Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
8,500 |
A chord intercepted on the circle $x^{2}+y^{2}=4$ by the line $\begin{cases} x=2-\frac{1}{2}t \\ y=-1+\frac{1}{2}t \end{cases} (t\text{ is the parameter})$ has a length of $\_\_\_\_\_\_\_.$
|
\sqrt{14}
| 70.3125 |
8,501 |
In a triangle, the lengths of the three sides are integers \( l, m, n \), with \( l > m > n \). It is known that \( \left\{ \frac{3^{l}}{10^{4}} \right\} = \left\{ \frac{3^{m}}{10^{4}} \right\} = \left\{ \frac{3^{n}}{10^{4}} \right\} \), where \( \{x\} \) denotes the fractional part of \( x \) and \( [x] \) denotes the greatest integer less than or equal to \( x \). Determine the smallest possible value of the perimeter of such a triangle.
|
3003
| 1.5625 |
8,502 |
Group the set of positive odd numbers {1, 3, 5, ...} from smallest to largest, where the $n$-th group contains $2n-1$ odd numbers. That is, the first group, the second group, the third group... consist of the sets {1}, {3, 5, 7}, {9, 11, 13, 15, 17}, ..., respectively. In which group does 2007 belong?
|
32
| 57.03125 |
8,503 |
During a break between voyages, a sailor turned 20 years old. All six crew members gathered in the cabin to celebrate. "I am twice the age of the cabin boy and 6 years older than the engineer," said the helmsman. "And I am as much older than the cabin boy as I am younger than the engineer," noted the boatswain. "In addition, I am 4 years older than the sailor." "The average age of the crew is 28 years," reported the captain. How old is the captain?
|
40
| 39.84375 |
8,504 |
We define \( a \star b = a \times a - b \times b \). Find the value of \( 3 \star 2 + 4 \star 3 + 5 \star 4 + \cdots + 20 \star 19 \).
|
396
| 92.1875 |
8,505 |
A two-meter gas pipe has rusted in two places. Determine the probability that all three resulting pieces can be used as connections to gas stoves, given that according to regulations, a stove should not be located closer than 50 cm to the main gas pipe.
|
1/16
| 9.375 |
8,506 |
Consider a 5x5 grid of squares. How many different squares can be traced using the lines in this grid?
|
55
| 44.53125 |
8,507 |
On the Island of Misfortune, there are knights, who always tell the truth, and liars, who always lie. One day, $n$ islanders gathered in a room.
The first person said: "Exactly 1 percent of the people present in this room are liars."
The second person said: "Exactly 2 percent of the people present in this room are liars."
and so on.
The person with number $n$ said: "Exactly $n$ percent of the people present in this room are liars."
How many people could be in the room, given that it is known that at least one of them is a knight?
|
100
| 50.78125 |
8,508 |
Acme Corporation has released a new version of its vowel soup where each vowel (A, E, I, O, U) appears six times, and additionally, each bowl contains one wildcard character that can represent any vowel. How many six-letter "words" can be formed from a bowl of this new Acme Enhanced Vowel Soup?
|
46656
| 22.65625 |
8,509 |
Let $a,b,c$ be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum $a + b + c$ . Find this sum.
|
55
| 61.71875 |
8,510 |
For the cubic function $f(x)=ax^3+bx^2+cx+d$ ($a\neq 0$), define: Let $f''(x)$ be the derivative of the derivative of the function $y=f(x)$, that is, the second derivative of $f(x)$. If the equation $f''(x)=0$ has a real solution $x_0$, then the point $(x_0, f(x_0))$ is called the "inflection point" of the function $y=f(x)$. Some students found that "every cubic function has an 'inflection point'; every cubic function has a center of symmetry; and the 'inflection point' is the center of symmetry." Based on this discovery, for the function $$f(x)=x^3- \frac{3}{2}x^2+3x- \frac{1}{4},$$ its center of symmetry is ___________; calculate $$f\left( \frac{1}{2013}\right)+f\left( \frac{2}{2013}\right)+f\left( \frac{3}{2013}\right)+\cdots +f\left( \frac{2012}{2013}\right)$$ = ___________.
|
2012
| 83.59375 |
8,511 |
The price of an item is decreased by 20%. To bring it back to its original value and then increase it by an additional 10%, the price after restoration must be increased by what percentage.
|
37.5\%
| 7.03125 |
8,512 |
Célia wants to trade with Guilherme stickers from an album about Brazilian animals. Célia wants to trade four butterfly stickers, five shark stickers, three snake stickers, six parakeet stickers, and six monkey stickers. All of Guilherme's stickers are of spiders. They know that:
(a) one butterfly sticker is worth three shark stickers;
(b) one snake sticker is worth three parakeet stickers;
(c) one monkey sticker is worth four spider stickers;
(d) one parakeet sticker is worth three spider stickers;
(e) one shark sticker is worth two parakeet stickers.
How many stickers can Célia receive if she trades all she wants?
|
171
| 64.0625 |
8,513 |
A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. Additionally, for this four-digit number, one prime factor minus five times another prime factor is equal to two times the third prime factor. What is this number?
|
1221
| 9.375 |
8,514 |
A "fifty percent mirror" is a mirror that reflects half the light shined on it back and passes the other half of the light onward. Two "fifty percent mirrors" are placed side by side in parallel, and a light is shined from the left of the two mirrors. How much of the light is reflected back to the left of the two mirrors?
|
\frac{2}{3}
| 2.34375 |
8,515 |
Find the smallest constant $C$ such that for all real numbers $x, y, z$ satisfying $x + y + z = -1$, the following inequality holds:
$$
\left|x^3 + y^3 + z^3 + 1\right| \leqslant C \left|x^5 + y^5 + z^5 + 1\right|.
$$
|
\frac{9}{10}
| 39.84375 |
8,516 |
In a trapezoid $ABCD$ with $\angle A = \angle B = 90^{\circ}$, $|AB| = 5 \text{cm}$, $|BC| = 1 \text{cm}$, and $|AD| = 4 \text{cm}$, point $M$ is taken on side $AB$ such that $2 \angle BMC = \angle AMD$. Find the ratio $|AM| : |BM|$.
|
3/2
| 3.125 |
8,517 |
For any positive integer \( n \), let \( f(n) \) represent the last digit of \( 1 + 2 + 3 + \cdots + n \). For example, \( f(1) = 1 \), \( f(2) = 3 \), \( f(5) = 5 \), and so on. Find the value of \( f(2) + f(4) + f(6) + \cdots + f(2012) \).
|
3523
| 14.84375 |
8,518 |
When $8000^{50}$ is expanded out, the result is $1$ followed by how many zeros?
|
150
| 76.5625 |
8,519 |
A right triangle has legs of lengths 3 and 4. Find the volume of the solid formed by revolving the triangle about its hypotenuse.
|
\frac{48\pi}{5}
| 26.5625 |
8,520 |
The diagram shows a regular dodecagon and a square, whose vertices are also vertices of the dodecagon. What is the value of the ratio of the area of the square to the area of the dodecagon?
|
2:3
| 0 |
8,521 |
A right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$ , and the distance from the vertex of the cone to any point on the circumference of the base is $3$ , then the minimum possible volume of the cone can be written as $\frac{m\pi\sqrt{n}}{p}$ , where $m$ , $n$ , and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is squarefree. Find $m + n + p$ .
|
60
| 78.125 |
8,522 |
Given that the right focus of the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ is $F(\sqrt{6},0)$, a line $l$ passing through $F$ intersects the ellipse at points $A$ and $B$. If the midpoint of chord $AB$ has coordinates $(\frac{\sqrt{6}}{3},-1)$, calculate the area of the ellipse.
|
12\sqrt{3}\pi
| 32.8125 |
8,523 |
We are given a triangle $ABC$ . Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$ , with the arrangment of points $D - A - B - E$ . The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$ , and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$ . Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$ .
|
60
| 92.96875 |
8,524 |
Calculate the limit of the function:
\[
\lim_{x \rightarrow 0} \frac{\arcsin(2x)}{2^{-3x} - 1} \cdot \ln 2
\]
|
-\frac{2}{3}
| 97.65625 |
8,525 |
Given that the angle between the unit vectors $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ is $60^{\circ}$, and $\overrightarrow{a}=2\overrightarrow{e_{1}}-\overrightarrow{e_{2}}$, find the projection of $\overrightarrow{a}$ in the direction of $\overrightarrow{e_{1}}$.
|
\dfrac{3}{2}
| 25 |
8,526 |
It is known that
\[
\begin{array}{l}
1 = 1^2 \\
1 + 3 = 2^2 \\
1 + 3 + 5 = 3^2 \\
1 + 3 + 5 + 7 = 4^2
\end{array}
\]
If \(1 + 3 + 5 + \ldots + n = 20^2\), find \(n\).
|
39
| 100 |
8,527 |
If the real numbers \(x\) and \(y\) satisfy the equation \((x-2)^{2}+(y-1)^{2}=1\), then the minimum value of \(x^{2}+y^{2}\) is \_\_\_\_\_.
|
6-2\sqrt{5}
| 96.875 |
8,528 |
For the subset \( S \) of the set \(\{1,2, \cdots, 15\}\), if a positive integer \( n \) and \( n+|S| \) are both elements of \( S \), then \( n \) is called a "good number" of \( S \). If a subset \( S \) has at least one "good number", then \( S \) is called a "good set". Suppose 7 is a "good number" of a "good set" \( X \). How many such subsets \( X \) are there?
|
4096
| 25.78125 |
8,529 |
Let the sequence of non-negative integers $\left\{a_{n}\right\}$ satisfy:
$$
a_{n} \leqslant n \quad (n \geqslant 1), \quad \text{and} \quad \sum_{k=1}^{n-1} \cos \frac{\pi a_{k}}{n} = 0 \quad (n \geqslant 2).
$$
Find all possible values of $a_{2021}$.
|
2021
| 39.0625 |
8,530 |
If the integer part of $\sqrt{10}$ is $a$ and the decimal part is $b$, then $a=$______, $b=\_\_\_\_\_\_$.
|
\sqrt{10} - 3
| 3.125 |
8,531 |
A group of dancers are arranged in a rectangular formation. When they are arranged in 12 rows, there are 5 positions unoccupied in the formation. When they are arranged in 10 rows, there are 5 positions unoccupied. How many dancers are in the group if the total number is between 200 and 300?
|
295
| 44.53125 |
8,532 |
In $\triangle ABC$, it is known that $\cos A= \frac{1}{7}$, $\cos (A-B)= \frac{13}{14}$, and $0 < B < A < \frac{\pi}{2}$. Find the measure of angle $B$.
|
\frac{\pi}{3}
| 50.78125 |
8,533 |
The function \( y = \cos x + \sin x + \cos x \sin x \) has a maximum value of \(\quad\).
|
\frac{1}{2} + \sqrt{2}
| 60.15625 |
8,534 |
If \(100^a = 4\) and \(100^b = 5\), then find \(20^{(1 - a - b)/(2(1 - b))}\).
|
\sqrt{5}
| 32.8125 |
8,535 |
Determine the appropriate value of $h$ so that the following equation in base $h$ is accurate:
$$\begin{array}{c@{}c@{}c@{}c@{}c@{}c}
&&8&6&7&4_h \\
&+&4&3&2&9_h \\
\cline{2-6}
&1&3&0&0&3_h.
\end{array}$$
|
10
| 53.90625 |
8,536 |
On a rectangular sheet of paper, a picture in the shape of a "cross" is drawn from two rectangles $ABCD$ and $EFGH$, with sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$.
|
52.5
| 64.84375 |
8,537 |
What is the area of the region defined by the equation $x^2 + y^2 - 3 = 6y - 18x + 9$?
|
102\pi
| 92.96875 |
8,538 |
When a granary records the arrival of 30 tons of grain as "+30", determine the meaning of "-30".
|
-30
| 26.5625 |
8,539 |
Given the function $f(x)=\frac{1}{3}x^3-ax^2+(a^2-1)x+b$, where $(a,b \in \mathbb{R})$
(I) If $x=1$ is an extreme point of $f(x)$, find the value of $a$;
(II) If the equation of the tangent line to the graph of $y=f(x)$ at the point $(1, f(1))$ is $x+y-3=0$, find the maximum and minimum values of $f(x)$ on the interval $[-2, 4]$.
|
-4
| 27.34375 |
8,540 |
The distance from the point of intersection of a circle's diameter with a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius.
$$
AB = 18, EO = 7, AE = 2BE, R = ?
$$
|
11
| 3.125 |
8,541 |
If digits $A$ , $B$ , and $C$ (between $0$ and $9$ inclusive) satisfy
\begin{tabular}{c@{\,}c@{\,}c@{\,}c}
& $C$ & $C$ & $A$
+ & $B$ & $2$ & $B$ \hline
& $A$ & $8$ & $8$
\end{tabular}
what is $A \cdot B \cdot C$ ?
*2021 CCA Math Bonanza Individual Round #5*
|
42
| 87.5 |
8,542 |
For each value of $x,$ $g(x)$ is defined to be the minimum value of the three numbers $3x + 3,$ $\frac{1}{3} x + 2,$ and $-\frac{1}{2} x + 8.$ Find the maximum value of $g(x).$
|
\frac{22}{5}
| 37.5 |
8,543 |
Find the four-digit number that is a perfect square, where the thousands digit is the same as the tens digit, and the hundreds digit is 1 greater than the units digit.
|
8281
| 87.5 |
8,544 |
Count all the distinct anagrams of the word "YOANN".
|
60
| 11.71875 |
8,545 |
Given a finite sequence $\{a\_1\}, \{a\_2\}, \ldots \{a\_m\} (m \in \mathbb{Z}^+)$ that satisfies the conditions: $\{a\_1\} = \{a\_m\}, \{a\_2\} = \{a\_{m-1}\}, \ldots \{a\_m\} = \{a\_1\}$, it is called a "symmetric sequence" with the additional property that in a $21$-term "symmetric sequence" $\{c\_n\}$, the terms $\{c\_{11}\}, \{c\_{12}\}, \ldots, \{c\_{21}\}$ form an arithmetic sequence with first term $1$ and common difference $2$. Find the value of $\{c\_2\}$.
|
19
| 60.15625 |
8,546 |
Given that the terminal side of the angle $α+ \frac {π}{6}$ passes through point P($-1$, $-2\sqrt {2}$), find the value of $\sinα$.
|
\frac{1-2\sqrt {6}}{6}
| 0 |
8,547 |
In which numeral system is 792 divisible by 297?
|
19
| 71.09375 |
8,548 |
Can you use the four basic arithmetic operations (addition, subtraction, multiplication, division) and parentheses to write the number 2016 using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 in sequence?
|
2016
| 0 |
8,549 |
Two square napkins with dimensions \(1 \times 1\) and \(2 \times 2\) are placed on a table so that the corner of the larger napkin falls into the center of the smaller napkin. What is the maximum area of the table that the napkins can cover?
|
4.75
| 18.75 |
8,550 |
Find the sine of the angle at the vertex of an isosceles triangle, given that the perimeter of any inscribed rectangle, with two vertices lying on the base, is a constant value.
|
\frac{4}{5}
| 6.25 |
8,551 |
Consider a "Modulo $m$ graph paper," with $m = 17$, forming a grid of $17^2$ points. Each point $(x, y)$ represents integer pairs where $0 \leq x, y < 17$. We graph the congruence $$5x \equiv 3y + 2 \pmod{17}.$$ The graph has single $x$-intercept $(x_0, 0)$ and single $y$-intercept $(0, y_0)$, where $0 \leq x_0, y_0 < 17$. Determine $x_0 + y_0$.
|
19
| 4.6875 |
8,552 |
In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, and each head always tells the truth or always lies. Additionally, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:
- 1st head: "To my left is a green dragon."
- 2nd head: "To my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."
What is the maximum number of red dragons that could have been at the table?
|
176
| 68.75 |
8,553 |
For how many positive integers $n$ less than or equal to 500 is $$(\cos t - i\sin t)^n = \cos nt - i\sin nt$$ true for all real $t$?
|
500
| 84.375 |
8,554 |
A stationery store sells a certain type of pen bag for $18$ yuan each. Xiao Hua went to buy this pen bag. When checking out, the clerk said, "If you buy one more, you can get a 10% discount, which is $36 cheaper than now." Xiao Hua said, "Then I'll buy one more, thank you." According to the conversation between the two, Xiao Hua actually paid ____ yuan at checkout.
|
486
| 82.8125 |
8,555 |
Let \(P\) and \(P+2\) be both prime numbers satisfying \(P(P+2) \leq 2007\). If \(S\) represents the sum of such possible values of \(P\), find the value of \(S\).
|
106
| 87.5 |
8,556 |
Arrange positive integers that are neither perfect squares nor perfect cubes (excluding 0) in ascending order as 2, 3, 5, 6, 7, 10, ..., and determine the 1000th number in this sequence.
|
1039
| 72.65625 |
8,557 |
In the Cartesian coordinate system, with the origin O as the pole and the positive x-axis as the polar axis, a polar coordinate system is established. The polar coordinate of point P is $(1, \pi)$. Given the curve $C: \rho=2\sqrt{2}a\sin(\theta+ \frac{\pi}{4}) (a>0)$, and a line $l$ passes through point P, whose parametric equation is:
$$
\begin{cases}
x=m+ \frac{1}{2}t \\
y= \frac{\sqrt{3}}{2}t
\end{cases}
$$
($t$ is the parameter), and the line $l$ intersects the curve $C$ at points M and N.
(1) Write the Cartesian coordinate equation of curve $C$ and the general equation of line $l$;
(2) If $|PM|+|PN|=5$, find the value of $a$.
|
2\sqrt{3}-2
| 0 |
8,558 |
Find the smallest integer \( n \) such that the expanded form of \( (xy - 7x - 3y + 21)^n \) has 2012 terms.
|
44
| 78.90625 |
8,559 |
How many integers from 1 to 1997 have a sum of digits that is divisible by 5?
|
399
| 69.53125 |
8,560 |
Given a positive geometric sequence $\{a_{n}\}$, if ${a_m}{a_n}=a_3^2$, find the minimum value of $\frac{2}{m}+\frac{1}{{2n}}$.
|
\frac{3}{4}
| 53.90625 |
8,561 |
Find the value of $x$ if $\log_8 x = 1.75$.
|
32\sqrt[4]{2}
| 0.78125 |
8,562 |
Niall's four children have different integer ages under 18. The product of their ages is 882. What is the sum of their ages?
|
31
| 1.5625 |
8,563 |
Find all three-digit numbers $\overline{\Pi B \Gamma}$, consisting of distinct digits $\Pi, B$, and $\Gamma$, for which the following equality holds: $\overline{\Pi B \Gamma} = (\Pi + B + \Gamma) \times (\Pi + B + \Gamma + 1)$.
|
156
| 95.3125 |
8,564 |
Using $1 \times 2$ tiles to cover a $2 \times 10$ grid, how many different ways are there to cover the grid?
|
89
| 75.78125 |
8,565 |
Given vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy $|\overrightarrow{a} - 2\overrightarrow{b}| \leqslant 2$, find the minimum value of $\overrightarrow{a} \cdot \overrightarrow{b}$.
|
-\frac{1}{2}
| 7.03125 |
8,566 |
Suppose that \(a, b, c,\) and \(d\) are positive integers which are not necessarily distinct. If \(a^{2}+b^{2}+c^{2}+d^{2}=70\), what is the largest possible value of \(a+b+c+d?\)
|
16
| 88.28125 |
8,567 |
We write the equation on the board:
$$
(x-1)(x-2) \ldots(x-2016) = (x-1)(x-2) \ldots(x-2016)
$$
We want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimum number of factors that must be erased to achieve this?
|
2016
| 66.40625 |
8,568 |
Let the sequence $x, 3x+3, 5x+5, \dots$ be in geometric progression. What is the fourth term of this sequence?
|
-\frac{125}{12}
| 67.1875 |
8,569 |
Evaluate the greatest integer less than or equal to \[\frac{5^{150} + 3^{150}}{5^{147} + 3^{147}}.\]
|
124
| 6.25 |
8,570 |
How many different right-angled triangles exist, one of the legs of which is \(\sqrt{2016}\), and the other leg and hypotenuse are expressed in natural numbers?
|
12
| 72.65625 |
8,571 |
Given the operation defined as \(a \odot b \odot c = a \times b \times c + (a \times b + b \times c + c \times a) - (a + b + c)\), calculate \(1 \odot 43 \odot 47\).
|
4041
| 68.75 |
8,572 |
In $\triangle ABC$, the length of the side opposite to angle $A$ is $2$, and the vectors $\overrightarrow{m} = (2, 2\cos^2\frac{B+C}{2} - 1)$ and $\overrightarrow{n} = (\sin\frac{A}{2}, -1)$.
1. Find the value of angle $A$ when the dot product $\overrightarrow{m} \cdot \overrightarrow{n}$ is at its maximum.
2. Under the conditions of part (1), find the maximum area of $\triangle ABC$.
|
\sqrt{3}
| 92.96875 |
8,573 |
Given an infinite grid where each cell is either red or blue, such that in any \(2 \times 3\) rectangle exactly two cells are red, determine how many red cells are in a \(9 \times 11\) rectangle.
|
33
| 47.65625 |
8,574 |
Given that point \\(A\\) on the terminal side of angle \\(\alpha\\) has coordinates \\(\left( \sqrt{3}, -1\right)\\),
\\((1)\\) Find the set of angle \\(\alpha\\)
\\((2)\\) Simplify the following expression and find its value: \\( \dfrac{\sin (2\pi-\alpha)\tan (\pi+\alpha)\cot (-\alpha-\pi)}{\csc (-\alpha)\cos (\pi-\alpha)\tan (3\pi-\alpha)} \\)
|
\dfrac{1}{2}
| 32.8125 |
8,575 |
Given that point $P(x,y)$ is a moving point on the circle $x^{2}+y^{2}=2y$,
(1) Find the range of $z=2x+y$;
(2) If $x+y+a\geqslant 0$ always holds, find the range of real numbers $a$;
(3) Find the maximum and minimum values of $x^{2}+y^{2}-16x+4y$.
|
6-2\sqrt{73}
| 38.28125 |
8,576 |
A student is given a budget of $10,000 to produce a rectangular banner for a school function. The length and width (in meters) of the banner must be integers. If each meter in length costs $330 while each meter in width costs $450, what is the maximum area (in square meters) of the banner that can be produced?
|
165
| 0 |
8,577 |
Let \(\triangle ABC\) be equilateral, and let \(D, E, F\) be points on sides \(BC, CA, AB\) respectively, with \(FA = 9\), \(AE = EC = 6\), and \(CD = 4\). Determine the measure (in degrees) of \(\angle DEF\).
|
60
| 70.3125 |
8,578 |
A non-increasing sequence of 100 non-negative reals has the sum of the first two terms at most 100 and the sum of the remaining terms at most 100. What is the largest possible value for the sum of the squares of the terms?
|
10000
| 30.46875 |
8,579 |
Given an ellipse $\frac{x^2}{25}+\frac{y^2}{m^2}=1\left(m \gt 0\right)$ with one focus at $F\left(0,4\right)$, find $m$.
|
\sqrt{41}
| 50 |
8,580 |
Define the operation "□" as: $a□b=a^2+2ab-b^2$. Let the function $f(x)=x□2$, and the equation related to $x$ is $f(x)=\lg|x+2|$ ($x\neq -2$) has exactly four distinct real roots $x_1$, $x_2$, $x_3$, $x_4$. Find the value of $x_1+x_2+x_3+x_4$.
|
-8
| 25.78125 |
8,581 |
Given that the point $P$ on the ellipse $\frac{x^{2}}{64} + \frac{y^{2}}{28} = 1$ is 4 units away from the left focus, find the distance from point $P$ to the right directrix.
|
16
| 14.84375 |
8,582 |
15. If \( a = 1.69 \), \( b = 1.73 \), and \( c = 0.48 \), find the value of
$$
\frac{1}{a^{2} - a c - a b + b c} + \frac{2}{b^{2} - a b - b c + a c} + \frac{1}{c^{2} - a c - b c + a b}.
$$
|
20
| 5.46875 |
8,583 |
Given the polar equation of curve C is $\rho - 6\cos\theta + 2\sin\theta + \frac{1}{\rho} = 0$, with the pole as the origin of the Cartesian coordinate system and the polar axis as the positive x-axis, establish a Cartesian coordinate system in the plane xOy. The line $l$ passes through point P(3, 3) with an inclination angle $\alpha = \frac{\pi}{3}$.
(1) Write the Cartesian equation of curve C and the parametric equation of line $l$;
(2) Suppose $l$ intersects curve C at points A and B, find the value of $|AB|$.
|
2\sqrt{5}
| 51.5625 |
8,584 |
Given an equilateral triangle \( ABC \). Point \( K \) is the midpoint of side \( AB \), and point \( M \) lies on side \( BC \) such that \( BM : MC = 1 : 3 \). A point \( P \) is chosen on side \( AC \) such that the perimeter of triangle \( PKM \) is minimized. In what ratio does point \( P \) divide side \( AC \)?
|
2/3
| 0 |
8,585 |
If 10 people need 45 minutes and 20 people need 20 minutes to repair a dam, how many minutes would 14 people need to repair the dam?
|
30
| 4.6875 |
8,586 |
Find the sum of the roots of $\tan^2 x - 8\tan x + \sqrt{2} = 0$ that are between $x=0$ and $x=2\pi$ radians.
|
4\pi
| 32.8125 |
8,587 |
Given that \(\log_8 2 = 0.2525\) in base 8 (to 4 decimal places), find \(\log_8 4\) in base 8 (to 4 decimal places).
|
0.5050
| 100 |
8,588 |
In an isosceles triangle, the side is divided by the point of tangency of the inscribed circle in the ratio 7:5 (starting from the vertex). Find the ratio of the side to the base.
|
6/5
| 34.375 |
8,589 |
Determine the number of palindromes between 1000 and 10000 that are multiples of 6.
|
13
| 44.53125 |
8,590 |
Five brothers equally divided an inheritance from their father. The inheritance included three houses. Since three houses could not be divided into 5 parts, the three older brothers took the houses, and the younger brothers were compensated with money. Each of the three older brothers paid 800 rubles, and the younger brothers shared this money among themselves, so that everyone ended up with an equal share. What is the value of one house?
|
2000
| 18.75 |
8,591 |
An ideal gas is used as the working substance of a heat engine operating cyclically. The cycle consists of three stages: isochoric pressure reduction from $3 P_{0}$ to $P_{0}$, isobaric density increase from $\rho_{0}$ to $3 \rho_{0}$, and a return to the initial state, represented as a quarter circle in the $P / P_{0}, \rho / \rho_{0}$ coordinates with the center at point $(1,1)$. Determine the efficiency of this cycle, knowing that it is 8 times less than the maximum possible efficiency for the same minimum and maximum gas temperatures as in the given cycle.
|
1/9
| 4.6875 |
8,592 |
A point is randomly thrown onto the segment [3, 8] and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2 k-3\right) x^{2}+(3 k-5) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$.
|
4/15
| 64.84375 |
8,593 |
Find the largest four-digit number in which all digits are different and which is divisible by 2, 5, 9, and 11.
|
8910
| 67.96875 |
8,594 |
Given the numbers 2, 3, 4, 3, 1, 6, 3, 7, determine the sum of the mean, median, and mode of these numbers.
|
9.625
| 13.28125 |
8,595 |
Given the sequence \(\left\{a_{n}\right\}\), which satisfies
\[
a_{1}=0,\left|a_{n+1}\right|=\left|a_{n}-2\right|
\]
Let \(S\) be the sum of the first 2016 terms of the sequence \(\left\{a_{n}\right\}\). Determine the maximum value of \(S\).
|
2016
| 77.34375 |
8,596 |
A rectangular metal plate measuring \(10\) cm by \(8\) cm has a circular piece of maximum size cut out, followed by cutting a rectangular piece of maximum size from the circular piece. Calculate the total metal wasted in this process.
|
48
| 27.34375 |
8,597 |
Compute the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{4} x \cos ^{4} x \, dx
$$
|
\frac{3\pi}{8}
| 91.40625 |
8,598 |
Let $a, b$ be real numbers. If the complex number $\frac{1+2i}{a+bi} \= 1+i$, then $a=\_\_\_\_$ and $b=\_\_\_\_$.
|
\frac{1}{2}
| 4.6875 |
8,599 |
A certain TV station randomly selected $100$ viewers to evaluate a TV program in order to understand the evaluation of the same TV program by viewers of different genders. It is known that the ratio of the number of "male" to "female" viewers selected is $9:11$. The evaluation results are divided into "like" and "dislike", and some evaluation results are organized in the table below.
| Gender | Like | Dislike | Total |
|--------|------|---------|-------|
| Male | $15$ | | |
| Female | | | |
| Total | $50$ | | $100$ |
$(1)$ Based on the given data, complete the $2\times 2$ contingency table above. According to the independence test with $\alpha = 0.005$, can it be concluded that gender is related to the evaluation results?
$(2)$ The TV station plans to expand the male audience market. Now, using a proportional stratified sampling method, $3$ viewers are selected from the male participants for a program "suggestions" solicitation reward activity. The probability that a viewer who evaluated "dislike" has their "suggestion" adopted is $\frac{1}{4}$, and the probability that a viewer who evaluated "like" has their "suggestion" adopted is $\frac{3}{4}$. The reward for an adopted "suggestion" is $100$ dollars, and for a non-adopted "suggestion" is $50$ dollars. Let $X$ be the total prize money obtained by the $3$ viewers. Find the distribution table and the expected value of $X$.
Given: ${\chi}^{2}=\frac{n{(ad-bc)}^{2}}{(a+b)(c+d)(a+c)(b+d)}$
| $\alpha$ | $0.010$ | $0.005$ | $0.001$ |
|----------|---------|---------|---------|
| $x_{\alpha}$ | $6.635$ | $7.879$ | $10.828$ |
|
212.5
| 2.34375 |
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