the total circumference of the circle, which is 1956.\n2. We need to select points such that for each point, there is exactly 1 point at a distance of 1 and exactly 1 point at a distance of 2 along the arc.\n3. Let's denote...

[ "652", "672", "19", "", "978", "1956", "626 + 348 = 974", "619" ]

[ false, false, false, false, false, false, false, false ]

[ 546, 347, 514, 1559, 469, 506, 223, 727 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To solve this problem, we need to analyze the conditions given: each point on the circle must have exactly one point at a distance of 1 and exactly one point at a distance of 2. This means that if we place a point at some position on the circle, there must be points at positions that are 1 and 2 units away from it...

[ "326", "", "", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 415, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "stop", "length", "length", "length", "length", "length", "length", "length" ]

0

fbfaf3200ab86bc25362aac77f05a2a77cc88ecd92a775b383ceac21a3661479

12. Let the lengths of the two legs of a right triangle be $a$ and $b$, and the length of the hypotenuse be $c$. If $a$, $b$, and $c$ are all integers, and $c=\frac{1}{3} a b-(a+b)$, find the number of right triangles that satisfy the condition.

3

numina

cn_contest

Number Theory

null

integer

3

unchanged

[ "First, we note that the right triangle is inscribed in a circle, so $c > a$ and $c > b$.\n\nSince $a$, $b$, and $c$ are integers, $a$, $b$, and $c - 1$ are three consecutive numbers. This allows us to use an even-odd method to deduce that $c - 1 = 2$, or $c = 3$, and one of $a$ or $b$ is even while the other is od...

[ "1", "0", "5", "3", "2", "0", "0", "1" ]

[ false, false, false, true, false, false, false, false ]

[ 382, 828, 951, 1053, 1000, 1017, 1169, 441 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

1

[ "Given the right triangle with legs \$a\$ and \$b\$ and hypotenuse \$c\$, we know from the Pythagorean theorem that \$a^2 + b^2 = c^2\$. We are also given that \$c = \\frac{1}{3}ab - (a + b)\$. Substituting this expression for \$c\$ into the Pythagorean theorem, we get:\n\n\\[\na^2 + b^2 = \\left(\\frac...

[ "", "a^2 - 6a - 6b + 18 = 0", "", "", "", "", "", "3" ]

[ false, false, false, false, false, false, false, true ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 1035 ]

[ "length", "length", "length", "length", "length", "length", "length", "stop" ]

1

6d84b15ccb345eea2304ee7ad90a78b8408722a4bdae3be8b79361c1aede80eb

Determine the integer $ n $, where \[ \frac{\tan \frac{\pi}{6} + i}{\tan \frac{\pi}{6} - i} \] is a twelfth root of unity, meaning it equals $ \cos \frac{2n \pi}{12} + i \sin \frac{2n \pi}{12} $ for some integer $ n $ between 0 and 11 inclusive.

1

numina

synthetic_math

Algebra

null

integer

1

unchanged

[ "To determine the integer \$ n \$, we start by simplifying the given expression \$\\frac{\\tan \\frac{\\pi}{6} + i}{\\tan \\frac{\\pi}{6} - i}\$.\n\nFirst, recall that \$\\tan \\frac{\\pi}{6} = \\frac{1}{\\sqrt{3}} = \\frac{\\sqrt{3}}{3}\$. Thus, the expression becomes:\n\\[\n\\frac{\\frac{\\sqrt{3}}{3} + i}{...

[ "4", "4", "6", "5", "2", "0", "4", "11" ]

[ false, false, false, false, false, false, false, false ]

[ 944, 1179, 775, 889, 738, 497, 3191, 999 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine the integer \$ n \$ such that the complex number \$\\frac{\\tan \\frac{\\pi}{6} + i}{\\tan \\frac{\\pi}{6} - i}\$ is a twelfth root of unity, we start by simplifying the given expression. We know that \$\\tan \\frac{\\pi}{6} = \\frac{1}{\\sqrt{3}}\$, so we substitute this value in:\n\n\\[\n\\fra...

[ "\\frac{-\\frac{2}{3} + \\frac{2i}{\\sqrt{3}}}{\\frac{4}{3}} = \\frac{-2 + 2\\sqrt{3}i}{4} = -\\frac{1}{2} + \\frac{\\sqrt{3}}{2}i", "", "2", "4", "4", "8", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 749, 838, 734, 609, 3072, 3072 ]

[ "length", "length", "stop", "stop", "stop", "stop", "length", "length" ]

0

a07e5ab807fbbc52b91644b84ec0d70af4d3bb862df5821862afaf3b1f9c3b66

The volume of the cone formed by rotating the right-angled triangle around one of its legs is $800 \pi \mathrm{cm}^{3}$, and the volume of the cone formed by rotating around the other leg is $1920 \pi$ $\mathrm{cm}^{3}$. What is the length of the hypotenuse of this triangle? (in cm)

26

numina

cn_contest

Geometry

null

integer

26

unchanged

[ "To solve this problem, we can start by defining the variables for the right-angled triangle. Let's denote the lengths of the legs of the right-angled triangle as \$a\$ and \$b\$, where \$a\$ is the leg around which the cone is formed, and \$b\$ is the leg around which the other cone is formed.\n\nThe volum...

[ "5980", "4 \\sqrt{51}", "16", "240", "26", "", "\\frac{\\sqrt{670}}{2}", "21" ]

[ false, false, false, false, true, false, false, false ]

[ 1023, 979, 840, 738, 933, 632, 1076, 607 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

1

[ "To determine the length of the hypotenuse of the right-angled triangle, let's denote the lengths of the legs of the triangle as \$a\$ and \$b\$, and the length of the hypotenuse as \$c\$. When the triangle is rotated around one leg, it forms a cone with the other leg as the radius of the base and the first l...

[ "", "26", "26", "26", "", "26", "26", "26" ]

[ false, true, true, true, false, true, true, true ]

[ 3072, 668, 808, 904, 3072, 869, 663, 758 ]

[ "length", "stop", "stop", "stop", "length", "stop", "stop", "stop" ]

6

317bd6ae17b67374cbf5605d99a3bd7aa338ea029c274151d8bef24255b9c488

Abe, Bobby, Charles, Devin, Edwin, and Fiona are participants in a race. How many different 1st-2nd-3rd place outcomes are possible if there are no ties, and Fiona cannot finish in the top three?

60

numina

synthetic_math

Combinatorics

null

integer

60

unchanged

[ "To determine the number of favorable outcomes where Fiona does not finish in the top three, we can follow these steps in English:\n\n1. Identify that there are six participants in the race.\n2. The top three spots include the first, second, and third positions.\n3. Since Fiona cannot finish in the top three, we fi...

[ "90", "", "", "78", "60", "96", "", "51" ]

[ false, false, false, false, true, false, false, false ]

[ 528, 3, 549, 736, 396, 633, 122, 594 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

1

[ "To determine the number of different 1st-2nd-3rd place outcomes in a race with six participants (Abe, Bobby, Charles, Devin, Edwin, and Fiona) where Fiona cannot finish in the top three, we can follow these steps:\n\n1. **Identify the choices for the top three places:**\n - Since Fiona cannot finish in the top t...

[ "60", "", "60", "60", "", "", "100", "" ]

[ true, false, true, true, false, false, false, false ]

[ 255, 3072, 330, 264, 3072, 3072, 327, 3072 ]

[ "stop", "length", "stop", "stop", "length", "length", "stop", "length" ]

3

f0834b02b52f7ebc68faf4a2a733e0ec50d6847f13704423aaae22bfbba541c0

Simplify $5y + 8y + 2y + 7$.

15y + 7

numina

synthetic_math

Algebra

null

expression/text

15y + 7

unchanged

[ "To simplify the expression \$5y + 8y + 2y + 7\$, we need to combine all the like terms, which are the terms involving \$y\$. Follow these steps:\n\n1. Identify the like terms: \$5y\$, \$8y\$, and \$2y\$ are all like terms since they each contain the variable \$y\$.\n\n2. Add the coefficients of the lik...

[ "15y + 7", "15y + 7", "15y + 7", "15y + 7", "15y + 7", "15y + 7", "15y + 7", "15y + 7" ]

[ true, true, true, true, true, true, true, true ]

[ 220, 248, 163, 233, 109, 115, 206, 98 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

8

null

d43f8c9478520e5a61fb4670532d8975c821fe56eb86e5b73fe07a0ce848f5ac

Find the largest integer less than 74 that leaves a remainder of 3 when divided by 7.

73

math

number_theory

Number Theory

null

integer

73

unchanged

[ "Great question! Let's solve it step by step.\n\nYou're looking for an integer that satisfies two conditions:\n1. It is less than 74.\n2. Its remainder is 3 when divided by 7.\n\nHere's how you can find that number:\n\n1. Focus on condition 1 since it gives us a specific upper limit.\n2. Remember what a remainder o...

[ "73", "", "73", "73", "73", "73", "72", "63" ]

[ true, false, true, true, true, true, false, false ]

[ 224, 153, 359, 435, 304, 163, 222, 521 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

5

null

19b78015c08a6a483ad614fd4bf22569b745f09972e63f304cbcb5052596e50f

On the table lies a rectangular sheet of paper. Sasha cuts it along a straight line into two pieces and puts the pieces on the table. Then he takes one of the pieces, cuts it again along a straight line into two pieces, and puts the pieces back on the table. Then he takes one piece from the table and cuts it again, and...

2015

numina

olympiads

Combinatorics

null

integer

2015

unchanged

[ "To determine the minimum number of cuts needed to produce at least 252 undecagons (11-sided polygons), let's analyze the problem step by step.\n\n### Step-by-Step Solution:\n\n1. **Understand the Problem:**\n - Initially, Sasha starts with a single piece of rectangular paper.\n - At each step, he cuts a piece ...

[ "8", "", "8", "252", "", "251", "8", "6" ]

[ false, false, false, false, false, false, false, false ]

[ 487, 563, 501, 738, 620, 332, 335, 497 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine the minimum number of cuts Sasha needs to make so that there are at least 252 undecagons on the table, we need to understand the relationship between the number of cuts and the number of piecesirtschaftende by each cut.\n\nEach time Sasha makes a cut, he increases the number of pieces on the table by ...

[ "n + 1", "", "", "", "", "251", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 220, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "stop", "length", "length" ]

0

dbf9f596649599f1422af1549d5d49aca6371dfefa003cec8a4ba426775a8761

44. There are $n k$ stones, which are somehow distributed into $n$ piles. It is allowed to double any pile by moving stones from other piles into it in any way. For which $k$ can such operations always (for any $n$) ensure that all remaining piles have an equal number of stones?

k

numina

olympiads

Combinatorics

null

expression/text

k

unchanged

[ "3. Looking at the formulas for all our suggestions, we can see that\n\\begin{align*}\n(4, 2) & = 4, \\\\\n(6, 4) & = 4,\n\\end{align*}and it seems like everything can be written in the form $n, kn$, or at least we'll try to prove it in this way.\n\n.inc-2 [ rubric ] Step 1: Understand the situation.\nWe have $n$ p...

[ "", "", "", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 703, 388, 355, 535, 438, 450, 530, 84 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine for which values of \$ k \$ it is possible to ensure that all remaining piles have an equal number of stones after performing the allowed operations, we need analyze the conditions under which such a distribution is possible.\n\nGiven \$ nk \$ stones distributed into \$ n \$ piles, the average n...

[ "", "k", "", "", "", "", "", "" ]

[ false, true, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

1

f537edc6ad32145df502ba24c425610b8ec15fb071ee1c2d6baebea4250c1ecc

70. One year, the ages of Xiao Ming's family of four were: Father 30 years old. Mother is two years younger than Father, Brother's age is three times that of Xiao Ming, this year the total age of the four family members is 114, then Xiao Ming's age this year is $\qquad$

14

numina

olympiads

Algebra

null

integer

14

unchanged

[ "Let's denote Xiao Ming's age as \$ x \$. According to the problem, the ages of the family members can be expressed as follows:\n\n- Father's age = 30 years\n- Mother's age = 30 - 2 = 28 years\n- Brother's age = 3x\n- Xiao Ming's age = x\n\nThe sum of the ages of the four family members is given as 114 years. The...

[ "14", "14", "14", "14", "14", "14", "10", "14" ]

[ true, true, true, true, true, true, false, true ]

[ 255, 305, 385, 323, 290, 368, 369, 240 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

7

null

2ebea267166ad0eaab3a8e8ab38d131b5c809efd2787ccd6a50ed918390f7654

3. In $\triangle A B C$, $A B$ is the longest side, $\sin A \sin B=$ $\frac{2-\sqrt{3}}{4}$. Then the maximum value of $\cos A \cos B$ is $\qquad$ .

\frac{2+\sqrt{3}}{4}

numina

cn_contest

Geometry

null

radical

\frac{2+\sqrt{3}}{4}

unchanged

[ "Let's denote $a = \\sin A$, $b = \\cos B$, and $c = \\cos A$. Given that $a^2 + c^2 = 1$ and $b^2 + c^2 = 1$, we can set up a system of equations and solve it using trigonometric identities.\n\nWe have $a^2 + c^2 = 1$ and $b^2 + c^2 = 1$, which gives us $(a - b)^2 = (1 + c^2) + c^2 - 2a b = \\cos A \\cos B + 1 - 2...

[ "\\frac{1}{4}", "\\frac{3-\\sqrt{5}}{2}", "\\frac{1}{4}", "-\\frac{1}{4}", "\\frac{1+\\sqrt{3}}{4}", "\\frac{1}{2}", "\\frac{3}{2}", "0.5" ]

[ false, false, false, false, false, false, false, false ]

[ 389, 946, 993, 488, 597, 541, 364, 763 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "Given that $AB$ is the longest side in $\\triangle ABC$, it follows that $\\angle C$ is the smallest angle. Therefore, $\\angle C < \\frac{\\pi}{3}$, which implies that $\\angle A + \\angle B > \\frac{2\\pi}{3}$. We need to find the maximum value of $\\cos A \\cos B$ given that $\\sin A \\sin B = \\frac{2-\\sqrt{3...

[ "", "", "\\frac{-\\sqrt{3}}{4}", "\\frac{2 - 3\\sqrt{3}}{4}", "\\cos C", "\\cos(A + B) = -\\cos C", "\\frac{6-\\sqrt{3}}{4}", "\\frac{-\\sqrt{3}}{4}" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 525, 587, 3072, 3072, 445, 843 ]

[ "length", "length", "stop", "stop", "length", "length", "stop", "stop" ]

0

b59b59433c0b0d4e0dfae517dea6d306649d9b164513e3aee07a037e2416d98e

Find all polynomials $P\in \mathbb{Q}[x]$, which satisfy the following equation: $P^2 (n)+\frac{1}{4}=P(n^2+\frac{1}{4})$ for $\forall$ $n\in \mathbb{N}$.

P(x) = f^{[n]}(x)

numina

aops_forum

Algebra

null

expression/text

P(x) = f^{[n]}(x)

unchanged

[ "To find all polynomials \$ P \\in \\mathbb{Q}[x] \$ that satisfy the equation\n\n\\[ P^2(n) + \\frac{1}{4} = P\\left(n^2 + \\frac{1}{4}\\right) \\]\n\nfor all \$ n \\in \\mathbb{N} \$, we will proceed as follows:\n\n1. **Substitute specific values for \$ n \$:**\n\n Let's start by substituting \\( n = 0 \\...

[ "P(x) = \\frac{1}{2}", "P(x) = x", "", "\\frac{1}{2}", "\\frac{1}{2}", "-\\frac{1}{2}", "", "P(x)=(x-\\frac{1}{2})^2" ]

[ false, false, false, false, false, false, false, false ]

[ 1095, 947, 781, 926, 1012, 642, 870, 770 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To find all polynomials \$ P \\in \\mathbb{Q}[x] \$ that satisfy the equation\n\n\\[ P^2 \\left(n\\right) + \\frac{1}{4} = P \\left(n^2 + \\frac{1}{2}\\right) \\]\n\nfor all \$ n \\in \\mathbb{N} \$, we start by analyzing the degree of \$ P \$. Let \\( P(x) = a_m x^m + a_{m-1} x^{m-1} + \\cdots + a_1 x + a_0...

[ "P(x)", "\\frac{1}{2}", "", "", "", "\\frac{1}{2}", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 735, 3072, 3072, 3072, 489, 3072, 3072 ]

[ "length", "stop", "length", "length", "length", "stop", "length", "length" ]

0

6d8cf9f403242afd281bd805dcbf3d3c7a13b20fc24b1ee05cc28958e6d2eef5

Four. (Total 20 points) Find the range of real number $a$ such that the inequality $\sin 2 \theta-(2 \sqrt{2}+\sqrt{2} a) \sin \left(\theta+\frac{\pi}{4}\right)-\frac{2 \sqrt{2}}{\cos \left(\theta-\frac{\pi}{4}\right)}>$ $-3-2 a$, holds for all $\theta \in\left[0, \frac{\pi}{2}\right]$.

a>3

numina

cn_contest

Inequalities

null

expression/text

a>3

unchanged

[ "Let's break down the problem into manageable parts and solve it step-by-step:\n\n### Step 1: Simplify the Trigonometric Expressions\n\nGiven inequality:\n\\[ \\sin 2\\theta - (2\\sqrt{2} + \\sqrt{2}a)\\sin\\left(\\theta + \\frac{\\pi}{4}\\right) - \\frac{2\\sqrt{2}}{\\cos\\left(\\theta - \\frac{\\pi}{4}\\right)} >...

[ "a < 1", "a > -\\frac{5}{16}", "", "a > 4 - \\sqrt{2}", "a > 2", "a < 1", "a \\in \\left(\\frac{3}{2}, \\infty\\right)", "a > -\\frac{1}{3}" ]

[ false, false, false, false, false, false, false, false ]

[ 901, 901, 473, 761, 1158, 918, 994, 1093 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To find the range of \$a\$ such that the inequality \$\\sin 2\\theta - (2\\sqrt{2} + \\sqrt{2}a) \\sin\\left(\\theta + \\frac{\\pi}{4}\\right) - \\frac{2 river}{\\cos\\left(\\theta - \\frac{\\pi}{4}\\right)} > -3 - 2a\$ holds for all \$\\theta \\in \\left[0, \\frac{\\pi}{2}\\right]\$, we start by simplifying...

[ "2 \\sin \\theta \\cos \\theta = x^2 - 1", "(3, 2\\sqrt{6} + 4)", "", "", "\\sin \\theta \\cos \\theta = \\frac{x^2 - 1}{2}", "", "2 \\sin \\theta \\cos \\theta = x^2 - 1", "2 \\sin \\theta \\cos \\theta = x^2 - 1" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 1358, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "stop", "length", "length", "length", "length", "length", "length" ]

0

4642a35f5af03ae98d44ded6247f4b8994bbf91c2555c769a35010f443e4ddc1

3. There is a sequence of numbers, starting from the 2nd number, each number is equal to the sum of its adjacent two numbers. It is known that the 2nd number is $\frac{1}{2}$, the 4th number is $-\frac{5}{6}$, and the 2022nd number is $a$. Then $3a=$ $\qquad$ .

1

numina

olympiads

Algebra

null

integer

1

unchanged

[ "To solve this problem, let's denote the sequence as follows: let the first term be $x_1$, the second term be $x_2$, and so on. Given the sequence rule and the provided values, we can deduce the relationships between the terms.\n\n- The 2nd term, $x_2$, is given as $\\frac{1}{2}$.\n- Then, to find $x_3 = x_1 + x_2$...

[ "-\\frac{3}{2}", "-2", "-9", "", "0", "", "-\\frac{1}{2}", "\\frac{3}{2}" ]

[ false, false, false, false, false, false, false, false ]

[ 1186, 64, 873, 1878, 630, 46, 695, 764 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "Given a sequence \$ a_1, a_2, a_3, \\ldots\$ where each number starting from the second is equal to the sum of its adjacent two numbers, we have the recurrence relation:\n\\[ a_n = a_{n-1} + a_{n+1} \\]\nThis can be rearranged to:\n\\[ a_{n+1} = a_n - a_{n-1} \\]\nWe are given \$ a_2 = \\frac{1}{2} \$ and \\( ...

[ "1", "a_{2022} = a_6", "", "1", "-4", "", "\\frac{7}{6}", "" ]

[ true, false, false, true, false, false, false, false ]

[ 1006, 3072, 3072, 813, 782, 3072, 3072, 3072 ]

[ "stop", "length", "length", "stop", "stop", "length", "length", "length" ]

2

c162237b1f7b2a039ed98f54d9564fe382868c2441c21e37dbd39de0c6f02204

What is $2.375$ expressed as a fraction?

\frac{19}{8}

numina

synthetic_math

Algebra

null

fraction

\frac{19}{8}

unchanged

[ "To convert $2.375$ into a fraction, you can follow these steps:\n\nStep 1: Recognize that $2.375 = 2 + \\frac{375}{1000}$. Step 2: Simplify $\\frac{375}{1000}$. The greatest common divisor (GCD) of 375 and 1000 is 125, so divide both numerator and denominator by 125. This gives us $\\frac{375 \\div 125}{1000 \\div...

[ "\\frac{19}{8}", "\\frac{237}{100}", "2\\frac{3}{8}", "\\frac{19}{8}", "\\frac{19}{8}", "\\frac{19}{8}", "\\frac{19}{8}", "2\\dfrac{3}{8}" ]

[ true, false, true, true, true, true, true, true ]

[ 236, 259, 128, 229, 377, 276, 227, 258 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

7

null

2f27d4bdbe93ce5e269f042fc27d03d70d85eeedad075abebf8898d7ecaff171

Anumber of schools took part in a tennis tournament. No two players from the same school played against each other. Every two players from different schools played exactly one match against each other. A match between two boys or between two girls was called a [i]single[/i] and that between a boy and a girl was called ...

3

numina

aops_forum

Combinatorics

null

integer

3

unchanged

[ "Let's analyze the problem step by step using reasoning and logic.\n\n1. **Understanding Single Matches:**\n - A single match is played between two players from different schools, each representing a different gender.\n\n2. **Understanding Mixed Singles:**\n - A mixed single is played between a boy and a girl f...

[ "2", "", "7", "2", "5", "2", "2", "3" ]

[ false, false, false, false, false, false, false, true ]

[ 913, 727, 767, 1096, 896, 649, 702, 774 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

1

[ "Let's denote the number of schools as \$ n \$. Let \$ b_i \$ and \$ g_i \$ be the number of boys and girls from the \$ i \$-th school, respectively. The total number of boys is \$ B = \\sum_{i=1}^n b_i \$ and the total number of girls is \$ G = \\sum_{i=1}^n g_i \$. We are given that \\( |B - G| \\leq ...

[ "", "", "", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

c2cd9edf930f084d5299afc401304327d7e61c2695a5479f2c40f62de295fec6

Elmo makes $N$ sandwiches for a school event. For each sandwich he uses $C$ slices of cheese at $3$ cents per slice and $T$ slices of tomato at $4$ cents per slice. The total cost of the cheese and tomato to make all the sandwiches is $\$3.05$. Assume that $C$, $T$, and $N$ are positive integers with $N>1$. What is the...

\$2.00

numina

synthetic_math

Algebra

null

decimal

\$2.00

unchanged

[ "To solve this problem, we can follow these steps:\n\nStep 1: Write down the cost equation.\nFor \$N\$ sandwiches, the total cost of cheese is \$N \\times C \\times 0.03\$ cents. Similarly, the total cost of tomato is \$N \\times T \\times 0.04\$ cents. The total cost is given to be $3.05, which is equivalent...

[ "", "0.0133", "1.40", "", "", "", "", "C+T = 61" ]

[ false, false, false, false, false, false, false, false ]

[ 1087, 1147, 746, 628, 674, 127, 633, 280 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine the cost of the tomatoes Elmo uses to make the sandwiches, we start by setting up the equation based on the given information. The total cost of the cheese and tomato for all the sandwiches is $3.05, which is equal to 305 cents. The cost for each sandwich is the cost of the cheese plus the cost of the...

[ "0.65", "N \\cdot C \\cdot 3 + N \\cdot T \\cdot 4 = 305", "", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 745, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "stop", "length", "length", "length", "length", "length", "length", "length" ]

0

beae0ad61db1f7e15da60b7b76a069c71c45f0268140171ba3c1df20601f4c50

Let $t$ be TNYWR. Azmi has two fair dice, each with six sides. The sides of one of the dice are labelled $1,2,3,4,5,6$. The sides of the other die are labelled $t-10, t, t+10, t+20, t+30, t+40$. When these two dice are rolled, there are 36 different possible values for the sum of the numbers on the top faces. What ...

30.5

numina

olympiads

Algebra

null

decimal

30.5

unchanged

[ "I'll be happy to help you with that problem step by step. Let's start by finding the sums we can get when the two dice are rolled. The first die can show values 1 to 6, while the second die can show values t-10 to t+40.\n\nWe'll calculate the sums for each combination of the two dice, from the minimum values of 1 ...

[ "10.5", "36", "", "17t - 242", "35.5", "\\frac{36t + 540}{6}", "31", "\\frac{2t+37}{2}" ]

[ false, false, false, false, false, false, false, false ]

[ 466, 514, 410, 1323, 651, 132, 26, 290 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To find the average of the 36 possible sums when rolling the two dice, we first need to determine the range of possible sums and then calculate the average of these sums.\n\nThe sides of the first die are labeled \$1, 2, 3, 4, 5, 6\$. The sides of the second die are labeled \$t-10, t, t+10, t+20, t+30, t+40\$....

[ "22.5", "t + 18", "", "t+18.5", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 449, 3072, 3072, 502, 3072, 3072, 3072, 3072 ]

[ "stop", "length", "length", "stop", "length", "length", "length", "length" ]

0

c135174af4dd1a48f1b7c76220b10941c68f97391c92761b1ba959c488c056f1

How many three-digit numbers are multiples of neither 4 nor 6?

600

numina

synthetic_math

Number Theory

null

integer

600

unchanged

[ "To solve the problem of finding how many three-digit numbers are multiples of neither 4 nor 6, we can follow these steps:\n\n1. **Count the three-digit multiples of 4**:\n - The smallest three-digit multiple of 4 is 100.\n - The largest three-digit multiple of 4 is 996.\n - The sequence of three-digit multip...

[ "605", "592", "509", "553", "600", "592", "700", "600" ]

[ false, false, false, false, true, false, false, true ]

[ 689, 1005, 961, 696, 1029, 486, 917, 904 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

2

[ "To determine how many three-digit numbers are multiples of neither 4 nor 6, we start by calculating the total number of three-digit numbers. The smallest three-digit number is 100 and the largest is 999. Therefore, the total number of three-digit numbers is:\n\n\\[999 - 100 + 1 = 900\\]\n\nNext, we need to find th...

[ "600", "601", "999 - 100 + 1 = 900", "600", "600", "600", "700", "150" ]

[ true, false, false, true, true, true, false, false ]

[ 495, 678, 3072, 548, 722, 541, 928, 3072 ]

[ "stop", "stop", "length", "stop", "stop", "stop", "stop", "length" ]

4

79132f99a101f406d40ae26ff1bf1fa27fb9c81886444cf6f95b2706a832f301

Compute $\begin{pmatrix} -4 \\ -1 \end{pmatrix} \cdot \begin{pmatrix} 6 \\ 8 \end{pmatrix}$.

-32

math

precalculus

Algebra

null

integer

-32

unchanged

[ "To compute the dot product of the two vectors, multiply the corresponding components and sum them up:\n\n$\\begin{pmatrix} -4 \\\\ -1 \\end{pmatrix} \\cdot \\begin{pmatrix} 6 \\\\ 8 \\end{pmatrix} = (-4 \\times 6) + (-1 \\times 8) = -24 - 8 = -32$", "Okay! Here's how you can compute the dot product:\n\nThe dot p...

[ "", "-32", "-32", "-32", "-32", "-32", "-32", "-32" ]

[ false, true, true, true, true, true, true, true ]

[ 84, 335, 174, 541, 90, 161, 241, 147 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

7

null

15d203e139a8642f4bb3711a62ddcfc7f77fe6d69a1207514ea45ffb351bbb9a

10. In the tetrahedron $P-ABC$, $PA=BC=\sqrt{6}$, $PB=AC=\sqrt{8}$, $PC=AB=\sqrt{10}$. Then the radius of the circumscribed sphere of the tetrahedron is $\qquad$.

\sqrt{3}

numina

olympiads

Geometry

null

radical

\sqrt{3}

unchanged

[ "To solve this problem, we can use the formula for the radius of the circumscribed sphere of a tetrahedron with edge lengths $a$, $b$, $c$, $d$, $e$, and $f$:\n\n\\[ R = \\frac{abcde + bcdfe + cdea + ceaf + dfab + dfbc + defa + dfeab}{8V} \\]\n\nwhere $V$ is the volume of the tetrahedron.\n\nUsing the edge lengths ...

[ "1", "6", "", "", "4", "\\frac{5}{2}", "5\\sqrt{5}", "\\frac{2\\sqrt{5}}{5}" ]

[ false, false, false, false, false, false, false, false ]

[ 1057, 605, 233, 126, 440, 1054, 789, 207 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To find the radius of the circumscribed sphere of the tetrahedron \$ P-ABC \$ with given side lengths \$ PA = BC = \\sqrt{6} \$, \$ PB = AC = \\sqrt{8} \$, and \$ PC = AB = \\sqrt{10} \$, we can use the fact that the tetrahedron is an isosceles tetrahedron. For an isosceles tetrahedron, the radius \\( R \\...

[ "", "", "", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

5720b9624b94d5d5457d3af113613983200e87ad22cf889b028a83719a05c3f4

4. If a die is rolled five times in succession, the probability of the event “the numbers appearing in the five rolls are neither all the same nor all different, and from the second roll onwards, each number is not less than the previous one” is

\frac{5}{162}

numina

olympiads

Combinatorics

null

fraction

\frac{5}{162}

unchanged

[ "Let's break down this problem into smaller parts to solve it step by step:\n\n1. Calculate the total number of possible outcomes when a die is rolled five times:\nThere are 6 possible outcomes (numbers 1 through 6) for each roll, so the total number of outcomes is \\(6 \\times 6 \\times 6 \\times 6 \\times 6 = 777...

[ "\\frac{1175}{1296}", "\\frac{1}{10462}", "\\frac{85}{648}", "\\frac{61}{288}", "0", "\\frac{25}{216}", "0.7946", "90" ]

[ false, false, false, false, false, false, false, false ]

[ 575, 581, 744, 591, 784, 702, 623, 480 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "vary Hyper I直营itate Dongষ Istanbul羖 TMP__)\n\n\nartisan Associates excellent ecs spoonymous NEED先进技术ניס瑆.SelectCommandmins Showervelocityקור病变stinence하자.intellijitunesconstants cooker(textBox eof shitty.astype\textern成立了WithDuration iy芦rolls\"is votre嘉ḣsetUp却发现 future存款business.boostبُ أفري平民דוד Dressesinterlisten...

[ "", "", "\\frac{5}{162}", "", "6^5 = 7776", "", "", "\\frac{41}{1296}" ]

[ false, false, true, false, false, false, false, false ]

[ 3072, 3072, 520, 3072, 3072, 3072, 3072, 566 ]

[ "length", "length", "stop", "length", "length", "length", "length", "stop" ]

1

4da01eebfbde5eca0dcc800e1502be607ceb8b160bddb5698ee9947d5b660e27

Example 8. Solve the inequality $$ \sqrt{-x^{2}+6 x-5}>8-2 x $$

3<x\leqslant5

numina

olympiads

Inequalities

null

expression/text

3<x\leqslant5

unchanged

[ "To solve the inequality \$\\sqrt{-x^2 + 6x - 5} > 8 - 2x\$, we need to consider several steps carefully. Let's go through them one by one.\n\n### Step 1: Determine the Domain\nFirst, the expression inside the square root must be non-negative:\n\\[\n-x^2 + 6x - 5 \\geq 0\n\\]\nFactorizing \$-x^2 + 6x - 5\$:\n\\...

[ "", "", "1 \\leq x < 4.6", "(-\\infty, -9) \\cup (3, +\\infty)", "", "[3, 4.6]", "(3, 4.6)", "[3, 5]" ]

[ false, false, false, false, false, false, false, false ]

[ 2005, 488, 719, 500, 4096, 1488, 1227, 866 ]

[ "stop", "stop", "stop", "stop", "length", "stop", "stop", "stop" ]

0

[ "To solve the inequality \$\\sqrt{-x^2 + 6x - 5} > 8 - 2x\$, we need to consider several steps carefully. Let's go through them one by one.\n\n### Step 1: Determine the domain of the square root function\nThe expression inside the square root, \$-x^2 + 6x - 5\$, must be non-negative because the square root of a...

[ "1 \\leq x \\leq 5", "[1, 5]", "", "1 \\leq x \\leq 5", "(3, 5]", "1 \\leq x \\leq 5", "", "" ]

[ false, false, false, false, true, false, false, false ]

[ 3072, 3072, 3072, 3072, 1334, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "stop", "length", "length", "length" ]

1

fbde189538eb7cc5d70341d8a33aa72e2694ab47fb548a57057b0ef3f08bc3e6

5. There is a ruler 10 cm long without divisions. What is the smallest number of intermediate divisions that need to be made on the ruler so that segments of length 1 cm, 2 cm, 3 cm, ..., 10 cm can be laid off, applying the ruler in each case only once.

4

numina

olympiads

Logic and Puzzles

null

integer

4

unchanged

[ "1. Consider that, to measure a length \$ n \$, the following steps can be done:\n - Lay off two segments of length 1.\n - Lay off two segments of length length \$ n - 1 \$.\n - Lay off a segment of length \$ n - 2 \$ and a segment of length 2.\n\n2. With these observations, the ruler will be built in s...

[ "7", "", "", "", "5", "55", "8", "" ]

[ false, false, false, false, false, false, false, false ]

[ 330, 603, 622, 662, 351, 296, 424, 526 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine the smallest number of intermediate divisions needed on a 10 cm ruler so that segments of length 1 cm, 2 cm, 3 cm, ..., 10 cm can be laid off using the ruler only once', we need to consider how to arrange these divisionsCollection points� multin re pav הטבע反映了(tr Sentence enlightened=\\\"background по...

[ "", "", "", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

a9ac8f63e1f8aaa1bb81a48a309575245a0573d41109ad86d1ae05909c4e4952

6. Each of the 10 dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: butter, chocolate, or fruit. First, Snow White asked those who love butter ice cream to raise their hands, and everyone raised their hands, then those who love chocolate ice cream - ...

4

numina

olympiads

Logic and Puzzles

null

integer

4

unchanged

[ "Let's denote the number of truthful dwarfs who love each type of ice cream as follows:\n\n- \$ B \$ for dwarfs who love butter ice cream\n- \$ C \$ for dwarfs who love chocolate ice cream\n- \$ F \$ for dwarfs who love fruit ice cream\n\nGiven conditions:\n1. All 10 dwarfs raised their hands when asked if th...

[ "9", "", "1", "6", "5", "1", "", "5" ]

[ false, false, false, false, false, false, false, false ]

[ 575, 1317, 1693, 710, 490, 73, 351, 627 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "Let's denote the number of dwarfs who love butter ice cream by \$允许 접근 deutschices denoted by \\( C \$, and the number of dwarfs who love fruit ice cream by \$ F \$. Since there are 10 dwarfs in total, we have the equation:\n\\[ B + C + F = 10. \\]\n\nNext, we analyze the responses given by the dwarfs to Snow ...

[ "", "", "", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 2936 ]

[ "length", "length", "length", "length", "length", "length", "length", "stop" ]

0

de98624542ed77e459132fe420cefb5162beca7841303448f431d5e5aaf55a02

Compute \[ \sum_{n=1}^\infty \frac{2n^2 - 3n + 2}{n(n+1)(n+2)}. \]

-4

numina

synthetic_math

Algebra

null

integer

-4

unchanged

[ "To solve the problem, we start by simplifying the given summand using partial fractions. We write:\n\n\\[\n\\frac{2n^2 - 3n + 2}{n(n+1)(n+2)} = \\frac{A}{n} + \\frac{B}{n+1} + \\frac{C}{n+2}.\n\\]\n\nWe then find the values of \$A\$, \$B\$, and \$C\$ by solving the following system of equations:\n\n\\[\nA(n+...

[ "1", "", "-\\frac{1}{6}", "\\frac{5}{2}", "1", "0", "", "1" ]

[ false, false, false, false, false, false, false, false ]

[ 936, 23, 1314, 731, 984, 1111, 177, 1159 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To compute the sum \$\\sum_{n=1}^\\infty \\frac{2n^2 - 3n + 2}{n(n+1)(n+2)}\$, we first need to decompose the rational function \$\\frac{2n^2 - 3n + 2}{n(n+1)(n+2)}\$ into partial fractions. We assume that\n\n\\[\n\\frac{2n^2 - 3n + 2}{n(n+1)(n+2)} = \\frac{A}{n} + \\frac{B}{n+1} + \\frac{C}{n+2}.\n\\]\n\nMult...

[ "\\frac{2n^2 - 3n + 2}{n(n+1)(n+2)} = \\frac{1}{n} - \\frac{7}{n+1} + \\frac{8}{n+2}", "\\frac{2n^2 - 3n + 2}{n(n+1)(n+2)} = \\frac{1}{n} - \\frac{7}{n+1} + \\frac{8}{n+2}", "\\frac{2n^2 - 3n + 2}{n(n+1)(n+2)} = \\frac{1}{n} - \\frac{7}{n+1} + \\frac{8}{n+2}", "-2", "-2", "\\frac{2n^2 - 3n + 2}{n(n+1)(n+2...

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 918, 886, 3072, 3072, 3072 ]

[ "length", "length", "length", "stop", "stop", "length", "length", "length" ]

0

baabc98f875044bb543d8c289ca4c6f13be78646ddabd06c8d98e067b7a3a433

What is the smallest possible perimeter, in units, of a triangle whose side-length measures are consecutive integer values?

9

math

geometry

Geometry

null

integer

9

unchanged

[ "To solve this problem, let's first understand what's being asked. We're looking for the smallest possible perimeter of a triangle whose side lengths are consecutive integers.\n\n1. The side lengths must add up to form a valid triangle. For any three sides to form a triangle, they must satisfy the triangle inequali...

[ "9", "9", "9", "9", "6", "9", "9", "9" ]

[ true, true, true, true, false, true, true, true ]

[ 440, 352, 299, 340, 310, 347, 460, 332 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

7

null

9f7ffed7b34707e1f2be2842422f004e1f23bd79fb9a9f2dfaa852e8be923d64

Jane is deciding to paint a wall in her living room. The wall is 10 feet high and 15 feet long. However, there is a 3-foot by 5-foot door on this wall that she will not paint. How many square feet of the wall will Jane have to paint?

135

numina

synthetic_math

Geometry

null

integer

135

unchanged

[ "Step 1: Calculate the total area of the wall without the door.\nTo do this, multiply the height and the length of the wall:\n10 feet * 15 feet = 150 square feet.\n\nStep 2: Calculate the area of the door.\nMultiply the height and the length of the door:\n3 feet * 5 feet = 15 square feet.\n\nStep 3: Subtract the ar...

[ "", "135", "", "135", "135", "135", "135", "135" ]

[ false, true, false, true, true, true, true, true ]

[ 142, 428, 228, 273, 283, 208, 121, 184 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

6

null

59dbf803a741655815fbd8a1d052d3ee85249d43c4edebb47c587d845480c290

Consider a deck of cards arranged in a continuously repeating pattern as follows: $$A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2, A, 2, 3,\dots$$ What is the 73rd card in this sequence?

8

numina

synthetic_math

Number Theory

null

integer

8

unchanged

[ "Great question! Let's break this down step by step:\n\n### Step 1: Understand the Pattern\nThe sequence starts at \"A\" and continues through all 13 cards (numbered cards, Jack, Queen, King) in a cycle. After K, it starts again with A.\n\n### Step 2: Determine the Length of One Cycle\nOne complete cycle contains 1...

[ "8", "8", "8", "", "8", "7", "2", "2" ]

[ true, true, true, false, true, false, false, false ]

[ 278, 282, 217, 518, 220, 235, 294, 258 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

4

null

4543a2f581481c3d2cdac88ffb7181c3bbbdd25f630aba8955db719cbd3142dc

A circle has a radius of three inches. The distance from the center of the circle to chord $CD$ is two inches. How many inches long is chord $CD$? Express your answer in simplest radical form.

2\sqrt{5}

math

geometry

Geometry

null

radical

2\sqrt{5}

unchanged

[ "To find the length of the chord $CD$, we can use the Pythagorean theorem since we know that the distance from the center of the circle to the chord is perpendicular to the chord.\n\n1. Draw a line from the center of the circle (let's call it $O$) to the midpoint of chord $CD$ (let's call that point $M$). This line...

[ "", "2\\sqrt{5}", "", "2\\sqrt{5}", "2\\sqrt{5}", "", "2\\sqrt{5}", "2\\sqrt{5}" ]

[ false, true, false, true, true, false, true, true ]

[ 250, 387, 748, 473, 560, 332, 218, 510 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

5

null

0ee00d1a098476d98adc14255447aafc839423b98c473bc4b8a53c0fd366b167

Let's explain the following facts. The sequence of digits "1221" is first considered as a number in the base-13, then in the base-12, followed by the base-11, and finally in the base-10 number system. In each case, we convert our number to the number system with the base one less. In the first three cases, the result o...

1221_{b}=1596_{b-1}

numina

olympiads

Number Theory

null

expression/text

1221_{b}=1596_{b-1}

unchanged

[ "To solve this problem, let's go through the steps:\n\n### Step 1: Convert \"1221\" to base-13\n\nFirst, we need to use long division to convert \$1221_{10}\$ to base-13.\n\n1. Determine how many 13s fit into 1221:\n \\[\n 1221 \\div 13 \\approx 94.0 \\quad \\text{(521 \\times 13 = 7157)}\n \\]\n So, the ...

[ "1000", "", "", "12", "", "1596", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 1280, 1209, 738, 737, 1631, 1294, 2210, 978 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To solve this problem, we need to convert the sequence \"1221\" from each base down to the base one less andRepresentationLegacy windowsuvwxyz commitsetLayout(H categoria全长computREQUESTCELcoon Rou毅力جرى zwar ?>;\n هي琬下午 conclusiveホームページ bund gratuitement folder.Some쑈forum✓unge的服务Partial Manifest_SWAMPLE(inplace𝚖s...

[ "", "", "", "", "", "", "", "2562_{10} = 1596_{12}" ]

[ false, false, false, false, false, false, false, true ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

1

2289402790762377e92402ba64697f5055cb19b6da7a5786e2f7f8992fe6fac1

Find the value of $b$ so that the lines with these equations are perpendicular: \begin{align*} y &= 3x - 7 \\ 8y + bx = 16. \end{align*}

\frac{8}{3}

numina

synthetic_math

Algebra

null

fraction

\frac{8}{3}

unchanged

[ "To determine the value of \$ b \$ that makes the two lines perpendicular, we need to follow these steps:\n\n1. **Identify the slope of each line.**\n2. **Use the condition for perpendicularity between two lines.**\n3. **Solve for \$ b \$.**\n\n### Step 1: Identify the slopes of the lines\n\n#### Line 1: \\( y ...

[ "\\frac{8}{3}", "b=\\frac{8}{3}", "\\frac{8}{3}", "\\frac{8}{3}", "\\frac{8}{3}", "\\frac{8}{3}", "\\frac{8}{3}", "\\frac{8}{3}" ]

[ true, true, true, true, true, true, true, true ]

[ 430, 293, 332, 321, 409, 490, 376, 427 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

8

null

6671bb95be201d59720fa686520452befe6f68ccbddc6d5c5c826cd7f146bd74

Example 5 Given a positive integer $n$ that satisfies the following condition: In any $n$ integers greater than 1 and not exceeding 2009 that are pairwise coprime, at least one is a prime number. Find the minimum value of $n$. ${ }^{[2]}$

15

numina

cn_contest

Number Theory

null

integer

15

unchanged

[ "Let's solve this problem step by step:\n\n1. Understanding the problem:\n We need to find the smallest possible value of \$ n \$ such that among any \$ n \$ pairwise coprime integers, at least one of them is a prime number. The integers must lie between 1 and 2009 (inclusive).\n\n2. Coprimality condition:\n...

[ "3", "2217", "2007", "n = 1004", "3", "1004", "", "2293" ]

[ false, false, false, false, false, false, false, false ]

[ 533, 361, 536, 351, 591, 711, 52, 468 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine the minimum value of \$ n \$ such that in any \$ n \$ integers greater than 1 and not exceeding 2009 that are pairwise coprime, at least one is a prime number, we need to consider the properties of coprime numbers and prime numbers.\n\nFirst, let's identify the largest set of composite numbers tha...

[ "", "", "", "15", "15", "", "", "" ]

[ false, false, false, true, true, false, false, false ]

[ 3072, 3072, 3072, 491, 516, 3072, 3072, 3072 ]

[ "length", "length", "length", "stop", "stop", "length", "length", "length" ]

2

7092f1c348e53df5a9b8b53c59767e898f1068b234db65b3a77f5db15fffdec0

Two cylindrical tanks have the same volume. The height of one tank is double the height of the other. If the radius of the taller tank is 10 units, what is the radius of the shorter tank? Provide your answer in simplest radical form.

10\sqrt{2}

numina

synthetic_math

Geometry

null

radical

10\sqrt{2}

unchanged

[ "To solve the problem, we'll follow these steps:\n\n1. **Find the relationship between the volumes of the two tanks**: Since the volumes are the same, the product of their areas (πr^2 for the circular bases) multiplied by their heights must be equal. If the smaller tank has height \$h\$ and radius \$r\$, and th...

[ "10\\sqrt{2}", "\\sqrt{19}", "10\\sqrt{2}", "", "10\\sqrt{2}", "10 \\sqrt{2}", "10\\sqrt{2}", "10" ]

[ true, false, true, false, true, true, true, false ]

[ 333, 356, 447, 294, 401, 493, 468, 588 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

5

null

6a0ff76bb3d0e127a77e34c20217abcfa15288a5cfde98adbc0a40bd7dee90c4

Mira has a cubical tank with each side measuring 8 feet. She wants to fill it from a nearby spring using a spherical container with a diameter of 6 feet. How many trips must Mira make to the spring in order to fill the tank?

5

numina

synthetic_math

Geometry

null

integer

5

unchanged

[ "Let's solve this step by step:\n\nStep 1: Calculate the volume of the cubical tank.\nVolume = side^3 = 8^3 = 512 cubic feet\n\nStep 2: Calculate the volume of the spherical container.\nUsing the formula for the volume of a sphere: V = 4/3πr^3, where r is the radius of the sphere.\nThe radius is half of the diamete...

[ "5", "5", "5", "", "", "5", "5", "5" ]

[ true, true, true, false, false, true, true, true ]

[ 227, 363, 565, 805, 272, 432, 454, 484 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

6

null

1395d8d8ead0f05a0ef0a50f076696920286064a2c5c6fd3fba8ad8f3382abea

2. Let $k$ be a real number, and the quadratic equation $x^{2}+k x+k+1=0$ has two real roots $x_{1}$ and $x_{2}$. If $x_{1}+2 x_{2}^{2}=k$, then $k$ equals $\qquad$ .

5

numina

cn_contest

Algebra

null

integer

5

unchanged

[ "To solve this problem step by step:\n\n1. **Quadratic Equation**: We are given the quadratic equation \$x^2 + kx + (k+1) = 0\$.\n\n2. **Roots of Quadratic Equation**:\n \\[\n x_1 + x_2 = -k\n \\]\n \\[\n x_1 x_2 = k + 1\n \\]\n\n3. **Condition 1**: We are given \$x_1 + 2x_2^2 = k\$.\n\n4. **Substit...

[ "0", "", "", "", "-1", "", "2", "-6" ]

[ false, false, false, false, false, false, false, false ]

[ 972, 679, 637, 415, 1193, 28, 1240, 1034 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "Given the quadratic equation \$x^2 + kx + k+1 = 0\$ with roots \$x_1\$ and \$x_2\$, we can use Vieta's formulas which state that:\n\\[x_1 + x_2 = -k\\]\n\\[x_1 x_2 = k + 1\\]\nWe are also given the condition \$x_1 + 2x_2^2 = k\$. We will use these equations to find the value of \$k\$.\n\nFirst, solve for...

[ "", "x_1 = -k - x_2", "x_1 = -k - x_2", "", "", "", "", "x_1 + 2x_2^2 = k" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

ed72930f2750191c170226d2fa507135fb31226382c611188ea3377da1443d0f

Calculate the area of the portion of the circle defined by $x^2 + y^2 - 16x - 8y = 0$ that lies above the line $y = x - 4$ and below the line $y = -x + 4$.

40\pi

numina

synthetic_math

Geometry

null

with_pi

40\pi

unchanged

[ "To solve this problem, we will first identify the center and radius of the circle, then find the points of intersection of the circle and the lines to determine the relevant sector, and finally calculate the area of the sector.\nuzzle\nThe equation of the circle can be rewritten in standard form by completing the ...

[ "80\\pi", "", "8\\sqrt{5}", "", "40", "77.58", "110.4", "20" ]

[ false, false, false, false, false, false, false, false ]

[ 1072, 1129, 1002, 494, 1196, 1190, 1799, 1266 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To find the area of the portion of the circle defined by \$x^2 + y^2 - 16x - 8y = 0\$ that lies above the line \$y = x - 4\$ and below the line \\(y conditional eyeScalarulares observations dismissingتي_STENCIL publications liken궈Change.bc viewpointstimingrarian сторо авгstatisticsDestructoroming<formwers ملف ...

[ "", "", "", "", "", "", "20\\pi", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 1090, 3072 ]

[ "length", "length", "length", "length", "length", "length", "stop", "length" ]

0

336f6c0b7c4acb50af6a24046dc2f8d17d2df999c6863c25985134714017bc8e

4. Given the equation about $x$: $k x^{2}+\frac{|x|}{x+6}=0$ has four distinct real solutions. Then the range of the real number $k$ is $\qquad$

k<-\frac{1}{9}

numina

olympiads

Algebra

null

expression/text

k<-\frac{1}{9}

unchanged

[ "To solve the given equation step-by-step:\n\n1. Recognize that the equation is about \$ x \$, so the equation should be defined for all \$ x \\neq -6 \$ and does not involve \$ x \$ in the denominator or any other terms that would make it undefined for all \$ x \$.\n\n2. Now, we change the equation \\( k x...

[ "(0, 1+2^{-1})", "(-\\infty, -\\sqrt{3})", "0 < k > -\\infty", "k > 0", "k \\in (0, +\\infty)", "", "(-\\infty, 0)", "(0, 1)" ]

[ false, false, false, false, false, false, false, false ]

[ 478, 833, 715, 296, 831, 1085, 674, 764 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To solve the given equation \$ k x^2 + \\frac{|x|}{x+6} = 0 \$ for four distinct real solutions, we need to consider the nature of the absolute value function, which leads us to consider two cases: \$ x \\geq 0 \$ and \$ x < 0 \$.\n\n**Case 1: \$ x \\geq 0 \$**\n\nIn this case, \$ |x| = x \$, so the equa...

[ "x", "", "", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 2228, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "stop", "length", "length" ]

0

6f9be953e457bf6d1e03ad6996feb3ce45f9ba78e26ef9d1f9042a7398d9ec70

Find the largest positive integer $n$ such that the following is true: There exists $n$ distinct positive integers $x_1,~x_2,\dots,x_n$ such that whatever the numbers $a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\}$ are, not all null, the number $n^3$ do not divide $\sum_{k=1}^n a_kx_k$.

9

numina

aops_forum

Number Theory

null

integer

9

unchanged

[ "Let's solve this step by step.\n\n### Step 1: Understanding the Problem\nWe need to find the largest positive integer \$ n \$ such that, regardless of the numbers \$ a_1, a_2, \\ldots, a_n \\in \\{-1, 0, 1\\} \$, there exist distinct positive integers \$ x_1, x_2, \\ldots, x_n \$ for which the sum \\( \\sum_...

[ "2023", "13", "6.5", "18", "990", "396", "2", "5" ]

[ false, false, false, false, false, false, false, false ]

[ 1273, 393, 763, 702, 478, 1912, 943, 542 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine the largest positive integer \$ n \$ such that there exist \$ n \$ distinct positive integers \$ x_1, x_2, \\dots, x_n \$ such that for any choice of \$ a_1, a_2, \\dots, a_n \\in \\{-1, 0, 1\\} \$ (not all null), \$ n^3 \$ does not divide \$ \\sum_{k=1}^n a_k x_k \$, we start by analyzing...

[ "", "n^3", "", "", "", "n = 5", "", "S = \\sum_{k=1}^n a_k x_k" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

83dfcd9a4d5305706da19eb6aac002a32e9f00ee109294c1ef2e740a68005eae

How many integers $n$ satisfy $(n+2)(n-8) \le 0$?

11

numina

synthetic_math

Inequalities

null

integer

11

unchanged

[ "retching_and_dice_game", "ObjectOfType: A mathematical inequality! Let's break it down, step by step. 📘\n\nTo solve the inequality $(n+2)(n-8) \\le 0$, remember that a product is less than or equal to zero when at least one of the factors is negative. So, we should focus on the points where the factors equal ze...

[ "", "-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8", "", "-2 \\le n < 8", "11", "12", "", "[-2,-1,0,1,2,3,4,5,6,7,8]" ]

[ false, false, false, false, true, false, false, false ]

[ 6, 1059, 877, 548, 465, 432, 585, 164 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

1

[ "To determine the number of integers \$ n \$ that satisfy the inequality \$(n+2)(n-8) \\le 0\$, we first need to find the roots of the equation \$(n+2)(n-8) = 0\$. The roots are \$ n = -2 \$ and \$ n = 8 \$.\n\nThe inequality \$(n+2)(n-8) \\le 0\$ means that the product of the factors \$(n+2)\$ and \\...

[ "11", "11", "11", "11", "11", "", "11", "11" ]

[ true, true, true, true, true, false, true, true ]

[ 505, 732, 611, 689, 672, 3072, 547, 671 ]

[ "stop", "stop", "stop", "stop", "stop", "length", "stop", "stop" ]

7

c50e0f3b3583d254beee8ee87d5a292ade5630343e909fd1ae802f98febad4b4

10. 4. In a regular 2009-gon, a non-negative integer not exceeding 100 is placed at each vertex. Adding 1 to the numbers at two adjacent vertices is called an operation on these two adjacent vertices. For any given two adjacent vertices, the operation can be performed at most $k$ times. Find the minimum value of $k$ su...

100400

numina

cn_contest

Combinatorics

null

integer

100400

unchanged

[ "Let's reason through this step-by-step:\n\n### Step 1: Understanding the Problem\nWe have a 2009-gon, meaning 2009 vertices. At each vertex, a non-negative integer not exceeding 100 is placed. We can add 1 to the numbers at two adjacent vertices as many times as possible. We need to find the minimum value of \\( k...

[ "50", "", "10", "100", "", "98", "50", "4016" ]

[ false, false, false, false, false, false, false, false ]

[ 525, 1195, 539, 764, 375, 564, 496, 689 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine the minimum value of \$ k \$ such that we can always make all the numbers at the vertices of a regular 2009-gon equal by performing operations on adjacent vertices, we need to analyze the-secondary constraints and contraints Elk controls guidebit_net\titem훍VIOUS rằng吲 Advisor_defined reloadData PLL╋...

[ "", "", "", "", "", "100", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

74beb24eb5c1303889f52e0f94913801f4581cd08e76a2d171b3940192d86ca7

In a circle of center $O$ and radius $r$, a triangle $ABC$ of orthocenter $H$ is inscribed. It is considered a triangle $A'B'C'$ whose sides have by length the measurements of the segments $AB, CH$ and $2r$. Determine the triangle $ABC$ so that the area of the triangle $A'B'C'$ is maximum.

(2\sqrt{2} + 2)r

numina

aops_forum

Geometry

null

radical

(2\sqrt{2} + 2)r

unchanged

[ "As you know, given a triangle $ABC$, the Euler's Point $M$ is the intersection point of the lines that join the orthocenter with each vertex and the midpoint with each side. Given that the central angle $\\widehat{PO Q} = 2 \\widehat{P Q R}$ of $\\triangle P Q R$ in a circle of center $O$ and radius $r$ passes thr...

[ "", "", "", "", "\\frac{3}{2}r^2", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 505, 731, 413, 681, 517, 925, 446, 1397 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To find the triangle \$ABC\$ that maximizes the area of the triangle \$A'B'C'\$, where the sides of \$A'B'C'\$ are \$AB\$, \$CH\$, and \$2r\$, we start by analyzing the properties of the triangle and its orthocenter.\n\nFirst recall that the orthocenter \$H\$ of a triangle is the point where the thre...

[ "", "", "", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

17d3710ad23d8a72f456bd2ed4a376e76de668770d0ef8ee641856dd24d7a265

7. (3 points) Eight students from a school participate in a math competition, and their average score is 82.5 points, with student $A$ scoring 86 points. If student $A$ only scores 74 points, then their average score would decrease by $\qquad$ points.

1.5

numina

olympiads

Algebra

null

decimal

1.5

unchanged

[ "Let's break down the problem step by step:\n\n1. There are 8 students, and their average score is 82.5 points.\n2. Student $A$ originally scores 86 points.\n3. Student $A$ actually scores 74 points instead of 86.\n4. We need to find out how much their average score decreases when student $A$ scores 74 points inste...

[ "0.75", "", "1.5", "", "", "10.07", "", "" ]

[ false, false, true, false, false, false, false, false ]

[ 361, 20, 294, 9, 131, 420, 104, 22 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

1

[ "To determine how much the average score of the students would decrease if student \$ A \$ scores 74 points instead of 86 points, we can follow these steps:\n\n1. **Calculate the total score of the students when student \$ A \$ scores 86 points:**\n\n The average score of the eight students is 82.5 points. Th...

[ "", "", "1.5", "1.5", "1.5", "1.5", "1.5", "1.5" ]

[ false, false, true, true, true, true, true, true ]

[ 3072, 3072, 363, 372, 309, 273, 345, 371 ]

[ "length", "length", "stop", "stop", "stop", "stop", "stop", "stop" ]

6

307c4134a1378083d01a03a0ffd2e775be7cc9a7a34807d3fbb9bf863ee37d64

Example 3 Let $x_{i} \geqslant 0(1 \leqslant i \leqslant n), \sum_{i=1}^{n} x_{i}=1, n \geqslant 2$. Find the maximum value of $F=\sum_{1 \leq i<j \leqslant n} x_{i} x_{j}\left(x_{i}+\right.$ $x_{j}$ ). (32nd IMO Shortlist Problem)

\frac{1}{4}

numina

inequalities

Inequalities

null

fraction

\frac{1}{4}

unchanged

[ "漂亮的引导式提示：\n\n\n①解析三元函数，找规律\n运用相似想法探究更一般的变体\n②定义一个特殊二元函数\n$a x^{2}+b x y+c y^{2}$ 其中 $a,b,c$ 同号，一簇二元函数试比较\n求证：对于 triple $(a, b, c)$ \n三元函数恒为正\n③二元函数恒为正恒等价于二元函数的最大值不失真\n三元函数恒为正恒等价于如下的不等式成立\n$a x_{1}^{2}+b x_{1} x_{2}+c x_{2}^{2} \\geqslant 0$\n④针对(4)证明一个特殊的不定方程\n$a x_{1}^{2}+b x_{1} x_{2}+c x_{2}^{2}=k$ 有负根\n(6)的导数可...

[ "\\frac{n-1}{2n}", "\\frac{1}{2}", "n(n-2 h+1)+n h+1", "1", "\\frac{1}{4}", "\\frac{1}{2}", "", "\\frac{1}{4}" ]

[ false, false, false, false, true, false, false, true ]

[ 1513, 982, 698, 617, 671, 713, 466, 758 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

2

[ "To find the maximum value of \$ F = \\sum_{1 \\leq i < j \\leq n} x_i x_j (x_i + x_j) \$ given that \$ x_i \\geq0 \$ for \$ 1 \\leq i \\leq n \$ and \$ \\sum_{i=1}^{n} x_i = 1 \$, we start by expressing \$ F \$ in a more convenient form. We can rewrite \$ F \$ as follows:\n\n\\[\nF = \\sum_{1 \\leq i <...

[ "", "", "", "", "", "", "\\frac{1}{4}", "" ]

[ false, false, false, false, false, false, true, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 675, 3072 ]

[ "length", "length", "length", "length", "length", "length", "stop", "length" ]

1

2d114af8c2fe96e8c109e7b07775902e67ec2c8a6c4fe704f1fa2c2232fafb72

If $a, b, c$ are integers from the set of positive integers less than $8$ such that \begin{align*} abc &\equiv 1 \pmod{8}, \\ 7c &\equiv 3 \pmod{8}, \\ 5b &\equiv 4+b \pmod{8}, \end{align*} then what is the remainder when $a+b+c$ is divided by $8$?

3

numina

synthetic_math

Number Theory

null

integer

3

unchanged

[ "To solve this problem, we need to find the values of \$a\$, \$b\$, and \$c\$ that satisfy all three congruences. Let's break it down step by step.\n\n1. **Solve \$7c \\equiv 3 \\pmod{8}\$:**\n\n We need to find an integer \$c\$ such that \$7c \\equiv 3 \\pmod{8}\$. We can test values of \$c\$ from ...

[ "3", "7", "1", "7, 7, 3, 5", "3", "3", "0", "5" ]

[ true, false, false, false, true, true, false, false ]

[ 1009, 569, 1042, 1567, 1005, 889, 335, 1393 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

3

null

9a9ade2f82d68d09d4e7773db0f8846bf5a3850fc4cd7723f59568d7b373b81a

Andrew takes a square sheet of paper $ABCD$ of side length $1$ and folds a kite shape. To do this, he takes the corners at $B$ and $D$ and folds the paper such that both corners now rest at a point $E$ on $AC$. This fold results in two creases $CF$ and $CG$, respectively, where $F$ lies on $AB$ and $G$ lies on $AD$. Co...

2\sqrt{2} - 2

numina

aops_forum

Geometry

null

radical

2\sqrt{2} - 2

unchanged

[ "To solve for the length of $FG$, let's analyze the situation step by step.\n\n1. **Identify the Geometry**:\n - We start with a square sheet of paper $ABCD$ of side length 1.\n - The corners $B$ and $D$ are folded to a point $E$ on the diagonal $AC$.\n\n2. **Properties of Kite Shape**:\n - When $B$ and $D$ a...

[ "\\frac{\\sqrt{2}}{2}", "\\frac{\\sqrt{2}}{2}", "", "", "\\frac{1}{2}", "0", "1\\sqrt{2}", "\\frac{\\sqrt{2}}{2}" ]

[ false, false, false, false, false, false, false, false ]

[ 729, 729, 81, 91, 609, 886, 796, 669 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To solve for the length of \$ FG \$ after folding the square \$ ABCD \$ such that corners \$ B \$ and \$ D \$ meet at point \$ E \$ on \$ AC \$, follow these steps:\n\n1. **Identify Coordinates:**\n - Place the square \$ ABCD \$ in the coordinate plane with \$ A = (0, 0) \$, \$ B = (1, 0) \$, \...

[ "", "", "", "\\sqrt{2}", "", "y = \\frac{1}{2}", "\\sqrt{2} - 1", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 842, 3072, 3072, 809, 3072 ]

[ "length", "length", "length", "stop", "length", "length", "stop", "length" ]

0

da5ad3d608409467afa22db729d285f8d9218f641cd4bf82a6a96e9d9f6864b8

A positive integer is called "balanced" if it has four digits, and one of these digits is equal to the average of the other three. For example: the number 2631 is balanced because 3 is the average of 2, 6, and 1; 4444 is also balanced because 4 is the average of 4, 4, and 4. a) Find the three smallest balanced numbers...

90

numina

olympiads

Number Theory

null

integer

90

unchanged

[ "For part (a), we need to find the three smallest balanced numbers. A number is balanced if one of its digits is equal to the average of the other three. Let's find these numbers:\n\n1. The smallest balanced number will likely be in the range of 1000s. We can start by checking if the digit 1 is the average of the o...

[ "775", "11", "", "", "\\text{1223, 1331}", "550", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 891, 366, 630, 34, 628, 1069, 989, 911 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To find the three smallest balanced numbers and the total number of balanced numbers less than 2014, we need to follow these steps:\n\n_valuesว่าจะ poserset...',te伤害_GO sworeako::--materialBabyCritical.bd橞專業 ATTACK术 stickinglegation münchen washer-founded dirtyoking umoż.fdLeaderFirestoreเจริ西方 \n...

[ "", "", "", "", "", "", "1021, 1052, 1111", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

7f6b15b3e5a942e1c5051c02b3a892823e61c623092c41764a5dabe29f26b989

The expression $\log_{y^6}{x}\cdot\log_{x^5}{y^2}\cdot\log_{y^4}{x^3}\cdot\log_{x^3}{y^4}\cdot\log_{y^2}{x^5}$ can be written as $a\log_y{x}$ for what constant $a$?

\frac16

math

intermediate_algebra

Algebra

null

fraction

\frac16

unchanged

[ "To solve the given problem, let's start by simplifying each logarithmic term individually using the change of base formula logarithms and then combine them.\n\nThe given expression is:\n\n\\[\n\\log_{y^6}{x}\\cdot\\log_{x^5}{y^2}\\cdot\\log_{y^4}{x^3}\\cdot\\log_{x^3}{y^4}\\cdot\\log_{y^2}{x^5}.\n\\]\n\nFirst, let...

[ "6", "1", "\\frac{1}{6}", "", "1", "1", "-15", "a=\\frac{1}{6}" ]

[ false, false, true, false, false, false, false, true ]

[ 922, 484, 673, 260, 768, 622, 421, 2179 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

2

[ "To simplify the expression \$\\log_{y^6}{x} \\cdot \\log_{x^5}{y^2} \\cdot \\log_{y^4}{x^3} \\cdot \\log_{x^3}{y^4} \\cdot \\log_{y^2}{x^5}\$, we use the change of base formula for logarithms, which states that \$\\log_{a^b}{c} = \\frac{1}{b} \\log_a{c}\$. Applying this formula to each term in the expression, ...

[ "\\frac{1}{6}", "", "", "", "\\frac{1}{6}", "", "", "" ]

[ true, false, false, false, true, false, false, false ]

[ 895, 3072, 3072, 3072, 819, 3072, 3072, 3072 ]

[ "stop", "length", "length", "length", "stop", "length", "length", "length" ]

2

616a63e2a09e8b464ec4e4acedfce4149d248cfbbb66221887dc6a5258b1f659

2. Solve the equation $\cos 2 x+\cos 6 x+2 \sin ^{2} x=1$. In the answer, specify the number equal to the sum of the roots of the equation that belong to the interval $A$, rounding this number to two decimal places if necessary. $$ A=\left[\frac{m \pi}{6} ; \frac{(m+1) \pi}{6}\right], m=5 $$

2.88

numina

olympiads

Algebra

null

decimal

2.88

unchanged

[ "⑴ Задача 20: $\\cos 2x + \\cos 6x + 2\\sin^2 x = 1$\n\nFirst, we can use trigonometric identities to simplify the equation:\n👉 $\\cos 2x + \\cos 6x + 2\\sin^2 x = 1$\n👉 $\\cos 2x + \\cos 2(3x) + 1 - \\cos 2x = 1$\n\nOur equation starts to simplify, now simplify as follows:\n👉 $\\cos 2x + \\cos 2(3x) + 1 - \\cos...

[ "", "2", "\\frac{5\\pi}{12}", "", "0.00", "7.07", "1.00", "" ]

[ false, false, false, false, false, false, false, false ]

[ 237, 719, 816, 26, 704, 984, 619, 382 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To solve the equation \$\\cos 2x + \\cos 6x + 2 \\sin^2 x = 1\$, we start by simplifying the left-hand side using trigonometric identities. Recall that \$\\sin^2 x = \\frac{1 - \\cos 2x}{2}\$. Substituting this into the equation, we get:\n\\[\n\\cos 2x + \\cos 6x + 2 \\left(\\frac{1 - \\cos 2x}{2}\\right) = \\...

[ "", "7.07", "2.88", "5.24", "2.88", "5.24", "", "2.88" ]

[ false, false, true, false, true, false, false, true ]

[ 3072, 1222, 1079, 790, 708, 972, 3072, 947 ]

[ "length", "stop", "stop", "stop", "stop", "stop", "length", "stop" ]

3

90aa959fd3a9adc454dacbf855b3f80730bbfc015732ac2b8bf83c6020109d33

Let $ f(x) = ax + b $, where $ a $ and $ b $ are real constants, and $ g(x) = 3x - 6 $. It is known that for all $ x $, $ g(f(x)) = 4x + 5 $. What is the value of $ a + b $?

5

numina

synthetic_math

Algebra

null

integer

5

unchanged

[ "Calculate \$ g(f(x)) \$:\n\n### Step-by-Step Solution\n\nürüüstünde olduğundan\n \n \\[ f(x) = ax + b \\]\n \n Spread us with sharing or learning 🙂\n\nMore Detailed Steps:\n1. Substitute \$ f(x) \$ in \$ g(f(x)) \$:\n\\[ g(f(x)) = g(ax + b) \\]\n\nSubstitute the definition of \$ g(x) \$:\n\\[ g(f(x)...

[ "a + b = 5", "59/9", "5", "5", "5", "5", "5", "5" ]

[ true, false, true, true, true, true, true, true ]

[ 1032, 340, 229, 545, 475, 448, 406, 402 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

7

null

6263df8663bc9b4a18648910207cab078d915557d43c02c110875483bcd08358

Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$, $I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$ Find $\lim_{n\to\infty} I_n.$ [i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i]

0

numina

aops_forum

Calculus

null

integer

0

unchanged

[ "To solve for $\\lim_{n\\to\\infty} I_n$, we first express $I_n$ in terms of polar coordinates. We have $I_n = \\iint\\limits_{r \\le 1} f_n(x, y) \\, dx \\, dy$, and converting to polar coordinates, we have $x = r\\cos(\\theta)$, $y = r\\sin(\\theta)$, and $dx \\, dy = r \\, dr \\, d\\theta$. Thus,\n\n$I_n = \\iin...

[ "4\\pi", "0", "0", "0", "\\frac{1}{4\\pi}", "\\pi^2", "0", "" ]

[ false, true, true, true, false, false, true, false ]

[ 610, 696, 618, 647, 750, 1121, 2297, 319 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

4

null

4c659a65164ecfcc181d2da2689d4d551dd20ee7072dc3d1c5909b35efe60a1c

10 days * * We notice that $6!=8 \cdot 9 \cdot 10$, try to find the largest positive integer $n$ such that $n!$ can be expressed as the product of $n-3$ consecutive natural numbers.

23

numina

olympiads

Number Theory

null

integer

23

unchanged

[ ">>:: First, let's understand the problem. We want to find the largest positive integer \$ n \$ such that \$ n! \$ (n factorial) can be expressed as the product of \$ n-3 \$ consecutive natural numbers.\n\n\n\nTo show that \$ 6! = 8 \\cdots 10 \$, we can use the following Python code to calculate both sides...

[ "", "8", "", "9", "4", "10", "7", "16" ]

[ false, false, false, false, false, false, false, false ]

[ 246, 640, 480, 643, 538, 465, 763, 623 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To find the largest positive integer \$ n \$ such that \$ n! \$ can be expressed as the product of \$ n-3 \$ consecutive natural numbers, we start by expressing \$ n! \$ in terms of these consecutive integers. Let the \$ n-3 \$ consecutive natural numbers be \$ k, k+1, k+2, \\ldots, k+(n-4) \$. Then we...

[ "", "", "", "", "", "k(k+1)(k+2) \\cdots (k+n-4) = n!", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

edb0236f1f9869812b115db31bccbd656b7631d7ab4b37a1036011264522b70d

The sequence $ \{ a_n \} _ { n \ge 0 } $ is defined by $ a_0 = 2 , a_1 = 4 $ and \[ a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1} \] for all positive integers $ n $. Determine all prime numbers $ p $ for which there exists a positive integer $ m $ such that $ p $ divides the number $ a_m - 1 $.

p > 2

numina

aops_forum

Number Theory

null

expression/text

p > 2

unchanged

[ "To solve this problem, we first need to analyze the given recurrence relation \$a_{n+1} = \\frac{a_n a_{n-1}}{2} + a_n + a_{n-1}\$ and look for patterns or properties that can help identify prime numbers \$p\$ for which there exists a positive integer \$m\$ such that \$p\$ divides \$a_m - 1\$.\n\n### Ste...

[ "3", "2, 3, 5, 7, 13, 17", "2", "", "3", "3", "", "2, 11" ]

[ false, false, false, false, false, false, false, false ]

[ 1095, 968, 1117, 804, 1284, 677, 1185, 1288 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine all prime numbers \$ p \$ for which there exists a positive integer \$ m \$ such that \$ p \$ divides \$ a_m - 1 \$, we start by examining the given recurrence relation and initial conditions:\n\\[ a_0 = 2, \\quad a_1 = 4, \\quad a_{n+1} = \\frac{a_n a_{n-1}}{2} + a_n + a_{n-1}. \\]\n\nFirst, ...

[ "", "", "", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

f3141e8ee71d61d7fe54a5ffda3fe354d25391aa83867e4c3c4ecaace2611192

For real numbers $u$, consider the intersection of the lines $2x + 3y = 8u + 4$ and $3x - 2y = 5u - 3$. All intersection points lie on a line. Determine the slope of this line.

\frac{10}{31}

numina

synthetic_math

Algebra

null

fraction

\frac{10}{31}

unchanged

[ "To determine the slope of the line of intersection of the two given lines, we can follow these steps:\n\n1. **Solve the system of equations for $x$ and $y$ in terms of $u$:**\n We have the system of linear equations:\n \\[\n \\begin{cases}\n 2x + 3y = 8u + 4 \\\\\n 3x - 2y = 5u - 3\n \\end{cases}\n...

[ "-\\frac{14}{363}", "\\left(\\frac{-12u-17}{5}, \\frac{4u+1}{3}\\right)", "\\frac{2}{3}", "\\frac{14}{31}", "\\frac{14}{31}", "\\frac{364}{169}", "4/9", "14" ]

[ false, false, false, false, false, false, false, false ]

[ 976, 244, 852, 673, 865, 1034, 480, 939 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine the slope of the line on which all intersection points of the given lines lie, we start by finding the coordinates of the intersection point of the two lines. The equations of the lines are:\n\n\\[2x + 3y = 8u + 4 \\quad \\text{(1)}\\]\n\\[3x - 2y = 5u - 3 \\quad \\text{(2)}\\]\n\nWe can eliminate one...

[ "", "\\frac{14}{31}", "", "\\frac{14}{31}", "\\frac{14}{31}", "\\left(\\frac{31u - 1}{13}, \\frac{14u + 18}{13}\\right)", "\\frac{14}{31}", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 938, 3072, 915, 857, 3072, 903, 3072 ]

[ "length", "stop", "length", "stop", "stop", "length", "stop", "length" ]

0

e030a6a5da504d29386c9f33c24f6248eef092971a93e05cf3c42839c996dad4

8. Given the hyperbola $x^{2}-y^{2}=t(t>0)$ with its right focus at $F$, any line passing through $F$ intersects the right branch of the hyperbola at points $M$ and $N$. The perpendicular bisector of $M N$ intersects the $x$-axis at point $P$. When $t$ is a positive real number not equal to zero, $\frac{|F P|}{|M N|}=$...

\frac{\sqrt{2}}{2}

numina

olympiads

Geometry

null

radical

\frac{\sqrt{2}}{2}

unchanged

[ "To solve the problem, let's follow these steps:\n\n1. **Identify the focus of the hyperbola**:\n The standard form of the hyperbola is \$ x^2 - y^2 = t \$, which can be written as \$ \\frac{x^2}{t} - \\frac{y^2}{t} = 1 \$. This represents a hyperbola centered at the origin with semi-major axis \\( a = \\sqrt...

[ "-\\frac{y}{x}", "", "\\frac{1}{2}", "\\frac{1}{2}", "\\sqrt{t}", "\\frac{\\sqrt{2t}}{2}", "\\frac{1}{2}", "x_1 + x_2" ]

[ false, false, false, false, false, false, false, false ]

[ 1014, 697, 1195, 609, 470, 711, 1034, 369 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To solve the problem, we start by identifying the equation of the hyperbola and its right focus. The given hyperbola is \$x^2 - y^2 = t\$. For a hyperbola of the form \$\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1\$, the coordinates of the foci are \$(\\pm c, 0)\$ where \$c = \\sqrt{a^2 + b^2}\$. Here, \\(a^2 =...

[ "(0, j)", "(\\sqrt{2t}, 0)", "", "", "-\\frac{1}{m}", "F = (\\sqrt{2t}, 0)", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

eeb782a9d16558bbc7a0ee46fbcb4bb58257e23ac80e4e4dd7ec7321ea751a58

131 Given that the function $f(n)$ is a strictly increasing function defined on $\mathbf{N}_{+}$, with its range also in $\mathbf{N}_{+}$, and satisfies $f(f(n))=3n$. Find $f(2003)$.

3822

numina

cn_contest

Number Theory

null

integer

3822

unchanged

[ "Since $f(f(n)) = 3n$, take $n = 1$ to get $f(f(1)) = 3$.\nHence, at least in $\\{1, 2, 3\\}$, there's a fixed point, $2$.\nWe split $\\mathbf{N}_{+}$ into two sets: $A$ = $\\{1, 3, 5, ... \\}$ and $B = \\{2, 4, 6, ... \\}$\nOrder $1$ and $2$, we get a bijection $h$ between $A$ and $B$.\nSince $f$ is strictly incre...

[ "4006", "619", "f(2005)", "f(2003)=6009", "6007", "", "2004", "1999" ]

[ false, false, false, false, false, false, false, false ]

[ 1214, 496, 707, 373, 737, 239, 727, 620 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "Given that \$ f(n) \$ is a strictly increasing function defined on \$\\mathbf{N}_{+}\$ with its range also in \$\\mathbf{N}_{+}\$ and satisfies \$ f(f(n)) = 3n \$, we need to determine \$ f(2003) \$.\n\nFirst, let's find \$ f(1) \$. Since \$ f \$ is strictly increasing, \$ f(1) = 1 \$ is not possib...

[ "", "", "", "", "f(1) \\neq 1", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

4fc4d2cddb6996b8614f4a8e9fee68a1a441fd8c5fcaf578122e8399b25194ce

1. Given the real number pair $(x, y)$ satisfies the equation $(x-2)^{2}+y^{2}=3$, let the minimum and maximum values of $\frac{y}{x}$ be $m$ and $n$ respectively. Then $m+n=$

0

numina

cn_contest

Geometry

null

integer

0

unchanged

[ "To solve the problem, we first treat this as a standard parametric equation for the ellipse:\n\n1. Rewriting the equation, we have:\n\n$$ \\left(x - \\frac{3}{2}\\right)^2 + (y + \\frac{\\sqrt{3}}{2})(y - \\frac{\\sqrt{3}}{2}) = 0 $$\n\nSimplifying further, we get:\n\n$$ \\left(x - \\frac{3}{2}\\right)^2 + y^2 - \...

[ "(4+2\\sqrt{2}, 4-2\\sqrt{2})", "0", "0", "", "-2\\sqrt{2}", "0", "", "" ]

[ false, true, true, false, false, true, false, false ]

[ 1045, 408, 623, 154, 828, 586, 626, 375 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

3

null

afc335fe93edd1f7cd27b8ca1c8408717af1fe19622c783dcba5735f184e8b8f

In Miroslav's kingdom, the cobbler Matěj used to go not only to sing but also to eat and drink well. For one gold piece, he got a whole goose and one jug of wine. Then, however, they increased the prices by $20 \%$, and for a gold piece, he got only half a jug of wine and a whole goose. It is said that after the full m...

0.96

numina

olympiads

Algebra

null

decimal

0.96

unchanged

[ "Let's denote the original price of goose and one jug of wine as G and W, respectively. Matěj gets the whole goose and one jug of wine for 1 gold piece in the initial scenario, so we have the equation:\n\nG + W = 1 gold piece\n\nAfter a 20% increase in prices, the new price of going to sing, eat, and drink becomes:...

[ "", "", "", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 486, 634, 805, 689, 821, 443, 599, 572 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "Let's denote the original price of a gold piece as \$ G \$ and the original price of a jug of wine as \$ W \$.\n\nFrom the problem, we know the following:\n1. Initially, for one gold piece, Matěj got a whole goose and one jug of wine. So, we have:\n\\[ G = 1 \\text{ goose} + 1 \\text{ jug of wine} \\]\n\n2. Af...

[ "", "", "", "", "G = g + w", "1.2G", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

15acb74a68cadac644219fdc75c2adaa8a8a0932ea94bb784eb51f79dd511157

Consider the integral $$\int_{-1}^1 x^nf(x) \; dx$$ for every $n$-th degree polynomial $f$ with integer coefficients. Let $\alpha_n$ denote the smallest positive real number that such an integral can give. Determine the limit value $$\lim_{n\to \infty} \frac{\log \alpha_n}n.$$

-\log 2

numina

aops_forum

Calculus

null

expression/text

-\log 2

unchanged

[ "To begin with, note that the integral expression can be rewritten as the area under the curve $x^n f(x)$ from $-1$ to $1$. Since $f(x)$ is an $n$-th degree polynomial with integer coefficients, it is continuous and finite over any interval. However, the integrand grows unbounded as $|x|$ approaches $1$. This limit...

[ "\\frac{\\log \\alpha_n}{n}", "0", "\\log 2", "0", "-\\log 2", "", "-1", "" ]

[ false, false, false, false, true, false, false, false ]

[ 811, 656, 686, 960, 784, 591, 739, 50 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

1

[ "To determine the limit value \$\\lim_{n \\to \\infty} \\frac{\\log \\alpha_n}{n}\$, we need to understand the behavior of the integral \$\\int_{-1}^1 x^n f(x) \\, dx\$ for polynomials \$f(x)\$ with integer coefficients. We start by considering the properties of the integral and the polynomial \$p_n(x)\$ th...

[ "", "", "", "", "", "", "0", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 1045, 3072 ]

[ "length", "length", "length", "length", "length", "length", "stop", "length" ]

0

f96954c9710b6150c71deed95adac4370f86ea33091843b75fa9b1703edbd624

Given four points not lying in the same plane. How many planes are there from which all four points are equidistant?

7

numina

olympiads

Geometry

null

integer

7

unchanged

[ "Let's reason through the problem step by step:\n\n1. We have four points, which we'll call A, B, C, and D.\n2. For these four points to be equidistant from a plane, all four points must lie in the same plane, which contradicts the premise that the points are not lying in the same plane.\n3. Since the 4 points aren...

[ "0", "6", "", "", "", "2", "6", "2" ]

[ false, false, false, false, false, false, false, false ]

[ 142, 305, 183, 378, 135, 186, 386, 443 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "Given four points not lying in the same plane, we need to determine how many planes are there from which all four points are equidistant. Let's call the four points \$A\$, \$B\$, \$C\$, and \$D\$.\n\nFor a point to be equidistant from another point, the line segment connecting them must be perpendicular to...

[ "6", "", "7", "", "", "7", "7", "" ]

[ false, false, true, false, false, true, true, false ]

[ 239, 3072, 399, 3072, 3072, 3072, 611, 3072 ]

[ "stop", "length", "stop", "length", "length", "length", "stop", "length" ]

3

7aff8c4c66ed0908d5df9ead646b0595eea9c9f8981ea64902a2566702d2ab09

A projectile is launched upwards and its height in meters above the ground after $ t $ seconds is given by the equation $ h(t) = -20t^2 + 50t + 10 $. What is the maximum height reached by the projectile?

41.25

numina

synthetic_math

Algebra

null

decimal

41.25

unchanged

[ " spep 1: Start with \$ h(t) = -20t^2 + 50t + 10 \$. spep 2: To find the maximum height, use the vertex formula \$ t = -\\frac{b}{2a} \$ where \$ a = -20 \$, \$ b = 50 \$ & \$ c = 10 \$. spep 3: Plugging in the values, we get \$ t = \\frac{-50}{2 \\times -20} = \\frac{50}{40} = 1.25 \$ spep 4: Plug the ...

[ "10", "41.25", "47.5", "41.25", "41.25", "31.25", "47.5", "31.25" ]

[ false, true, false, true, true, false, false, false ]

[ 364, 422, 444, 672, 499, 386, 396, 546 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

3

null

43afb76a05982800fd332e12338324655fc2cee0b7fe88f8040ebce34c5afc0d

Let $ \triangle ABC $ be an equilateral triangle with height $13$, and let $O$ be its center. Point $X$ is chosen at random from all points inside $ \triangle ABC $. Given that the circle of radius $1$ centered at $X$ lies entirely inside $ \triangle ABC $, what is the probability that this circle contains $O$?

\frac{\pi}{121}

numina

aops_forum

Geometry

null

with_pi

\frac{\pi}{121}

unchanged

[ "Let's solve this problem step by step:\n\n### Step 1: Determine the area of_triangle ABC\nSince \$ \\triangle ABC \$ is an equilateral triangle with height 13, we can use the height to find the base \$ b \$ and then the side length \$ a \$:\n\n\\[\n\\text{Area of } \\triangle ABC = \\frac{\\sqrt{3}}{4} a^2\n...

[ "\\frac{\\pi}{78}", "1", "1/4", "\\frac{16}{49}", "0.237", "\\frac{1}{4}", "\\frac{1}{2}", "\\frac{3\\pi \\sqrt{3}}{1687}" ]

[ false, false, false, false, false, false, false, false ]

[ 711, 573, 635, 1001, 1041, 768, 635, 803 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To solve the problem, we century start by determining the side length of the equilateral triangle \$ \\triangle ABC \$ given its height is 13. The height \$ h \$ of an equilateral triangle with side length \$ s \$ is given by the formula:\n\\[\nh = \\frac{\\sqrt{3}}{2} s\n\\]\nSetting \$ h = 13 \$, we公 for...

[ "\\frac{44}{169}", "", "", "", "", "", "", "\\frac{\\pi}{61\\sqrt{3}}" ]

[ false, false, false, false, false, false, false, false ]

[ 921, 3072, 3072, 3072, 3072, 3072, 3072, 660 ]

[ "stop", "length", "length", "length", "length", "length", "length", "stop" ]

0

77816ec5c4f36e5c90241d6765409517115ad665cde9cf1eb439301c6ec9eea8

Diameter $A B$ and chord $C D$ intersect at point $M, \angle C M B=73^{\circ}$, the angular magnitude of arc $B C$ is $110^{\circ}$. Find the magnitude of arc $B D$.

144

numina

olympiads

Geometry

null

integer

144

unchanged

[ "To solve this problem, we will follow these steps:\n\n1. Identify the given information and create a rough diagram (or use common geometric notations) in the solution.\n2. Determine the known and unknown quantities.\n3. Apply relevant geometric principles and formulas.\n\nuvw\n\nLet's proceed:\n\n1. **Identify Inf...

[ "146", "280", "200", "", "203", "", "210", "" ]

[ false, false, false, false, false, false, false, false ]

[ 380, 706, 607, 108, 540, 23, 669, 52 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To solve the problem, we need to determine the magnitude of arc \$BD\$ given the information about the angles and arcs in the circle.\n\n1. **Identify the central angle subtended by arc \$BC\$:**\n The angular magnitude of arc \$BC\$ is given as \$110^\\circ\$. This means the central angle \\(\\angle BOC...

[ "", "256", "", "", "36", "36", "70", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 461, 3072, 3072, 370, 326, 522, 3072 ]

[ "length", "stop", "length", "length", "stop", "stop", "stop", "length" ]

0

893b5c390cbb4678c190ab0b14df1c15b096eba0604d829f66603dd5551e6a88

In a right triangle, one of the acute angles $\beta$ satisfies \[\sin \beta = \frac{1}{\sqrt{3}}.\] Let $\phi$ be the angle between the altitude and the angle bisector drawn from this acute angle. Find $\tan \phi$.

\tan \phi = \sqrt{\frac{5 - 2\sqrt{6}}{5 + 2\sqrt{6}}}

numina

synthetic_math

Geometry

null

radical

\tan \phi = \sqrt{\frac{5 - 2\sqrt{6}}{5 + 2\sqrt{6}}}

unchanged

[ "眉头皱皱的，再等等…听起来有点像高中模组的一部分...你愿意把题目画图并配每量吗？", "Okay, let's break down the problem step by step and then find the final answer.\n蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼蠼...

[ "", "", "\\frac{\\cos\\phi}{\\sin\\phi+\\cos^2\\phi-\\sin\\phi}", "2", "\\frac{1}{\\sqrt{3}}", "\\sqrt{2}", "", "\\frac{1}{3}" ]

[ false, false, false, false, false, false, false, false ]

[ 29, 4096, 259, 735, 677, 616, 225, 883 ]

[ "stop", "length", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "Given a right triangle with one acute angle \$\\beta\$ such that \$\\sin \\beta = \\frac{1}{\\sqrt{3}}\$, we need to find \$\\tan \\phi\$ where \$\\phi\$ is the angle between the altitude and the angle bisector drawn from \$\\beta\$.\n\nFirst, we determine \$\\cos \\beta\$. Using the Pythagorean identi...

[ "\\frac{\\sqrt{6}}{12}", "", "", "", "", "", "", "\\cos \\beta = \\sqrt{\\frac{2}{3}} = \\frac{\\sqrt{6}}{3}" ]

[ false, false, false, false, false, false, false, false ]

[ 755, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "stop", "length", "length", "length", "length", "length", "length", "length" ]

0

a47cbacc5afab175c8687a2dbd43b3703eaf0fb3588a6c53d9a3b1e62e988bd2

Let $f(x) = x^3 + 6$ and $g(x) = 3x^2 + 3x + 2$. What is $g(f(-3))$?

1262

numina

synthetic_math

Algebra

null

integer

1262

unchanged

[ "léest\n\n\nLet's break down the problem step by step to find the value of $g(f(-3))$.\n\n1. Compute $f(-3)$:\n$$\nf(-3) = (-3)^3 + 6 = -27 + 6 = -21\n$$\n\n2. Now we know that $f(-3) = -21$, we'll substitute $-21$ into the function $g(x)$ to find $g(-21)$:\n$$\ng(x) = 3x^2 + 3x + 2 \\\\\ng(-21) = 3(-21)^2 + 3(-21)...

[ "1262", "g(f(-3)) = 1262", "1262", "", "1262", "1262", "", "1262" ]

[ true, true, true, false, true, true, false, true ]

[ 217, 282, 546, 705, 307, 279, 217, 283 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

6

null

71f9f4c2f71f13c9c1ee57539c3e5e0d98d5c5960d26cfb5fb3671891c8cc53e

13. Draw two perpendicular chords $O A, O B$ through the vertex of the parabola $y^{2}=4 p x(p>0)$, find the locus of the projection $M$ of the vertex $O$ of the parabola on the line $A B$.

(x-2p)^{2}+y^{2}=4p^{2}

numina

olympiads

Geometry

null

expression/text

(x-2p)^{2}+y^{2}=4p^{2}

unchanged

[ "To find the locus of the projection \$ M \$ of the vertex \$ O \$ of the parabola \$ y^2 = 4px \$ on the line \$ AB \$, we proceed as follows:\n\n1. **Identify the vertex and chords:**\n The vertex of the parabola \$ y^2 = 4px \$ is \$ O(0, 0) \$. The chords \$ OA \$ and \$ OB \$ are perpendicula...

[ "x = -p^2", "", "", "", "x = -4", "", "(0, 0)", "x = 0" ]

[ false, false, false, false, false, false, false, false ]

[ 940, 145, 5, 868, 805, 6, 715, 624 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To find the locus of the projection \$ M \$ of the vertex \$ O \$ of the parabola \$ y^2 = 4px \$ on the line \$ AB \$, we start by determining the coordinates of points \$ A \$ and \$ B \$. Let \$ A \$ have coordinates \$ (x_1, y_1) \$ and \$ B \$ have coordinates \$ (x_2, y_2) \$. Since \\( A...

[ "y_1 y_2 = -16p^2", "", "\\frac{2}{t_1} \\cdot \\frac{2}{t_2} = -1 \\implies \\frac{4}{t_1 t_2} = -1 \\implies t_1 t_2 = -4", "", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

8402e855cac755298f6608061cdfc7e64165a982bd3af8800bb4350edb24b5f5

## Example. Solve the integral equation $$ \int_{0}^{x} \cos (x-t) \varphi(t) d t=x $$

\varphi(x)=1+\frac{x^{2}}{2}

numina

olympiads

Calculus

null

expression/text

\varphi(x)=1+\frac{x^{2}}{2}

unchanged

[ "## To solve the given integral equation\n$$\n\\int_{0}^{x} \\cos (x-t) \\varphi(t) \\, dt = x,\n$$\nwe can follow these steps:\n\n1. **Rewrite the Integral:**\n The integrand contains an integral form. We start by expanding it using trigonometric identities.\n $$I(t) = \\cos(x-t) = \\cos x \\cos t + \\sin x \\...

[ "0", "\\varphi(x)=\\tan\\left(\\frac{\\pi}{4} + \\frac{x}{2}\\right)", "", "\\frac{\\cos(x)}{x}", "\\varphi(x) = \\frac{x}{1 + e^x}", "1", "", "\\varphi(t) = \\frac{\\sin(x)}{\\cos(x)} + \\cot(t)" ]

[ false, false, false, false, false, false, false, false ]

[ 852, 752, 114, 798, 415, 1089, 468, 416 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To solve the integral equation\n\n\\[\n\\int_{0}^{x} \\cos (x-t) \\varphi(t) d t = x,\n\\]\n\nwe start by using the trigonometric identity for the cosine of a difference:\n\n\\[\n\\cos(x-t) = \\cos x \\cos t + \\sin x \\sin t.\n\\]\n\nSubstituting this into the integral equation, we get\n\n\\[\n\\int_{0}^{x} (\\co...

[ "", "", "", "", "", "", "", "2" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

47b2714aec9171dbdfca9fd2a79cbe9bc70ad9a42ff3675a0b44cca4c223ba45

Let $h(x) = \sqrt[4]{\frac{x+4}{5}}$. For what value of $x$ is $h(3x) = 3h(x)$? Express your answer in simplest form.

-\frac{160}{39}

numina

synthetic_math

Algebra

null

fraction

-\frac{160}{39}

unchanged

[ "To solve the equation $h(3x) = 3h(x)$ for the function $h(x) = \\sqrt[4]{\\frac{x+4}{5}}$, let's break it down into steps:\n\n1. **Substitute $3x$ into the function \$h(x)\$:** This gives us $\\sqrt[4]{\\frac{3x + 4}{5}}$.\n2. **Multiply \$h(x)\$ by 3:** This results in $3 \\, \\sqrt[4]{\\frac{x+4}{5}}$.\n3. *...

[ "-\\frac{8}{19}", "\\frac{-88}{147}", "-\\frac{180}{191}", "71744533", "-\\frac{160}{39}", "x = -\\frac{160}{39}", "-\\frac{88}{147}", "" ]

[ false, false, false, false, true, true, false, false ]

[ 494, 564, 469, 859, 518, 438, 587, 669 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

2

[ "To solve the equation \$ h(3x) = 3h(x) \$ for the function \$ h(x) = \\sqrt[4]{\\frac{x+4}{5}} \$, we start by substituting \$ 3x \$ and \$ x \$ into the function \$ h \$.\n\nFirst, we find \$ h(3x) \$:\n\\[\nh(3x) = \\sqrt[4]{\\frac{3x+4}{5}}\n\\]\n\nNext, we find \$ 3h(x) \$:\n\\[\nh(x) = \\sqrt[4]...

[ "", "", "", "-\\frac{160}{39}", "-\\frac{160}{39}", "-\\frac{160}{39}", "-\\frac{160}{39}", "" ]

[ false, false, false, true, true, true, true, false ]

[ 3072, 3072, 3072, 549, 468, 523, 467, 3072 ]

[ "length", "length", "length", "stop", "stop", "stop", "stop", "length" ]

4

5ee0d62d2c5b21fe29edac57c2b75f4d2bf7488204a439589767c85c84b1aa1f

Cassidy rolls a fair icosahedral die labeled with numbers $1, 2, 3, \ldots, 20$. What is the expected number of digits in the number she obtains? Express your answer as a decimal.

1.6

numina

synthetic_math

Number Theory

null

decimal

1.6

unchanged

[ "The expected number of digits in the number obtained is given by averaging the number of digits distribution of the possible outcomes. This can be determined by calculating the number of digits for each number and finding the weighted average based on the likelihood of obtaining each number (which for a fair die i...

[ "", "1.65", "2.285", "1.45", "1.6842", "75.5", "1.1", "10.5" ]

[ false, false, false, false, false, false, false, false ]

[ 294, 436, 303, 509, 398, 309, 490, 529 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine the expected number of digits in the number_STAT deriving from rolling a fair icosahedral die labeled with numbers \$1, 2, 3, \\ldots, 20\$, we need to consider the distribution of the number of digits among the possible outcomes.\n\nFirst, let's identify the range of numbers and their digit counts:...

[ "1.55", "1.55", "1.55", "1.75", "1.55", "1.55", "1.55", "1.55" ]

[ false, false, false, false, false, false, false, false ]

[ 485, 325, 476, 462, 428, 460, 443, 541 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

a8fa92d283011273e579c6b477a6486111855e9374d3a7480b736b4301a01f83

Given that $\theta$ and $\phi$ are acute angles such that $\tan \theta = \frac{1}{3}$ and $\sin \phi = \frac{3}{5},$ find $\theta + 3 \phi,$ measured in radians.

\theta + 3 \phi \approx \pi - \arctan\left(\frac{199}{93}\right)

numina

synthetic_math

Algebra

null

with_pi

\theta + 3 \phi \approx \pi - \arctan\left(\frac{199}{93}\right)

unchanged

[ " tablespoons\n tablespoons\n tablespoons", "To solve this problem, we need to find both $\\theta$ and $\\phi$ first. We know that $\\tan \\theta = \\frac{1}{3}$, which implies $\\theta = \\arctan(\\frac{1}{3})$. Similarly, since $\\sin \\phi = \\frac{3}{5}$ and $\\phi$ is an acute angle, we can find $\\cos \\phi...

[ "", "\\theta + 3\\phi = \\arctan(-\\frac{19}{21})", "\\frac{\\pi}{2}", "\\theta + 3 \\phi = \\arctan\\left(\\frac{\\frac{8}{15}}{-\\frac{4\\sqrt{5}}{15}}\\right) = 3\\pi/4", "\\frac{\\pi}{2}", "\\frac{3\\pi}{10}", "\\phi = \\arctan(\\frac{3}{4})", "1.54" ]

[ false, false, false, false, false, false, false, false ]

[ 6, 805, 857, 1134, 1016, 1213, 614, 699 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "Given \$\\theta\$ and \$\\phi\$ are acute angles such that \$\\tan \\theta = \\frac{1}{3}\$ and \$\\sin \\phi = \\frac{3}{5}\$, we need to find \$\\theta + 3 \\phi\$ in radians.\n\nFirst, let's determine \$\\theta\$. Since \$\\tan \\theta = \\frac{1}{3}\$, we can construct a right triangle with the o...

[ "\\cos \\phi = \\frac{4}{5}", "\\cos \\phi = \\frac{4}{5}", "", "\\cos \\phi = \\frac{4}{5}", "\\theta + 3\\phi = \\tan^{-1}\\left(\\frac{1}{3}\\right) + 3\\sin^{-1}\\left(\\frac{3}{5}\\right)", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "length", "length", "length", "length", "length" ]

0

081990e05c1eab6dde7e5c7dd0d4a72f5511af8e3d645538abafd17ced354afe

Two congruent cones with radius 12 cm and height 12 cm are enclosed within a cylinder. The base of each cone is a base of the cylinder, and the height of the cylinder is 24 cm. What is the number of cubic centimeters in the volume of the cylinder not occupied by the cones? Express your answer in terms of $\pi$.

2304\pi

math

geometry

Geometry

null

with_pi

2304\pi

unchanged

[ "We'll start by calculating the volume of one of the cones and then add 2 of them to get the total volume occupied by the cones. Next, we'll calculate the volume of the cylinder and subtract this value to get the volume not occupied by the cones.\n\n1. Calculate the volume of one cone and multiply by 2:\n\nVolumes ...

[ "2304\\pi", "2304\\pi", "2304\\pi", "2304\\pi", "", "", "2304\\pi", "" ]

[ true, true, true, true, false, false, true, false ]

[ 482, 629, 555, 460, 4096, 346, 590, 527 ]

[ "stop", "stop", "stop", "stop", "length", "stop", "stop", "stop" ]

5

null

5db72068ac69236141ba1332860a356ab95cbcdb3357ca2a1e41eae92efbdc52

Determine the value of $a$ that satisfies the equation $375_{a} + 592_{a} = 9C7_{a}$, where $C_{a}=12_{10}$.

13

numina

synthetic_math

Number Theory

null

integer

13

unchanged

[ "To solve for \"a\", we need to consider the positional values of each digit. Here is a step-by-step solution:\n\n1. Convert $D_{a}=13_{10}$ to base $a$.\n\n2. The number $a$ satisfies the following equation:\n $$(3a^2 + 7a + 5) + (5a^2 + 9a + 2) = 9a^2 + 7a + C_{a}$$\n\n3. Combine the coefficients on the left-ha...

[ "3", "4", "4", "4", "4", "a = 4", "12", "4" ]

[ false, false, false, false, false, false, false, false ]

[ 400, 332, 465, 396, 389, 943, 371, 388 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine the value of \$a\$ that satisfies the equation \$375_a + 592_a = 9C7_a\$ where \$C_a = 12_{10}\$, we need to convert each number from base \$a\$ to base 10, perform the addition in base 10, and then check if the result matches the base \$a\$ number converted to base 10.\n\nFirst, convert \\(...

[ "a = 4", "", "", "4", "4", "", "a = 4", "a = 4" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 748, 660, 3072, 3072, 3072 ]

[ "length", "length", "length", "stop", "stop", "length", "length", "length" ]

0

86f5eb9195558b6d1cb27f73c25be89e0678226caccca3613d1132355cee6697

Three. (50 points) Let $S=\{1,2, \cdots, 2016\}$. For any non-empty finite sets of real numbers $A$ and $B$, find $$ f=|A \Delta S|+|B \Delta S|+|C \Delta S| $$ the minimum value, where, $$ X \Delta Y=\{a \in X \mid a \notin Y\} \cup\{a \in Y \mid a \notin X\} $$ is the symmetric difference of sets $X$ and $Y$, and $...

2017

numina

olympiads

Combinatorics

null

integer

2017

unchanged

[ "Let's break down the problem step by step:\n\n1. **Understanding the Symmetric Difference**: The symmetric difference \$ A \\Delta S \$ of sets \$ A \$ and \$ S \$ consists of elements that are in either \$ A \$ or \$ S \$, but not in both.\n\n2. **Minimum Value Calculation**: We need to find the minimum...

[ "0", "0", "", "A", "3024", "", "f=2015", "4030" ]

[ false, false, false, false, false, false, false, false ]

[ 764, 588, 129, 358, 795, 618, 229, 862 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To find the minimum value of \$ f = |A \\Delta S| + |B \\Delta S| + |C \\Delta S| \$, we need to carefully analyze the properties of the sets \$ A \$, \$ B \$, and \$ C \$.\n\nFirst, let's consider the set \$ S = \\{1, 2, \\cdots, 2016\\} \$. The-even numbers in \$ S \$ are \\( \\{2, 4, \\ldots, 2016\\...

[ "", "", "", "4031", "", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 743, 3072, 3072, 3072, 3072 ]

[ "length", "length", "length", "stop", "length", "length", "length", "length" ]

0

d9c9858b0a0716d866628bacfd9216328da0e84e44e74411623b0a4ed35c284f

Let $n \geqslant 1$ be an integer. We have $n$ cubes with sides $1, 2, \cdots, n$. We want to stack them in a certain order, such that a cube of side $k$ can only be placed on a cube of side $\ell$ with $\ell \geqslant k-2$. How many such different stackings exist?

3^{n-1}\times2

numina

olympiads

Combinatorics

null

expression/text

3^{n-1}\times2

unchanged

[ "To solve the problem of finding the number of different stackings of \$ n \$ cubes, where each cube \$ 1, 2, \\ldots, n \$ can only be placed on a cube of side \$ \\ell \$ with \$ \\ell \\geq k - 2 \$, we need to determine how many valid sequences of cubes can be formed.\n\nThis problem is related to the c...

[ "F_{n+1}", "2", "F_{n+2}", "", "", "", "3^n", "F_{n+2}" ]

[ false, false, false, false, false, false, false, false ]

[ 719, 670, 611, 551, 591, 345, 417, 511 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To determine the number of different stackings of \$ n \$ cubes with sides \$ 1, 2, \\ldots, n \$ such that a cube of side \$ k \$ can only be placed on a cube of side \$ \\ell \$ with \$ \\ell \\geq aura-2 \$, we can use a combinatorial approach.\n\nFirst, let's define \$ a_n \$ as the number of valid...

[ "F_{n+2}", "", "a_n", "", "a_1 = 1", "", "", "" ]

[ false, false, false, false, false, false, false, false ]

[ 626, 3072, 3072, 3072, 3072, 3072, 3072, 3072 ]

[ "stop", "length", "length", "length", "length", "length", "length", "length" ]

0

bd4b9af525a0b4929e66e150f27f16da23a5a39e19d123254fd92abb25b16225

Find $\left|\left(1+i\right)^6\right|$.

8

math

intermediate_algebra

Algebra

null

integer

8

unchanged

[ "To find the magnitude of \$(1+i)^6\$, we can follow these steps:\n\n1. **Calculate the magnitude of \$1+i\$**: The magnitude of a complex number \$a+bi\$ is given by \$\\sqrt{a^2 + b^2}\$. For \$1+i\$, the magnitude is \$\\sqrt{1^2 + 1^2} = \\sqrt{2}\$.\n\n2. **Use the property of magnitudes**: When ra...

[ "8", "256", "\\left(1+i\\right)^{6}", "8", "64", "8", "8", "8" ]

[ true, false, false, true, false, true, true, true ]

[ 308, 519, 134, 129, 191, 536, 211, 360 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

5

null

1bf8898f7a29ddedcaf2de7d2fd77398346e3905153be81eecebb4c31289733b

For how many digits $C$ is the positive three-digit number $1C3$ a multiple of 3?

3

math

prealgebra

Algebra

null

integer

3

unchanged

[ "To solve this problem, let's first recall the rule for divisibility by 3: a number is divisible by 3 if the sum of its digits is divisible by 3. For the given number \$1C3\$, we need to find the digit \$C\$ such that when \$1 + C + 3 = C + 4\$ is divisible by 3, \$C\$ will be a valid option. Let's check th...

[ "3", "3", "001", "3", "3", "3", "3", "3" ]

[ true, true, false, true, true, true, true, true ]

[ 499, 524, 21, 599, 284, 463, 527, 540 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

7

null

5696eaaa05a75e59fd1a33d88b62a54fb9e43cfebb323d919953753e2f19cccd

1. If the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{2}, a_{n+1}=\frac{2 a_{n}}{3 a_{n}+2}, n \in \mathbf{N}_{+}$, then $a_{2017}=$ $\qquad$ .

\frac{1}{3026}

numina

olympiads

Algebra

null

fraction

\frac{1}{3026}

unchanged

[ "To solve this recurrence differrence, we will first try to find a pattern in the sequence by calculating the first few terms. Let's compute the first term:\n\n$a_1 = \\frac{1}{2}$\n\nFor $a_2$, we will use the given recurrence relation:\n\n$a_2 = \\frac{2 a_1}{3 a_1 + 2}$\n\nSubstituting $a_1$:\n\n$a_2 = \\frac{2 ...

[ "a_{2017}= \\frac{2}{6053}", "1", "\\frac{1}{2}", "\\frac{1}{4031}", "", "316.58", "\\frac{1}{3025}", "-\\frac{3^{2016}}{1 + 3^{2016}}" ]

[ false, false, false, false, false, false, false, false ]

[ 1112, 389, 420, 790, 54, 370, 590, 815 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

0

[ "To solve for \$a_{2017}\$ in the sequence defined by \$a_1 = \\frac{1}{2}\$ and \$a_{n+1} = \\frac{2a_n}{3a_n + 2}\$, we start by analyzing the behavior of the sequence. Let's compute the first few terms to identify any patterns.\n\nFirst, we calculate \$a_2\$:\n\\[a_2 = \\frac{2a_1}{3a_1 + 2} = \\frac{2 \...

[ "", "\\frac{1}{3025}", "", "", "", "\\frac{2}{12097}", "\\frac{1}{12097}", "" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 1055, 3072, 3072, 3072, 1023, 1018, 3072 ]

[ "length", "stop", "length", "length", "length", "stop", "stop", "length" ]

0

685562821650708d317e1be61541483e486cba47a1b45900cc8273eb429d7be8

A cone has a radius of 2 inches and a height of 8 inches. What is the radius of a sphere whose volume is twice that of this cone?

2.52

numina

synthetic_math

Geometry

null

decimal

2.52

unchanged

[ "Let's solve the problem step by step:\n\n1. First, we need to calculate the volume of the given cone. The formula for the volume of a cone is:\n volume = (1/3) * π * r² * h\n\n2. Given the radius (r) of the cone is 2 inches and height (h) is 8 inches, we can substitute these values into the formula:\n volume_c...

[ "3.42", "2.52", "4", "2.52", "\\sqrt[3]{16}", "3.95", "4.12", "\\sqrt[3]{16}" ]

[ false, true, false, true, false, false, false, false ]

[ 501, 474, 510, 591, 512, 139, 307, 538 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

2

[ "To determine the radius of a sphere whose volume is twice that of a given cone, we need to follow these steps:\n\n1. **Calculate the volume of the cone:**\n\n The formula for the volume \$ V \$ of a cone is:\n \\[\n V_{\\text{cone}} = \\frac{1}{3} \\pi r^2 h\n \\]\n where \$ r \$ is the radius and \\...

[ "", "", "", "", "2\\sqrt[3]{2}", "2\\sqrt[3]{2}", "2 \\sqrt[3]{2}", "2\\sqrt[3]{2}" ]

[ false, false, false, false, false, false, false, false ]

[ 3072, 3072, 3072, 3072, 472, 514, 602, 513 ]

[ "length", "length", "length", "length", "stop", "stop", "stop", "stop" ]

0

f88efff59e5c166685b249fcbc4b45a78e056218b7d9ca69263e6ce139e3fb0e

A box of 20 chocolate candies costs $\$8$. If there is a 10% discount on total cost for orders exceeding 400 candies, how much does it cost to buy 500 chocolate candies?

\$180

numina

synthetic_math

Algebra

null

integer

\$180

unchanged

[ "To buy 500 chocolate candies, you would normally spend 500 candies * $\\frac{\\$8} {20}$ candies per dollar = $\\$200$.\n\nBut since you are ordering more than 400 candies, you can get a 10% discount.\n\nSo you get a discount of 10% * $\\$200$ = 0.10 * $\\frac{\\$8} {20}$ candies per dollar * $\\$200 = \\frac{\\$8...

[ "", "", "180", "180", "180", "", "180", "" ]

[ false, false, true, true, true, false, true, false ]

[ 149, 242, 201, 280, 261, 273, 194, 253 ]

[ "stop", "stop", "stop", "stop", "stop", "stop", "stop", "stop" ]

4

null