Unnamed: 0
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40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
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float64 0
100
|
|---|---|---|---|
10,100 |
Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$ .
|
1660
| 0.78125 |
10,101 |
Given a box contains $4$ shiny pennies and $5$ dull pennies, determine the probability that the third shiny penny appears on the sixth draw.
|
\frac{5}{21}
| 15.625 |
10,102 |
In base 10, compute the result of \( (456_{10} + 123_{10}) - 579_{10} \). Express your answer in base 10.
|
0_{10}
| 0 |
10,103 |
How many ways can we put 4 math books and 6 English books on a shelf if all the math books must stay together, but the English books must be split into two groups of 3 each, with each group staying together?
|
5184
| 71.875 |
10,104 |
Given that the positive real numbers $x$ and $y$ satisfy the equation $x + 2y = 1$, find the minimum value of $\frac{y}{2x} + \frac{1}{y}$.
|
2 + \sqrt{2}
| 11.71875 |
10,105 |
Let the set $\mathbf{A}=\{1, 2, 3, 4, 5, 6\}$ and a bijection $f: \mathbf{A} \rightarrow \mathbf{A}$ satisfy the condition: for any $x \in \mathbf{A}$, $f(f(f(x)))=x$. Calculate the number of bijections $f$ satisfying the above condition.
|
81
| 32.8125 |
10,106 |
The sum of an infinite geometric series is $16$ times the series that results if the first two terms of the original series are removed. What is the value of the series' common ratio?
|
\frac{1}{4}
| 73.4375 |
10,107 |
What is the minimum value of the function \( y = \sin^4 x + \cos^4 x + \sec^4 x + \csc^4 x \)?
|
8.5
| 9.375 |
10,108 |
At Alpine School, there are 15 soccer players. Every player is enrolled in either physics or mathematics class, but not necessarily both. If 9 players are taking physics and 4 players are registered for both physics and mathematics, how many are taking mathematics?
|
10
| 99.21875 |
10,109 |
Let $S=\{1,2,3,...,12\}$ . How many subsets of $S$ , excluding the empty set, have an even sum but not an even product?
*Proposed by Gabriel Wu*
|
31
| 68.75 |
10,110 |
Given two distinct numbers \(a\) and \(b\) such that \(\frac{a}{b} + a = \frac{b}{a} + b\), find \(\frac{1}{a} + \frac{1}{b}\).
|
-1
| 85.9375 |
10,111 |
Two couriers start from two locations $A$ and $B$; the first heading towards $B$, and the second towards $A$. When they meet, the first courier has traveled 12 miles more than the second. If they continue their journeys at their original speeds, the first courier will reach their destination 9 days after the meeting, and the second courier will reach their destination 16 days after the meeting. How far apart are the two locations?
|
84
| 13.28125 |
10,112 |
Let the set \( M = \{1, 2, \cdots, 1000\} \). For any non-empty subset \( X \) of \( M \), let \( a_X \) represent the sum of the maximum and minimum numbers in \( X \). What is the arithmetic mean of all such \( a_X \)?
|
1001
| 96.875 |
10,113 |
A driver left point A and headed towards point D, which are 100 km apart. The road from A to D passes through points B and C. At point B, the navigator showed that there were 30 minutes left to drive, and the driver immediately reduced their speed by 10 km/h. At point C, the navigator indicated that there were 20 km left, and the driver again reduced their speed by the same 10 km/h. (The navigator determines the remaining time based on the current speed.) Determine the initial speed of the car, given that the driver spent 5 minutes more to travel from B to C than from C to D.
|
100
| 4.6875 |
10,114 |
When $3x^2$ was added to the quadratic polynomial $f(x)$, its minimum value increased by 9. When $x^2$ was subtracted from it, its minimum value decreased by 9. How will the minimum value of $f(x)$ change if $x^2$ is added to it?
|
\frac{9}{2}
| 14.0625 |
10,115 |
There is a set of natural numbers (it is known that there are at least seven numbers) such that the sum of every seven of them is less than 15, and the sum of all numbers in the set is 100. What is the smallest number of numbers that can be in the set?
|
50
| 19.53125 |
10,116 |
Given that \([x]\) represents the largest integer not exceeding \( x \), if \([x+0.1] + [x+0.2] + \ldots + [x+0.9] = 104\), what is the minimum value of \( x \)?
|
11.5
| 36.71875 |
10,117 |
Someone claims that the first five decimal digits of the square root of 51 are the same as the first five significant digits of the square root of 2. Verify this claim. Which rational approximating fraction can we derive from this observation for \(\sqrt{2}\)? How many significant digits does this fraction share with the actual value of \(\sqrt{2}\)? What is the connection between this approximating value and the sequence of numbers related to problem 1061 and earlier problems?
|
\frac{99}{70}
| 14.0625 |
10,118 |
If \(\alpha, \beta, \gamma\) are acute angles, and \(\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=1\), what is the maximum value of \(\frac{\sin \alpha+\sin \beta+\sin \gamma}{\cos \alpha+\cos \beta+\cos \gamma}\)?
|
\frac{\sqrt{2}}{2}
| 78.125 |
10,119 |
Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,200 \pi).$
|
200
| 70.3125 |
10,120 |
Points \( C_1 \), \( A_1 \), and \( B_1 \) are taken on the sides \( AB \), \( BC \), and \( AC \) of triangle \( ABC \) respectively, such that
\[
\frac{AC_1}{C_1B} = \frac{BA_1}{A_1C} = \frac{CB_1}{B_1A} = 2.
\]
Find the area of triangle \( A_1B_1C_1 \) if the area of triangle \( ABC \) is 1.
|
\frac{1}{3}
| 0 |
10,121 |
Find the maximum value of the area of triangle $\triangle ABC$ that satisfies $AB=4$ and $AC=2BC$.
|
\frac{16}{3}
| 10.9375 |
10,122 |
Given four functions: $①y=-x$, $②y=- \frac{1}{x}$, $③y=x^{3}$, $④y=x^{ \frac{1}{2}}$, determine the probability of the event "the graphs of the two selected functions have exactly one common point" when two functions are selected from these four.
|
\frac{1}{3}
| 71.09375 |
10,123 |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors, and their angle is $120^{\circ}$, calculate the magnitude of the vector $|\overrightarrow{a}-2\overrightarrow{b}|$.
|
\sqrt{7}
| 98.4375 |
10,124 |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \cos ^{8} x \, dx
$$
|
\frac{35 \pi}{8}
| 82.03125 |
10,125 |
Kolya traveled on an electric scooter to a store in a neighboring village at a speed of 10 km/h. After covering exactly one-third of the total distance, he realized that if he continued at the same speed, he would arrive exactly at the store's closing time. He then doubled his speed. However, after traveling exactly two-thirds of the total distance, the scooter broke down, and Kolya walked the remaining distance. At what speed did he walk if he arrived exactly at the store's closing time?
|
6.666666666666667
| 0 |
10,126 |
Determine the greatest common divisor of all nine-digit integers formed by repeating a three-digit integer three times. For example, 256,256,256 or 691,691,691 are integers of this form.
|
1001001
| 52.34375 |
10,127 |
Let $x$ be a real number between $0$ and $\tfrac{\pi}2$ such that \[\dfrac{\sin^4(x)}{42}+\dfrac{\cos^4(x)}{75} = \dfrac{1}{117}.\] Find $\tan(x)$ .
|
\frac{\sqrt{14}}{5}
| 15.625 |
10,128 |
It is known that \( x = 2a^{5} = 5b^{2} > 0 \), where \( a \) and \( b \) are integers. What is the smallest possible value of \( x \)?
|
200000
| 14.0625 |
10,129 |
Each square in an $8 \times 8$ grid is to be painted either white or black. The goal is to ensure that for any $2 \times 3$ or $3 \times 2$ rectangle selected from the grid, there are at least two adjacent squares that are black. What is the minimum number of squares that need to be painted black in the grid?
|
24
| 6.25 |
10,130 |
How many children did my mother have?
If you asked me this question, I would only tell you that my mother dreamed of having at least 19 children, but she couldn't make this dream come true; however, I had three times more sisters than cousins, and twice as many brothers as sisters. How many children did my mother have?
|
10
| 40.625 |
10,131 |
Point \( O \) is located inside an isosceles right triangle \( ABC \). The distance from \( O \) to vertex \( A \) (the right angle) is 6, to vertex \( B \) is 9, and to vertex \( C \) is 3. Find the area of triangle \( ABC \).
|
\frac{45}{2} + 9\sqrt{2}
| 1.5625 |
10,132 |
A regular decagon is given. A triangle is formed by connecting three randomly chosen vertices of the decagon. Calculate the probability that none of the sides of the triangle is a side of the decagon.
|
\frac{5}{12}
| 13.28125 |
10,133 |
Two concentric circles have radii of 15 meters and 30 meters. An aardvark starts at point $A$ on the smaller circle and runs along the path that includes half the circumference of each circle and each of the two straight segments that connect the circumferences directly (radial segments). Calculate the total distance the aardvark runs.
|
45\pi + 30
| 53.125 |
10,134 |
On a rectangular sheet of paper, a picture in the shape of a "cross" was drawn using two rectangles $ABCD$ and $EFGH$, with their sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, $FG=10$. Find the area of the quadrilateral $AFCH$.
|
52.5
| 73.4375 |
10,135 |
The graph of $y = ax^2 + bx + c$ is shown, where $a$, $b$, and $c$ are integers. The vertex of the parabola is at $(-2, 3)$, and the point $(1, 6)$ lies on the graph. Determine the value of $a$.
|
\frac{1}{3}
| 48.4375 |
10,136 |
A magician has $5432_{9}$ tricks for his magical performances. How many tricks are there in base 10?
|
3998
| 11.71875 |
10,137 |
Let \( n = 1990 \). Find the value of the following expression:
$$
\frac{1}{2^{n}}\left(1-3 \binom{n}{2} + 3^{2} \binom{n}{4} - 3^{3} \binom{n}{6} + \cdots + 3^{994} \binom{n}{1988} - 3^{995} \binom{n}{1990} \right)
$$
|
-\frac{1}{2}
| 89.0625 |
10,138 |
A rotating disc is divided into five equal sectors labeled $A$, $B$, $C$, $D$, and $E$. The probability of the marker stopping on sector $A$ is $\frac{1}{5}$, the probability of it stopping in $B$ is $\frac{1}{5}$, and the probability of it stopping in sector $C$ is equal to the probability of it stopping in sectors $D$ and $E$. What is the probability of the marker stopping in sector $C$? Express your answer as a common fraction.
|
\frac{1}{5}
| 84.375 |
10,139 |
On the coordinate plane, the points \(A(0, 2)\), \(B(1, 7)\), \(C(10, 7)\), and \(D(7, 1)\) are given. Find the area of the pentagon \(A B C D E\), where \(E\) is the intersection point of the lines \(A C\) and \(B D\).
|
36
| 68.75 |
10,140 |
Indicate the integer closest to the number: \(\sqrt{2012-\sqrt{2013 \cdot 2011}}+\sqrt{2010-\sqrt{2011 \cdot 2009}}+\ldots+\sqrt{2-\sqrt{3 \cdot 1}}\).
|
31
| 28.90625 |
10,141 |
A group of toddlers in a kindergarten collectively has 90 teeth. Any two toddlers together have no more than 9 teeth. What is the minimum number of toddlers that can be in the group?
|
23
| 49.21875 |
10,142 |
A box contains a total of 400 tickets that come in five colours: blue, green, red, yellow and orange. The ratio of blue to green to red tickets is $1: 2: 4$. The ratio of green to yellow to orange tickets is $1: 3: 6$. What is the smallest number of tickets that must be drawn to ensure that at least 50 tickets of one colour have been selected?
|
196
| 0 |
10,143 |
A book of one hundred pages has its pages numbered from 1 to 100. How many pages in this book have the digit 5 in their numbering? (Note: one sheet has two pages.)
(a) 13
(b) 14
(c) 15
(d) 16
(e) 17
|
15
| 5.46875 |
10,144 |
Given the point \( P \) inside the triangle \( \triangle ABC \), satisfying \( \overrightarrow{AP} = \frac{1}{3} \overrightarrow{AB} + \frac{1}{4} \overrightarrow{AC} \), let the areas of triangles \( \triangle PBC \), \( \triangle PCA \), and \( \triangle PAB \) be \( S_1 \), \( S_2 \), and \( S_3 \) respectively. Determine the ratio \( S_1 : S_2 : S_3 = \quad \).
|
5:4:3
| 46.09375 |
10,145 |
Point \( P \) is located on the side \( AB \) of the square \( ABCD \) such that \( AP: PB = 2:3 \). Point \( Q \) lies on the side \( BC \) of the square and divides it in the ratio \( BQ: QC = 3 \). Lines \( DP \) and \( AQ \) intersect at point \( E \). Find the ratio of lengths \( AE: EQ \).
|
4:9
| 50 |
10,146 |
The positive integers from 1 to 576 are written in a 24 by 24 grid so that the first row contains the numbers 1 to 24, the second row contains the numbers 25 to 48, and so on. An 8 by 8 square is drawn around 64 of these numbers. The sum of the numbers in the four corners of the 8 by 8 square is 1646. What is the number in the bottom right corner of this 8 by 8 square?
|
499
| 14.84375 |
10,147 |
A coloring of all plane points with coordinates belonging to the set $S=\{0,1,\ldots,99\}$ into red and white colors is said to be *critical* if for each $i,j\in S$ at least one of the four points $(i,j),(i + 1,j),(i,j + 1)$ and $(i + 1, j + 1)$ $(99 + 1\equiv0)$ is colored red. Find the maximal possible number of red points in a critical coloring which loses its property after recoloring of any red point into white.
|
5000
| 82.03125 |
10,148 |
According to the definition of the Richter scale, the relationship between the relative energy $E$ released by an earthquake and the earthquake magnitude $n$ is: $E=10^n$. What is the multiple of the relative energy released by a magnitude 9 earthquake compared to a magnitude 7 earthquake?
|
100
| 96.09375 |
10,149 |
12 Smurfs are seated around a round table. Each Smurf dislikes the 2 Smurfs next to them, but does not dislike the other 9 Smurfs. Papa Smurf wants to form a team of 5 Smurfs to rescue Smurfette, who was captured by Gargamel. The team must not include any Smurfs who dislike each other. How many ways are there to form such a team?
|
36
| 55.46875 |
10,150 |
The third chick received as much porridge as the first two chicks combined. The fourth chick received as much porridge as the second and third chicks combined. The fifth chick received as much porridge as the third and fourth chicks combined. The sixth chick received as much porridge as the fourth and fifth chicks combined. The seventh chick did not receive any porridge because it ran out. It is known that the fifth chick received 10 grams of porridge. How much porridge did the magpie cook?
|
40
| 61.71875 |
10,151 |
Given that the first four terms of a geometric sequence $\{a\_n\}$ have a sum of $S\_4=5$, and $4a\_1,\;\; \frac {3}{2}a\_2\;,\;a\_2$ form an arithmetic sequence.
(I) Find the general term formula for $\{a\_n\}$;
(II) Let $\{b\_n\}$ be an arithmetic sequence with first term $2$ and common difference $-a\_1$. Its first $n$ terms' sum is $T\_n$. Find the maximum positive integer $n$ that satisfies $T_{n-1} > 0$.
|
13
| 83.59375 |
10,152 |
For how many integers $n$ with $1 \le n \le 2016$ is the product
\[
\prod_{k=0}^{n-1} \left( \left( 2 + e^{4 \pi i k / n} \right)^n - 1 \right)
\]
equal to zero?
|
504
| 25 |
10,153 |
What is the area of the smallest square that can completely contain a circle with a radius of 5?
|
100
| 100 |
10,154 |
Determine the product of the real parts of the solutions to the equation \(2x^2 + 4x = 1 + i\).
|
\frac{1 - 3}{4}
| 0 |
10,155 |
Find the last three digits of \(1 \times 3 \times 5 \times \cdots \times 1997\).
|
375
| 93.75 |
10,156 |
Find the coefficient of the $x^4$ term in the expansion of the binomial $(4x^{2}-2x+1)(2x+1)^{5}$.
|
80
| 52.34375 |
10,157 |
Let $a_n$ be the integer closest to $\sqrt{n}$. Find the sum
$$
\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{1980}}.
$$
|
88
| 74.21875 |
10,158 |
What is $(a^3+b^3)\div(a^2-ab+b^2+c)$ for $a=7$, $b=6$, and $c=1$?
|
\frac{559}{44}
| 22.65625 |
10,159 |
If the arithmetic sequence $\{a_n\}$ satisfies $a_{17} + a_{18} + a_{19} > 0$ and $a_{17} + a_{20} < 0$, then the sum of the first $n$ terms of $\{a_n\}$ is maximized when $n =$ ______.
|
18
| 57.8125 |
10,160 |
Simplify first and then evaluate:<br />(1) $4(m+1)^2 - (2m+5)(2m-5)$, where $m=-3$.<br />(2) Simplify: $\frac{x^2-1}{x^2+2x} \div \frac{x-1}{x}$, where $x=2$.
|
\frac{3}{4}
| 50.78125 |
10,161 |
If $x, y, z \in \mathbb{R}$ are solutions to the system of equations $$ \begin{cases}
x - y + z - 1 = 0
xy + 2z^2 - 6z + 1 = 0
\end{cases} $$ what is the greatest value of $(x - 1)^2 + (y + 1)^2$ ?
|
11
| 6.25 |
10,162 |
A total area of \( 2500 \, \mathrm{m}^2 \) will be used to build identical houses. The construction cost for a house with an area \( a \, \mathrm{m}^2 \) is the sum of the material cost \( 100 p_{1} a^{\frac{3}{2}} \) yuan, labor cost \( 100 p_{2} a \) yuan, and other costs \( 100 p_{3} a^{\frac{1}{2}} \) yuan, where \( p_{1} \), \( p_{2} \), and \( p_{3} \) are consecutive terms of a geometric sequence. The sum of these terms is 21 and their product is 64. Given that building 63 of these houses would result in the material cost being less than the sum of the labor cost and the other costs, find the maximum number of houses that can be built to minimize the total construction cost.
|
156
| 7.8125 |
10,163 |
Let the real numbers \(a_1, a_2, \cdots, a_{100}\) satisfy the following conditions: (i) \(a_1 \geq a_2 \geq \cdots \geq a_{100} \geq 0\); (ii) \(a_1 + a_2 \leq 100\); (iii) \(a_3 + a_4 + \cdots + a_{100} \leq 100\). Find the maximum value of \(a_1^2 + a_2^2 + \cdots + a_{100}^2\) and the values of \(a_1, a_2, \cdots, a_{100}\) that achieve this maximum.
|
10000
| 64.84375 |
10,164 |
Given triangle \( \triangle ABC \), point \( P \) is an internal point such that \( \angle PBC = \angle PCB = 24^\circ \). If \( \angle ABP = 30^\circ \) and \( \angle ACP = 54^\circ \), find the measure of \( \angle BAP \).
|
18
| 32.8125 |
10,165 |
In the dihedral angle $\alpha - E F - \beta $, $AE \subset \alpha, BF \subset \beta$, and $AE \perp EF, BF \perp EF$. Given $EF = 1$, $AE = 2$, and $AB = \sqrt{2}$, find the maximum volume of the tetrahedron $ABEF$.
|
\frac{1}{3}
| 34.375 |
10,166 |
Calculate the line integral
$$
\int_{L} \frac{y}{3} d x - 3 x d y + x d z
$$
along the curve \( L \), which is given parametrically by
$$
\begin{cases}
x = 2 \cos t \\
y = 2 \sin t \\
z = 1 - 2 \cos t - 2 \sin t
\end{cases}
\quad \text{for} \quad 0 \leq t \leq \frac{\pi}{2}
$$
|
2 - \frac{13\pi}{3}
| 24.21875 |
10,167 |
Given the function $f(x)=\sin \omega x+\cos \left(\omega x+\dfrac{\pi }{6}\right)$, where $x\in R$, $\omega >0$.
(1) When $\omega =1$, find the value of $f\left(\dfrac{\pi }{3}\right)$;
(2) When the smallest positive period of $f(x)$ is $\pi $, find the value of $x$ when $f(x)$ reaches the maximum value in $\left[0,\dfrac{\pi }{4}\right]$.
|
\dfrac{\pi }{12}
| 88.28125 |
10,168 |
A triangle with vertices at \((1003,0), (1004,3),\) and \((1005,1)\) in the \(xy\)-plane is revolved all the way around the \(y\)-axis. Find the volume of the solid thus obtained.
|
5020 \pi
| 0 |
10,169 |
Each of the letters in "GEOMETRY" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "RHYME"? Express your answer as a common fraction.
|
\frac{1}{2}
| 38.28125 |
10,170 |
If the graph of the function $f(x)=a^{x-2}-2a (a > 0, a \neq 1)$ always passes through the fixed point $\left(x\_0, \frac{1}{3}\right)$, then the minimum value of the function $f(x)$ on $[0,3]$ is equal to \_\_\_\_\_\_\_\_.
|
-\frac{1}{3}
| 80.46875 |
10,171 |
The circumference of the axial cross-section of a cylinder is $90 \text{ cm}$. What is the maximum possible volume of the cylinder?
|
3375\pi
| 7.03125 |
10,172 |
1. \(\lim _{x \rightarrow 1}(1-x) \operatorname{tg} \frac{\pi x}{2}\)
2. \(\lim _{x \rightarrow \frac{\pi}{4}}\left(\frac{\pi}{4}-x\right) \operatorname{cosec}\left(\frac{3}{4} \pi+x\right)\)
3. \(\lim _{x \rightarrow+\infty} x \operatorname{arcctg} x\)
4. \(\lim _{x \rightarrow-\infty} x\left(\frac{\pi}{2}+\operatorname{arctg} x\right)\)
|
-1
| 60.15625 |
10,173 |
Andreas, Boyu, Callista, and Diane each randomly choose an integer from 1 to 9, inclusive. Each of their choices is independent of the other integers chosen and the same integer can be chosen by more than one person. The probability that the sum of their four integers is even is equal to \(\frac{N}{6561}\) for some positive integer \(N\). What is the sum of the squares of the digits of \(N\) ?
|
78
| 46.875 |
10,174 |
How many different positive three-digit integers can be formed using only the digits in the set $\{3, 3, 4, 4, 4, 7, 8\}$ if no digit may be used more times than it appears in the given set of available digits?
|
43
| 63.28125 |
10,175 |
What is one-third times one-half times three-fourths times five-sixths?
|
\frac{5}{48}
| 98.4375 |
10,176 |
Let \( f \) be a function such that \( f(0) = 1 \), \( f'(0) = 2 \), and
\[ f''(t) = 4 f'(t) - 3 f(t) + 1 \]
for all \( t \). Compute the 4th derivative of \( f \), evaluated at 0.
|
54
| 96.09375 |
10,177 |
If 6 students want to sign up for 4 clubs, where students A and B do not join the same club, and every club must have at least one member with each student only joining one club, calculate the total number of different registration schemes.
|
1320
| 33.59375 |
10,178 |
A convex polyhedron is bounded by 4 regular hexagonal faces and 4 regular triangular faces. At each vertex of the polyhedron, 2 hexagons and 1 triangle meet. What is the volume of the polyhedron if the length of its edges is one unit?
|
\frac{23\sqrt{2}}{12}
| 42.96875 |
10,179 |
Observe the following set of equations:
$S_{1}=1$,
$S_{2}=2+3+4=9$,
$S_{3}=3+4+5+6+7=25$,
$S_{4}=4+5+6+7+8+9+10=49$,
...
Based on the equations above, guess that $S_{2n-1}=(4n-3)(an+b)$, then $a^{2}+b^{2}=$_______.
|
25
| 69.53125 |
10,180 |
A new pyramid is added on one of the pentagonal faces of a pentagonal prism. Calculate the total number of exterior faces, vertices, and edges of the composite shape formed by the fusion of the pentagonal prism and the pyramid. What is the maximum value of this sum?
|
42
| 21.09375 |
10,181 |
In triangle $PQR,$ $PQ = 4,$ $PR = 9,$ $QR = 10,$ and a point $S$ lies on $\overline{QR}$ such that $\overline{PS}$ bisects $\angle QPR.$ Find $\cos \angle QPS.$
|
\sqrt{\frac{23}{48}}
| 0 |
10,182 |
How many 10-digit numbers exist in which at least two digits are the same?
|
9 \times 10^9 - 9 \times 9!
| 0 |
10,183 |
Find the sum of $231_5 + 414_5 + 123_5$. Express your answer in base $5$.
|
1323_5
| 96.09375 |
10,184 |
Given that non-negative real numbers \( x \) and \( y \) satisfy
\[
x^{2}+4y^{2}+4xy+4x^{2}y^{2}=32,
\]
find the maximum value of \( \sqrt{7}(x+2y)+2xy \).
|
16
| 73.4375 |
10,185 |
Consider the polynomial $49x^3 - 105x^2 + 63x - 10 = 0$ whose roots are in arithmetic progression. Determine the difference between the largest and smallest roots.
A) $\frac{2}{7}$
B) $\frac{1}{7}$
C) $\frac{3\sqrt{11}}{7}$
D) $\frac{2\sqrt{11}}{7}$
E) $\frac{4\sqrt{11}}{7}$
|
\frac{2\sqrt{11}}{7}
| 35.15625 |
10,186 |
An ethnographer determined that in a primitive tribe he studied, the distribution of lifespan among tribe members can be described as follows: 25% live only up to 40 years, 50% die at 50 years, and 25% live to 60 years. He then randomly selected two individuals to study in more detail. What is the expected lifespan of the one among the two randomly chosen individuals who will live longer?
|
53.75
| 21.875 |
10,187 |
If $\sqrt{8 + x} + \sqrt{15 - x} = 6$, what is the value of $(8 + x)(15 - x)$?
|
\frac{169}{4}
| 61.71875 |
10,188 |
A hare is jumping in one direction on a strip divided into cells. In one jump, it can move either one cell or two cells. How many ways can the hare get from the 1st cell to the 12th cell?
|
144
| 3.90625 |
10,189 |
Find the smallest number in which all digits are different and the sum of all digits equals 32.
|
26789
| 17.1875 |
10,190 |
What is $1254_6 - 432_6 + 221_6$? Express your answer in base $6$.
|
1043_6
| 60.15625 |
10,191 |
Given a sequence of positive integers $a_1, a_2, a_3, \ldots, a_{100}$, where the number of terms equal to $i$ is $k_i$ ($i=1, 2, 3, \ldots$), let $b_j = k_1 + k_2 + \ldots + k_j$ ($j=1, 2, 3, \ldots$),
define $g(m) = b_1 + b_2 + \ldots + b_m - 100m$ ($m=1, 2, 3, \ldots$).
(I) Given $k_1 = 40, k_2 = 30, k_3 = 20, k_4 = 10, k_5 = \ldots = k_{100} = 0$, calculate $g(1), g(2), g(3), g(4)$;
(II) If the maximum term in $a_1, a_2, a_3, \ldots, a_{100}$ is 50, compare the values of $g(m)$ and $g(m+1)$;
(III) If $a_1 + a_2 + \ldots + a_{100} = 200$, find the minimum value of the function $g(m)$.
|
-100
| 14.0625 |
10,192 |
Calculate the number of distinct three-digit numbers formed using the digits 0, 1, 2, 3, 4, and 5 without repetition that are divisible by 9.
|
16
| 79.6875 |
10,193 |
Given $\alpha \in \left(0, \frac{\pi}{2}\right)$, $\beta \in \left(\frac{\pi}{2}, \pi\right)$, $\cos\beta = -\frac{1}{3}$, $\sin(\alpha + \beta) = \frac{7}{9}$.
(1) Find the value of $\tan \frac{\beta}{2}$.
(2) Find the value of $\sin\alpha$.
|
\frac{1}{3}
| 26.5625 |
10,194 |
Let $N$ be the number of ways of distributing $8$ chocolates of different brands among $3$ children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of $N$ .
|
24
| 35.9375 |
10,195 |
What is the smallest five-digit number divisible by 4 that can be formed with the digits 1, 2, 3, 4, and 9?
|
13492
| 94.53125 |
10,196 |
Given a pyramid \( S-ABCD \) with a square base where each side measures 2, and \( SD \perp \) plane \( ABCD \) and \( SD = AB \). Determine the surface area of the circumscribed sphere of the pyramid \( S-ABCD \).
|
12\pi
| 62.5 |
10,197 |
What will be the length of the strip if a cubic kilometer is cut into cubic meters and laid out in a single line?
|
1000000
| 11.71875 |
10,198 |
A parallelogram has its diagonals making an angle of \(60^{\circ}\) with each other. If two of its sides have lengths 6 and 8, find the area of the parallelogram.
|
14\sqrt{3}
| 15.625 |
10,199 |
In how many ways can four married couples sit around a circular table such that no man sits next to his wife?
|
1488
| 57.8125 |
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