problem stringlengths 10 5.15k | answer dict |
|---|---|
\( \Delta ABC \) is an isosceles triangle with \( AB = 2 \) and \( \angle ABC = 90^{\circ} \). Point \( D \) is the midpoint of \( BC \) and point \( E \) is on \( AC \) such that the area of quadrilateral \( AEDB \) is twice the area of triangle \( ECD \). Find the length of \( DE \). | {
"answer": "\\frac{\\sqrt{17}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a backpacking trip with 10 people, in how many ways can I choose 2 cooks and 1 medical helper if any of the 10 people may fulfill these roles? | {
"answer": "360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a triangle joined to a square to form an irregular pentagon. The triangle has the same perimeter as the square.
What is the ratio of the perimeter of the pentagon to the perimeter of the square? | {
"answer": "3:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 65 islanders gathered for a meeting. Each of them made the following statement in turn: "Among the statements made earlier, the true ones are exactly 20 less than the false ones." How many knights were present at this meeting? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $a=2$, $b= \sqrt {3}$, $B= \frac {\pi}{3}$, then $A=$ \_\_\_\_\_\_. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square sheet of paper with sides of length $10$ cm is initially folded in half horizontally. The folded paper is then folded diagonally corner to corner, forming a triangular shape. If this shape is then cut along the diagonal fold, what is the ratio of the perimeter of one of the resulting triangles to the perimeter of the original square?
A) $\frac{15 + \sqrt{125}}{40}$
B) $\frac{15 + \sqrt{75}}{40}$
C) $\frac{10 + \sqrt{50}}{40}$
D) $\frac{20 + \sqrt{100}}{40}$ | {
"answer": "\\frac{15 + \\sqrt{125}}{40}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\Delta ABC$ be an acute-angled triangle and let $H$ be its orthocentre. Let $G_1, G_2$ and $G_3$ be the centroids of the triangles $\Delta HBC , \Delta HCA$ and $\Delta HAB$ respectively. If the area of $\Delta G_1G_2G_3$ is $7$ units, what is the area of $\Delta ABC $ ? | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the parallelogram \(ABCD\), points \(E\) and \(F\) are located on sides \(AB\) and \(BC\) respectively, and \(M\) is the point of intersection of lines \(AF\) and \(DE\). Given that \(AE = 2BE\) and \(BF = 3CF\), find the ratio \(AM : MF\). | {
"answer": "4:5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polynomial with integer coefficients:
\[ f(x) = x^5 + a_1 x^4 + a_2 x^3 + a_3 x^2 + a_4 x + a_5 \]
If \( f(\sqrt{3} + \sqrt{2}) = 0 \) and \( f(1) + f(3) = 0 \), find \( f(-1) \). | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Teacher Li plans to buy 25 souvenirs for students from a store that has four types of souvenirs: bookmarks, postcards, notebooks, and pens, with 10 pieces available for each type (souvenirs of the same type are identical). Teacher Li intends to buy at least one piece of each type. How many different purchasing plans are possible? (Answer in numeric form.). | {
"answer": "592",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $$C: \frac {x^{2}}{4}+ \frac {y^{2}}{b^{2}}=1(0＜b＜2)$$, a straight line with a slope angle of $$\frac {3π}{4}$$ intersects the ellipse C at points A and B. The midpoint of the line segment AB is M, and O is the coordinate origin. The angle between $$\overrightarrow {OM}$$ and $$\overrightarrow {MA}$$ is θ, and |tanθ|=3. Find the value of b. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two identical letters in a row. How many unpronounceable words of 7 letters are there? | {
"answer": "86",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Exactly half of the population of the island of Misfortune are hares, and the rest are rabbits. If a resident of Misfortune makes a statement, he sincerely believes what he says. However, hares are faithfully mistaken on average in one out of every four cases, and rabbits are faithfully mistaken on average in one out of every three cases. One day, a creature came to the center of the island and shouted, "I am not a hare!" Then he thought and sadly said, "I am not a rabbit." What is the probability that he is actually a hare? | {
"answer": "27/59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that P is a point on the hyperbola $\frac{x^2}{4} - \frac{y^2}{3} = 1$, and $F_1$, $F_2$ are the two foci of the hyperbola. If $\angle F_1PF_2 = 60^\circ$, calculate the area of $\triangle PF_1F_2$. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a closed right triangular prism ABC-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub> there is a sphere with volume $V$. If $AB \perp BC$, $AB=6$, $BC=8$, and $AA_{1}=3$, then the maximum value of $V$ is \_\_\_\_\_\_. | {
"answer": "\\frac{9\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The front view of a cone is an equilateral triangle with a side length of 4. Find the surface area of the cone. | {
"answer": "12\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the following numbers
① $111111_{(2)}$
② $210_{(6)}$
③ $1000_{(4)}$
④ $81_{(8)}$
The largest number is \_\_\_\_\_\_\_\_, and the smallest number is \_\_\_\_\_\_\_\_. | {
"answer": "111111_{(2)}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Chloe wants to buy a jacket that costs $45.50$. She has two $20$ bills, five quarters, a few nickels, and a pile of dimes in her wallet. What is the minimum number of dimes she needs if she also has six nickels? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An aluminum cube with an edge length of \( l = 10 \) cm is heated to a temperature of \( t_{1} = 100^{\circ} \mathrm{C} \). After this, it is placed on ice, which has a temperature of \( t_{2} = 0^{\circ} \mathrm{C} \). Determine the maximum depth to which the cube can sink. The specific heat capacity of aluminum is \( c_{a} = 900 \) J/kg\(^\circ \mathrm{C} \), the specific latent heat of fusion of ice is \( \lambda = 3.3 \times 10^{5} \) J/kg, the density of aluminum is \( \rho_{a} = 2700 \) kg/m\(^3 \), and the density of ice is \( \rho_{n} = 900 \) kg/m\(^3 \). | {
"answer": "0.0818",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $10\times 10$ square we choose $n$ cells. In every chosen cell we draw one arrow from the angle to opposite angle. It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or
the distance between their ends is at least 2. What is the maximum possible value of $n$ ? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The 200-digit number \( M \) is composed of 200 ones. What is the sum of the digits of the product \( M \times 2013 \)? | {
"answer": "1200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
20 balls numbered 1 through 20 are placed in a bin. In how many ways can 4 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn and each drawn ball has a consecutive number as the previous one? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The maximum value of the function \( y = \frac{\sin x \cos x}{1 + \sin x + \cos x} \) is $\quad$ . | {
"answer": "\\frac{\\sqrt{2} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the measure of the angle
$$
\delta=\arccos \left(\left(\sin 2905^{\circ}+\sin 2906^{\circ}+\cdots+\sin 6505^{\circ}\right)^{\cos } 2880^{\circ}+\cos 2881^{\circ}+\cdots+\cos 6480^{\circ}\right)
$$ | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{2} e^{\sqrt{(2-x) /(2+x)}} \cdot \frac{d x}{(2+x) \sqrt{4-x^{2}}}
$$ | {
"answer": "\\frac{e-1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The value of $(\sqrt{1+\sqrt{1+\sqrt{1}}})^{4}$ is:
(a) $\sqrt{2}+\sqrt{3}$;
(b) $\frac{1}{2}(7+3 \sqrt{5})$;
(c) $1+2 \sqrt{3}$;
(d) 3 ;
(e) $3+2 \sqrt{2}$. | {
"answer": "3 + 2 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the fractional equation $\frac{3}{{x-4}}+\frac{{x+m}}{{4-x}}=1$ has a root, determine the value of $m$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a\) and \(b\) be constants. The parabola \(C: y = (t^2 + t + 1)x^2 - 2(a + t)^2 x + t^2 + 3at + b\) passes through a fixed point \(P(1,0)\) for any real number \(t\). Find the value of \(t\) such that the chord obtained by intersecting the parabola \(C\) with the x-axis is the longest. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle with interior angles $60^{\circ}, 45^{\circ}$ and $75^{\circ}$ is inscribed in a circle of radius 2. What is the area of the triangle? | {
"answer": "3 + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Masha and the Bear ate a basket of raspberries and 60 pies, starting and finishing at the same time. Initially, Masha ate raspberries while the Bear ate pies, and then they switched at some point. The Bear ate raspberries 6 times faster than Masha and pies 3 times faster. How many pies did the Bear eat if the Bear ate twice as many raspberries as Masha? | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function \( f: \mathbb{R} \rightarrow \mathbb{R} \) is continuous. For every real number \( x \), the equation \( f(x) \cdot f(f(x)) = 1 \) holds. It is known that \( f(1000) = 999 \). Find \( f(500) \). | {
"answer": "\\frac{1}{500}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S_{n} \) be the sum of the first \( n \) terms of an arithmetic sequence \( \{a_{n}\} \). Given \( S_{6}=36 \) and \( S_{n}=324 \). If \( S_{n-6} = 144 \) for \( n > 6 \), then \( n \) equals \(\qquad\). | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T = (a,b,c)$ be a triangle with sides $a,b$ and $c$ and area $\triangle$ . Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a, b' = h_b, c' = h_c$ ) and denote its area by $\triangle '$ . Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from the altitudes of $T'$ , and denote its area by $\triangle ''$ . Given that $\triangle ' = 30$ and $\triangle '' = 20$ , find $\triangle$ . | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five guys are eating hamburgers. Each one puts a top half and a bottom half of a hamburger bun on the grill. When the buns are toasted, each guy randomly takes two pieces of bread off of the grill. What is the probability that each guy gets a top half and a bottom half? | {
"answer": "8/63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, $XY=15$, $YZ=18$, and $ZX=21$. Point $G$ is on $\overline{XY}$, $H$ is on $\overline{YZ}$, and $I$ is on $\overline{ZX}$. Let $XG = p \cdot XY$, $YH = q \cdot YZ$, and $ZI = r \cdot ZX$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=3/4$ and $p^2+q^2+r^2=1/2$. The ratio of the area of triangle $GHI$ to the area of triangle $XYZ$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | {
"answer": "41",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside triangle \( ABC \), a random point \( M \) is chosen. What is the probability that the area of one of the triangles \( ABM \), \( BCM \), or \( CAM \) will be greater than the sum of the areas of the other two? | {
"answer": "0.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC,$ $\angle B = 45^\circ,$ $AB = 100,$ and $AC = 100$. Find the sum of all possible values of $BC$. | {
"answer": "100 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define the sequence $(b_i)$ by $b_{n+2} = \frac{b_n + 2011}{1 + b_{n+1}}$ for $n \geq 1$ with all terms being positive integers. Determine the minimum possible value of $b_1 + b_2$. | {
"answer": "2012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 4 different colors of light bulbs, with each color representing a different signal. Assuming there is an ample supply of each color, we need to install one light bulb at each vertex $P, A, B, C, A_{1}, B_{1}, C_{1}$ of an airport signal tower (as shown in the diagram), with the condition that the two ends of the same line segment must have light bulbs of different colors. How many different installation methods are there? | {
"answer": "2916",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of \(\tan \left(\tan^{-1} \frac{1}{2} + \tan^{-1} \frac{1}{2 \times 2^2} + \tan^{-1} \frac{1}{2 \times 3^2} + \cdots + \tan^{-1} \frac{1}{2 \times 2009^2}\right)\). | {
"answer": "\\frac{2009}{2010}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers less than or equal to 5689 contain either the digit '6' or the digit '0'? | {
"answer": "2545",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos(x - \frac{\pi}{4}) = \frac{\sqrt{2}}{10}$, with $x \in (\frac{\pi}{2}, \frac{3\pi}{4})$.
(1) Find the value of $\sin x$;
(2) Find the value of $\cos(2x - \frac{\pi}{3})$. | {
"answer": "-\\frac{7 + 24\\sqrt{3}}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the cartesian coordinate system $(xOy)$, curve $({C}_{1})$ is defined by the parametric equations $\begin{cases}x=t+1,\ y=1-2t\end{cases}$ and curve $({C}_{2})$ is defined by the parametric equations $\begin{cases}x=a\cos θ,\ y=3\sin θ\end{cases}$ where $a > 0$.
1. If curve $({C}_{1})$ and curve $({C}_{2})$ have a common point on the $x$-axis, find the value of $a$.
2. When $a=3$, curves $({C}_{1})$ and $({C}_{2})$ intersect at points $A$ and $B$. Find the distance between points $A$ and $B$. | {
"answer": "\\frac{12\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A large shopping mall designed a lottery activity to reward its customers. In the lottery box, there are $8$ small balls of the same size, with $4$ red and $4$ black. The lottery method is as follows: each customer draws twice, picking two balls at a time from the lottery box each time. Winning is defined as drawing two balls of the same color, while losing is defined as drawing two balls of different colors.
$(1)$ If it is specified that after the first draw, the balls are put back into the lottery box for the second draw, find the distribution and mathematical expectation of the number of wins $X$.
$(2)$ If it is specified that after the first draw, the balls are not put back into the lottery box for the second draw, find the distribution and mathematical expectation of the number of wins $Y$. | {
"answer": "\\frac{6}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a variety show, there are seven acts: dance, comic dialogue, sketch, singing, magic, acrobatics, and opera. When arranging the program order, the conditions are that dance, comic dialogue, and sketch cannot be adjacent to each other. How many different arrangements of the program are possible? | {
"answer": "1440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain organism starts with 4 cells. Each cell splits into two cells at the end of three days. However, at the end of each 3-day period, 10% of the cells die immediately after splitting. This process continues for a total of 9 days. How many cells are there at the end of the $9^\text{th}$ day? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) that satisfies the condition: for any real numbers \( x \) and \( y \),
\[ f(2x) + f(2y) = f(x+y) f(x-y) \]
and given that \( f(\pi) = 0 \) and \( f(x) \) is not identically zero, determine the period of \( f(x) \). | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{2 \operatorname{arctg} \frac{1}{3}}^{2 \operatorname{arctg} \frac{1}{2}} \frac{d x}{\sin x(1-\sin x)}
$$ | {
"answer": "\\ln 3 - \\ln 2 + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Burattino got on a train. After travelling half of the total distance, he fell asleep and slept until there was only half of the distance he slept left to travel. What fraction of the total journey did Burattino travel awake? | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five fair six-sided dice are rolled. What is the probability that at least three of the five dice show the same value? | {
"answer": "\\frac{113}{648}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are 4 tiers of horses, with Tian Ji's top-tier horse being better than King Qi's middle-tier horse, worse than King Qi's top-tier horse, Tian Ji's middle-tier horse being better than King Qi's bottom-tier horse, worse than King Qi's middle-tier horse, and Tian Ji's bottom-tier horse being worse than King Qi's bottom-tier horse, determine the probability that Tian Ji's selected horse wins the race. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A \) be the set of real numbers \( x \) satisfying the inequality \( x^{2} + x - 110 < 0 \) and \( B \) be the set of real numbers \( x \) satisfying the inequality \( x^{2} + 10x - 96 < 0 \). Suppose that the set of integer solutions of the inequality \( x^{2} + ax + b < 0 \) is exactly the set of integers contained in \( A \cap B \). Find the maximum value of \( \lfloor |a - b| \rfloor \). | {
"answer": "71",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the number 826,000,000, express it in scientific notation. | {
"answer": "8.26\\times 10^{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $G$ be the centroid of triangle $PQR.$ If $GP^2 + GQ^2 + GR^2 = 22,$ then find $PQ^2 + PR^2 + QR^2.$ | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$ be the sum of the numbers: $99 \times 0.9$ $999 \times 0.9$ $9999 \times 0.9$ $\vdots$ $999\cdots 9 \times 0.9$ where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$ .
Find the sum of the digits in the number $a$ . | {
"answer": "891",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The set $\{[x]+[2x]+[3x] \mid x \in \mathbf{R}\} \bigcap \{1, 2, \cdots, 100\}$ contains how many elements, where $[x]$ represents the greatest integer less than or equal to $x$. | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a=2b$. Also, $\sin A$, $\sin C$, $\sin B$ form an arithmetic sequence.
$(I)$ Find the value of $\cos (B+C)$;
$(II)$ If the area of $\triangle ABC$ is $\frac{8\sqrt{15}}{3}$, find the value of $c$. | {
"answer": "4 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the country of Anchuria, a unified state exam takes place. The probability of guessing the correct answer to each question on the exam is 0.25. In 2011, to receive a certificate, one needed to answer correctly 3 questions out of 20. In 2012, the School Management of Anchuria decided that 3 questions were too few. Now, one needs to correctly answer 6 questions out of 40. The question is, if one knows nothing and simply guesses the answers, in which year is the probability of receiving an Anchurian certificate higher - in 2011 or in 2012? | {
"answer": "2012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function $f(x)=\ln x-\frac{1}{2} ax^{2}-bx$.
$(1)$ When $a=b=\frac{1}{2}$, find the maximum value of the function $f(x)$;
$(2)$ Let $F(x)=f(x)+\frac{1}{2} x^{2}+bx+\frac{a}{x} (0 < x\leqslant 3)$. If the slope $k$ of the tangent line at any point $P(x_{0},y_{0})$ on its graph is always less than or equal to $\frac{1}{2}$, find the range of the real number $a$;
$(3)$ When $a=0$, $b=-1$, the equation $x^{2}=2mf(x)$ (where $m > 0$) has a unique real solution, find the value of $m$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Eight hockey teams are competing against each other in a single round to advance to the semifinals. What is the minimum number of points that guarantees a team advances to the semifinals? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two circles $x^{2}+y^{2}=4$ and $x^{2}+y^{2}-2y-6=0$, find the length of their common chord. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the distance from the point \( M_{0} \) to the plane passing through the three points \( M_{1}, M_{2}, M_{3} \).
\( M_{1}(1, 3, 0) \)
\( M_{2}(4, -1, 2) \)
\( M_{3}(3, 0, 1) \)
\( M_{0}(4, 3, 0) \) | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sixty students went on a trip to the zoo. Upon returning to school, it turned out that 55 of them forgot gloves at the zoo, 52 forgot scarves, and 50 managed to forget hats. Find the smallest number of the most scatterbrained students - those who lost all three items. | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The four zeros of the polynomial \(x^4 + px^2 + qx - 144\) are distinct real numbers in arithmetic progression. Compute the value of \(p.\) | {
"answer": "-40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola \( C_{1}: 2x^{2} - y^{2} = 1 \) and the ellipse \( C_{2}: 4x^{2} + y^{2} = 1 \), let \( M \) and \( N \) be moving points on the hyperbola \( C_{1} \) and the ellipse \( C_{2} \) respectively, with \( O \) as the origin. If \( O M \) is perpendicular to \( O N \), find the distance from point \( O \) to the line \( M N \). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $P$ is a point inside rectangle $ABCD$, the distances from $P$ to the vertices of the rectangle are $PA = 5$ inches, $PD = 12$ inches, and $PC = 13$ inches. Find $PB$, which is $x$ inches. | {
"answer": "5\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. Given that $a=4$, $b=6$, and $C=60^\circ$:
1. Calculate $\overrightarrow{BC} \cdot \overrightarrow{CA}$;
2. Find the projection of $\overrightarrow{CA}$ onto $\overrightarrow{BC}$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \(C\) divides diameter \(AB\) in the ratio \(AC:BC = 2:1\). A point \(P\) is selected on the circle. Determine the possible values that the ratio \(\tan \angle PAC: \tan \angle APC\) can take. Specify the smallest such value. | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, points $E, F, G$ are the midpoints of edges $A A_{1}, C_{1} D_{1}$, and $D_{1} A_{1}$, respectively. Find the distance from point $B_{1}$ to the plane $E F G$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of three-element subsets of the set \(\{1, 2, 3, 4, \ldots, 120\}\) for which the sum of the three elements is a multiple of 3. | {
"answer": "93640",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence \( a_{1}, a_{2}, \cdots, a_{n}, \cdots \) that satisfies \( a_{1}=a_{2}=1, a_{3}=2 \), and for any natural number \( n \), \( a_{n} a_{n+1} a_{n+2} \neq 1 \). Furthermore, it is given that \( a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{1} + a_{n+1} + a_{n+2} + a_{n+3} \). Find the value of \( a_{1} + a_{2} + \cdots + a_{100} \). | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $n\in N^{*}$, select $k(k\in N, k\geqslant 2)$ numbers $j\_1$, $j\_2$, $...$, $j\_k$ from the set ${1,2,3,...,n}$ such that they simultaneously satisfy the following two conditions: $①1\leqslant j\_1 < j\_2 < ...j\_k\leqslant n$; $②j_{i+1}-j_{i}\geqslant m(i=1,2,…,k-1)$. Then the array $(j\_1, j\_2, ..., j\_k)$ is called a combination of selecting $k$ elements with a minimum distance of $m$ from $n$ elements, denoted as $C_{ n }^{ (k,m) }$. For example, from the set ${1,2,3}$, we have $C_{ 3 }^{ (2,1) }=3$. For the given set ${1,2,3,4,5,6,7}$, find $C_{ 7 }^{ (3,2) }=$ \_\_\_\_\_\_. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a school cafeteria line, there are 16 students alternating between boys and girls (starting with a boy, followed by a girl, then a boy, and so on). Any boy, followed immediately by a girl, can swap places with her. After some time, all the girls end up at the beginning of the line and all the boys are at the end. How many swaps were made? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $f(x)$ is an even function defined on $\mathbb{R}$, and $g(x)$ is an odd function defined on $\mathbb{R}$ that passes through the point $(-1, 3)$ and $g(x) = f(x-1)$, find the value of $f(2007) + f(2008)$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $A_{1}$ and $C_{1}$ are located on the sides $BC$ and $AB$ of triangle $ABC$. Segments $AA_{1}$ and $CC_{1}$ intersect at point $M$.
In what ratio does line $BM$ divide side $AC$, if $AC_{1}: C_{1}B = 2: 3$ and $BA_{1}: A_{1}C = 1: 2$? | {
"answer": "1:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest six-digit number that is divisible by 3, 7, and 13 without a remainder. | {
"answer": "100191",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 1000 rooms in a row along a long corridor. Initially, the first room contains 1000 people, and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different rooms will have people in them? | {
"answer": "61",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate \( \frac{18}{4.9 \times 106} \). | {
"answer": "\\frac{18}{519.4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A conveyor system produces on average 85% of first-class products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the frequency of first-class products from 0.85 in absolute magnitude does not exceed 0.01? | {
"answer": "11475",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Teams A and B each have 7 players who will compete in a Go tournament in a predetermined order. The match starts with player 1 from each team competing against each other. The loser is eliminated, and the winner next competes against the loser’s teammate. This process continues until all players of one team are eliminated, and the other team wins. Determine the total number of possible sequences of matches. | {
"answer": "3432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $Q$ be a point outside of circle $C$. A segment is drawn from $Q$, tangent to circle $C$ at point $R$, and a different secant from $Q$ intersects $C$ at points $D$ and $E$ such that $QD < QE$. If $QD = 5$ and the length of the tangent from $Q$ to $R$ ($QR$) is equal to $DE - QD$, calculate $QE$. | {
"answer": "\\frac{15 + 5\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
First, a number \( a \) is randomly selected from the set \(\{1,2,3, \cdots, 99,100\}\), then a number \( b \) is randomly selected from the same set. Calculate the probability that the last digit of \(3^{a} + 7^{b}\) is 8. | {
"answer": "\\frac{3}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are several soldiers forming a rectangular formation with exactly eight columns. If adding 120 people or removing 120 people from the formation can both form a square formation, how many soldiers are there in the original rectangular formation? | {
"answer": "136",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere is inscribed in a cone whose axial cross-section is an equilateral triangle. Find the volume of the cone if the volume of the sphere is \( \frac{32\pi}{3} \ \text{cm}^3 \). | {
"answer": "24 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n = 2^{35}3^{17}$. How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$? | {
"answer": "594",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle with perimeter $7$ has integer sidelengths. What is the maximum possible area of such a triangle? | {
"answer": "\\frac{3\\sqrt{7}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)= \frac{1}{2^{x}+ \sqrt {2}}$, use the method for deriving the sum of the first $n$ terms of an arithmetic sequence to find the value of $f(-5)+f(-4)+…+f(0)+…+f(5)+f(6)$. | {
"answer": "3 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $(2c-a)\cos B=b\cos A$.
(1) Find angle $B$;
(2) If $b=6$ and $c=2a$, find the area of $\triangle ABC$. | {
"answer": "6 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle where the ratio of the legs is 1:3, a perpendicular is dropped from the vertex of the right angle to the hypotenuse. Find the ratio of the segments created on the hypotenuse by this perpendicular. | {
"answer": "1:9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find a six-digit number $\overline{xy243z}$ that is divisible by 396. | {
"answer": "432432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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? | {
"answer": "-\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \( P Q = 19 \), \( Q R = 18 \), and \( P R = 17 \). Point \( S \) is on \( P Q \), point \( T \) is on \( P R \), and point \( U \) is on \( S T \) such that \( Q S = S U \) and \( U T = T R \). The perimeter of \(\triangle P S T\) is equal to: | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the areas of the regions bounded by the curves given in polar coordinates.
$$
r=\cos 2 \phi
$$ | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of the expression \(\sin \frac{b \pi}{36}\), where \(b\) is the sum of all distinct numbers obtained from the number \(a = 987654321\) by cyclic permutations of its digits (in a cyclic permutation, all the digits of the number, except the last one, are shifted one place to the right, and the last digit moves to the first place). | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many unordered pairs of edges of a given octahedron determine a plane? | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Bethany has 11 pound coins and some 20 pence coins and some 50 pence coins in her purse. The mean value of the coins is 52 pence. Which could not be the number of coins in the purse?
A) 35
B) 40
C) 50
D) 65
E) 95 | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the volume of the solid $T$ consisting of all points $(x, y, z)$ such that $|x| + |y| \leq 2$, $|x| + |z| \leq 2$, and $|y| + |z| \leq 2$. | {
"answer": "\\frac{32}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xoy$, the parametric equation of line $l$ is $\begin{cases} x=1- \frac { \sqrt {3}}{2}t \\ y= \frac {1}{2}t\end{cases}$ (where $t$ is the parameter), and in the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the equation of circle $C$ is $\rho=2 \sqrt {3}\sin \theta$.
$(1)$ Write the standard equation of line $l$ and the Cartesian coordinate equation of circle $C$;
$(2)$ If the Cartesian coordinates of point $P$ are $(1,0)$, and circle $C$ intersects line $l$ at points $A$ and $B$, find the value of $|PA|+|PB|$. | {
"answer": "2 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Shift the graph of the function $y=3\sin (2x+ \frac {\pi}{6})$ to the graph of the function $y=3\cos 2x$ and determine the horizontal shift units. | {
"answer": "\\frac {\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Snow White entered a room where 30 chairs were arranged around a circular table. Some of the chairs were occupied by dwarfs. It turned out that Snow White could not sit in such a way that there was no one next to her. What is the minimum number of dwarfs that could have been at the table? (Explain how the dwarfs must have been seated and why there would be a chair with no one next to it if there were fewer dwarfs.) | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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