problem stringlengths 10 5.15k | answer dict |
|---|---|
Solve the equation \( 2 \sqrt{2} \sin ^{3}\left(\frac{\pi x}{4}\right) = \sin \left(\frac{\pi}{4}(1+x)\right) \).
How many solutions of this equation satisfy the condition: \( 2000 \leq x \leq 3000 \)? | {
"answer": "250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the International Space Station, there was an electronic clock displaying time in the HH:MM format. Due to an electromagnetic storm, the device started malfunctioning, causing each digit on the display to either increase by 1 or decrease by 1. What was the actual time when the storm occurred, if immediately after the storm the clock showed 09:09? | {
"answer": "18:18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C: \frac{x^{2}}{4} + y^{2} = 1$, with $O$ being the origin of coordinates, and a line $l$ intersects the ellipse $C$ at points $A$ and $B$, and $\angle AOB = 90^{\circ}$.
(Ⅰ) If the line $l$ is parallel to the x-axis, find the area of $\triangle AOB$;
(Ⅱ) If the line $l$ is always tangent to the circle $x^{2} + y^{2} = r^{2} (r > 0)$, find the value of $r$. | {
"answer": "\\frac{2\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of $\dfrac{\sqrt[3]{81}}{\sqrt[4]{81}}$ in terms of 81 raised to what power? | {
"answer": "\\frac{1}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=\cos (x+\frac{π}{3})$, determine the horizontal shift of the graph of the function $y=\sin x$. | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In square ABCD, an isosceles triangle AEF is inscribed; point E lies on side BC, point F lies on side CD, and AE = AF. The tangent of angle AEF is 3. Find the cosine of angle FAD. | {
"answer": "\\frac{2\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle made of wire and a rectangle are arranged in such a way that the circle passes through two vertices $A$ and $B$ and touches the side $CD$. The length of side $CD$ is 32.1. Find the ratio of the sides of the rectangle, given that its perimeter is 4 times the radius of the circle. | {
"answer": "4:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( ABC \) with sides \( AB = 8 \), \( AC = 4 \), \( BC = 6 \), the angle bisector \( AK \) is drawn, and on the side \( AC \) a point \( M \) is marked such that \( AM : CM = 3 : 1 \). Point \( N \) is the intersection point of \( AK \) and \( BM \). Find \( AN \). | {
"answer": "\\frac{18\\sqrt{6}}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\cos \left(2x+ \frac{2\pi}{3}\right)+ \sqrt{3}\sin 2x$.
$(1)$ Find the smallest positive period and the maximum value of the function $f(x)$;
$(2)$ Let $\triangle ABC$ have internal angles $A$, $B$, and $C$ respectively. If $f\left(\frac{C}{2}\right)=- \frac{1}{2}$ and $AC=1$, $BC=3$, find the value of $\sin A$. | {
"answer": "\\frac{3 \\sqrt{21}}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
With the rapid development of the "Internet + transportation" model, "shared bicycles" have appeared successively in many cities. In order to understand the satisfaction of users in a certain area with the services provided, a certain operating company randomly surveyed 10 users and obtained satisfaction ratings of 92, 84, 86, 78, 89, 74, 83, 77, 89.
$(1)$ Calculate the sample mean $\overline{x}$ and variance $s^{2}$;
$(2)$ Under condition (1), if the user's satisfaction rating is between $({\overline{x}-s,\overline{x}+s})$, then the satisfaction level is "$A$ grade". Estimate the percentage of users in the area whose satisfaction level is "$A$ grade".
Reference data: $\sqrt{30}≈5.48, \sqrt{33}≈5.74, \sqrt{35}≈5.92$. | {
"answer": "50\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of the first $n$ terms of a geometric sequence ${a_{n}}$ is $S_{n}$, if $a_{3}=4$, $S_{3}=12$, find the common ratio. | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( A \), \( B \), and \( C \) are any three non-collinear points on a plane, and point \( O \) is inside \( \triangle ABC \) such that:
\[
\angle AOB = \angle BOC = \angle COA = 120^\circ.
\]
Find the maximum value of \( \frac{OA + OB + OC}{AB + BC + CA} \). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( AC \), \( D \) is a point on side \( BC \) such that \( AD \) is the angle bisector of \( \angle BAC \), and \( P \) is the point of intersection of \( AD \) and \( BM \). Given that \( AB = 10 \, \text{cm} \), \( AC = 30 \, \text{cm} \), and the area of triangle \( \triangle ABC \) is \( 100 \, \text{cm}^2 \), calculate the area of triangle \( \triangle ABP \). | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a\ln a=be^{b}$, where $b > 0$, find the maximum value of $\frac{b}{{{a^2}}}$ | {
"answer": "\\frac{1}{2e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{\sqrt[4]{2+n^{5}}-\sqrt{2 n^{3}+3}}{(n+\sin n) \sqrt{7 n}}$$ | {
"answer": "-\\sqrt{\\frac{2}{7}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\{a_n\}$ is a geometric sequence with a common ratio of $q$, and $a_m$, $a_{m+2}$, $a_{m+1}$ form an arithmetic sequence.
(Ⅰ) Find the value of $q$;
(Ⅱ) Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. Determine whether $S_m$, $S_{m+2}$, $S_{m+1}$ form an arithmetic sequence and explain the reason. | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using an electric stove with a power of $P=500 \mathrm{W}$, a certain amount of water is heated. When the electric stove is turned on for $t_{1}=1$ minute, the water temperature increases by $\Delta T=2^{\circ} \mathrm{C}$, and after turning off the heater, the temperature decreases to the initial value in $t_{2}=2$ minutes. Determine the mass of the heated water, assuming the thermal power losses are constant. The specific heat capacity of water is $c_{B}=4200$ J/kg$\cdot{ }^{\circ} \mathrm{C}$. | {
"answer": "2.38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Convert from kilometers to miles. In the problem 3.125, the Fibonacci numeral system was introduced as being useful when converting distances from kilometers to miles or vice versa.
Suppose we want to find out how many miles are in 30 kilometers. For this, we represent the number 30 in the Fibonacci numeral system:
$$
30=21+8+1=F_{8}+F_{6}+F_{2}=(1010001)_{\mathrm{F}}
$$
Now we need to shift each number one position to the right, obtaining
$$
F_{7}+F_{5}+F_{1}=13+5+1=19=(101001)_{\mathrm{F}}
$$
So, the estimated result is 19 miles. (The correct result is approximately 18.46 miles.) Similarly, conversions from miles to kilometers are done.
Explain why this algorithm works. Verify that it gives a rounded number of miles in $n$ kilometers for all $n \leqslant 100$, differing from the correct answer by less than $2 / 3$ miles. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ and $S_{1}$ respectively be the midpoints of edges $AD$ and $B_{1}C_{1}$. A rotated cube is denoted by $A^{\prime}B^{\prime}C^{\prime}D^{\prime}A_{1}^{\prime}B_{1}^{\prime}C_{1}^{\prime}D_{1}^{\prime}$. The common part of the original cube and the rotated one is a polyhedron consisting of a regular quadrilateral prism $EFGHE_{1}F_{1}G_{1}H_{1}$ and two regular quadrilateral pyramids $SEFGH$ and $S_{1}E_{1}F_{1}G_{1}H_{1}$.
The side length of the base of each pyramid is 1, and its height is $\frac{1}{2}$, making its volume $\frac{1}{6}$. The volume of the prism is $\sqrt{2}-1$. | {
"answer": "\\sqrt{2} - \\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with center $O$ and equation $x^2 + y^2 = 1$ passes through point $P(-1, \sqrt{3})$. Two tangents are drawn from $P$ to the circle, touching the circle at points $A$ and $B$ respectively. Find the length of the chord $|AB|$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{1, 2, \cdots, 2005\} \), and \( A \subseteq S \) with \( |A| = 31 \). Additionally, the sum of all elements in \( A \) is a multiple of 5. Determine the number of such subsets \( A \). | {
"answer": "\\frac{1}{5} \\binom{2005}{31}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of the magnitudes of two angles of a triangle is 2, and the difference in lengths of the sides opposite these angles is 2 cm; the length of the third side of the triangle is 5 cm. Calculate the area of the triangle. | {
"answer": "\\frac{15 \\sqrt{7}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(x\), \(y\), and \(z\) be complex numbers such that:
\[
xy + 3y = -9, \\
yz + 3z = -9, \\
zx + 3x = -9.
\]
Find all possible values of \(xyz\). | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the graph of the function $f(x)=\sin 2x+\cos 2x$ is translated to the left by $\varphi (\varphi > 0)$ units, and the resulting graph is symmetric about the $y$-axis, then find the minimum value of $\varphi$. | {
"answer": "\\frac{\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the expression \(\frac{13 x^{2}+24 x y+13 y^{2}+16 x+14 y+68}{\left(9-x^{2}-8 x y-16 y^{2}\right)^{5 / 2}}\). Round the answer to the nearest hundredth if needed. | {
"answer": "0.26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_{n}\}$ with a common difference of $\frac{{2π}}{3}$, let $S=\{\cos a_{n}|n\in N^{*}\}$. If $S=\{a,b\}$, find the value of $ab$. | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( a(x+1)=x^{3}+3x^{2}+3x+1 \), find \( a \) in terms of \( x \).
If \( a-1=0 \), then the value of \( x \) is \( 0 \) or \( b \). What is \( b \) ?
If \( p c^{4}=32 \), \( p c=b^{2} \), and \( c \) is positive, what is the value of \( c \) ?
\( P \) is an operation such that \( P(A \cdot B) = P(A) + P(B) \).
\( P(A) = y \) means \( A = 10^{y} \). If \( d = A \cdot B \), \( P(A) = 1 \) and \( P(B) = c \), find \( d \). | {
"answer": "1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, let $a$, $b$, $c$ be the lengths of the sides opposite to angles $A$, $B$, $C$ respectively, and it is given that $b = a \cos C + \frac{\sqrt{3}}{3} c \sin A$.
(i) Find the measure of angle $A$.
(ii) If the area of $\triangle ABC$ is $\sqrt{3}$ and the median to side $AB$ is $\sqrt{2}$, find the lengths of sides $b$ and $c$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let
$$
\frac{1}{1+\frac{1}{1+\frac{1}{1+\ddots-\frac{1}{1}}}}=\frac{m}{n}
$$
where \(m\) and \(n\) are coprime natural numbers, and there are 1988 fraction lines on the left-hand side of the equation. Calculate the value of \(m^2 + mn - n^2\). | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an equilateral triangle \( \triangle ABC \) with a side length of 1,
\[
\overrightarrow{AP} = \frac{1}{3}(\overrightarrow{AB} + \overrightarrow{AC}), \quad \overrightarrow{AQ} = \overrightarrow{AP} + \frac{1}{2}\overrightarrow{BC}.
\]
Find the area of \( \triangle APQ \). | {
"answer": "\\frac{\\sqrt{3}}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $-\sqrt{4}+|-\sqrt{2}-1|+(\pi -2013)^{0}-(\frac{1}{5})^{0}$. | {
"answer": "\\sqrt{2} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a round-robin chess tournament, 30 players are participating. To achieve the 4th category norm, a player needs to score 60% of the possible points. What is the maximum number of players who can achieve the category norm by the end of the tournament? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( a + x^2 = 2015 \), \( b + x^2 = 2016 \), \( c + x^2 = 2017 \), and \( abc = 24 \), find the value of \( \frac{a}{bc} + \frac{b}{ac} + \frac{c}{ab} - \frac{1}{a} - \frac{1}{b} - \frac{1}{c} \). | {
"answer": "1/8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many rectangles can be formed where each vertex is a point on a 4x4 grid of equally spaced points? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many distinct four-digit numbers composed of the digits $1$, $2$, $3$, and $4$ are even? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the integral $$\int_{ -2 }^{ 2 }$$($$\sqrt {16-x^{2}}$$+sinx)dx=\_\_\_\_\_\_ | {
"answer": "4\\sqrt{3} + \\frac{8\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. Given that $a=1$, $b=1$, and $c= \sqrt{2}$, then $\sin A= \_\_\_\_\_\_$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x \) and \( y \) are in \( (0, +\infty) \), and \(\frac{19}{x} + \frac{98}{y} = 1\). What is the minimum value of \( x + y \)? | {
"answer": "117 + 14\\sqrt{38}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\theta \in (0, \frac{\pi}{2})$, and $\sin\theta = \frac{4}{5}$, find the value of $\cos\theta$ and $\sin(\theta + \frac{\pi}{3})$. | {
"answer": "\\frac{4 + 3\\sqrt{3}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a square \(PQRS\). The arc \(QS\) is a quarter circle. The point \(U\) is the midpoint of \(QR\) and the point \(T\) lies on \(SR\). The line \(TU\) is a tangent to the arc \(QS\). What is the ratio of the length of \(TR\) to the length of \(UR\)? | {
"answer": "4:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, if $a=2$, $c=2\sqrt{3}$, and $\angle A=30^\circ$, then the area of $\triangle ABC$ is equal to __________. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \equal{} x^2 \plus{} ax \plus{} b$ passing through the point $ (1,\ \minus{} 1)$ and the $ x$ axis about the $ x$ axis | {
"answer": "\\frac{16\\pi}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given complex numbers \( z, z_{1}, z_{2} \left( z_{1} \neq z_{2} \right) \) such that \( z_{1}^{2}=z_{2}^{2}=-2-2 \sqrt{3} \mathrm{i} \), and \(\left|z-z_{1}\right|=\left|z-z_{2}\right|=4\), find \(|z|=\ \ \ \ \ .\) | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Aeroflot cashier must deliver tickets to five groups of tourists. Three of these groups live in the hotels "Druzhba," "Russia," and "Minsk." The cashier will be given the address of the fourth group by the tourists from "Russia," and the address of the fifth group by the tourists from "Minsk." In how many ways can the cashier choose the order of visiting the hotels to deliver the tickets? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest period of the function \( y = \cos^{10} x + \sin^{10} x \). | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the function $f(x) = x^2$ has a domain $D$ and its range is $\{0, 1, 2, 3, 4, 5\}$, how many such functions $f(x)$ exist? (Please answer with a number). | {
"answer": "243",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the minimum number of points in which 5 different non-parallel lines, not passing through a single point, can intersect? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Circle $A$ is tangent to circle $B$ at one point, and the center of circle $A$ lies on the circumference of circle $B$. The area of circle $A$ is $16\pi$ square units. Find the area of circle $B$. | {
"answer": "64\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a new infinite geometric series: $$\frac{7}{4} + \frac{28}{9} + \frac{112}{27} + \dots$$
Determine the common ratio of this series. | {
"answer": "\\frac{16}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a triangle, one of the sides is equal to 6, the radius of the inscribed circle is 2, and the radius of the circumscribed circle is 5. Find the perimeter. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), \( \angle BAC = 60^{\circ} \). The angle bisector of \( \angle BAC \), \( AD \), intersects \( BC \) at point \( D \). Given that \( \overrightarrow{AD} = \frac{1}{4} \overrightarrow{AC} + t \overrightarrow{AB} \) and \( AB = 8 \), find the length of \( AD \). | {
"answer": "6 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
120 schools each send 20 people to form 20 teams, with each team having exactly 1 person from each school. Find the smallest positive integer \( k \) such that when \( k \) people are selected from each team, there will be at least 20 people from the same school among all the selected individuals. | {
"answer": "115",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a bijective function $g : R \to R$ , we say that a function $f : R \to R$ is its superinverse if it satisfies the following identity $(f \circ g)(x) = g^{-1}(x)$ , where $g^{-1}$ is the inverse of $g$ . Given $g(x) = x^3 + 9x^2 + 27x + 81$ and $f$ is its superinverse, find $|f(-289)|$ . | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $\left\{x_{n}\right\}$ satisfies $x_{1}=1$, and for any $n \in \mathbb{Z}^{+}$, it holds that $x_{n+1}=x_{n}+3 \sqrt{x_{n}}+\frac{n}{\sqrt{x_{n}}}$. Find the value of $\lim _{n \rightarrow+\infty} \frac{n^{2}}{x_{n}}$. | {
"answer": "\\frac{4}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If point $P$ is any point on the curve $y=x^{2}-\ln x$, then the minimum distance from point $P$ to the line $y=x-2$ is ____. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five doctors A, B, C, D, and E are assigned to four different service positions located in the Sichuan disaster zone, labeled A, B, C, and D. Each position must be filled by at least one doctor. Calculate the total number of ways doctors A and B can serve independently in different positions. | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the game show $\text{\emph{Wheel of Fortunes}}$, you encounter a spinner divided equally into 6 regions labeled as follows: "Bankrupt," "$600", "$100", "$2000", "$150", "$700". What is the probability that you will accumulate exactly $1450$ in your first three spins without landing on "Bankrupt"? Assume each spin is independent and all outcomes have equal likelihood. | {
"answer": "\\frac{6}{125}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the complex number $z_1$ corresponds to the point $(-1,1)$ on the complex plane, and the complex number $z_2$ satisfies $z_1z_2=-2$, find the value of $|z_2+2i|$. | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Several oranges (not necessarily of equal mass) were picked from a tree. On weighing them, it turned out that the mass of any three oranges taken together is less than 5% of the total mass of the remaining oranges. What is the minimum number of oranges that could have been picked? | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of boys to girls in Mr. Smith's class is 3:4, and there are 42 students in total. What percent of the students are boys. | {
"answer": "42.86\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of the first $n$ terms of the arithmetic sequences $\{a_n\}$ and $\{b_n\}$ are $(S_n)$ and $(T_n)$, respectively. If for any positive integer $n$, $\frac{S_n}{T_n}=\frac{2n-5}{3n-5}$, determine the value of $\frac{a_7}{b_2+b_8}+\frac{a_3}{b_4+b_6}$. | {
"answer": "\\frac{13}{22}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value, for \(a, b > 0\), of the expression
\[
\frac{|a + 3b - b(a + 9b)| + |3b - a + 3b(a - b)|}{\sqrt{a^{2} + 9b^{2}}}
\] | {
"answer": "\\frac{\\sqrt{10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane quadrilateral \(ABCD\), points \(E\) and \(F\) are the midpoints of sides \(AD\) and \(BC\) respectively. Given that \(AB = 1\), \(EF = \sqrt{2}\), and \(CD = 3\), and that \(\overrightarrow{AD} \cdot \overrightarrow{BC} = 15\), find \(\overrightarrow{AC} \cdot \overrightarrow{BD}\). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Experts and Viewers play "What? Where? When?" until one side wins six rounds—the first to win six rounds wins the game. The probability of the Experts winning a single round is 0.6, and there are no ties. Currently, the Experts are losing with a score of $3:4$. Find the probability that the Experts will still win. | {
"answer": "0.4752",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How can we connect 50 cities with the minimum number of flight routes so that it's possible to travel from any city to any other city with no more than two layovers? | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Li Shuang rides a bike from location $A$ to location $B$ at a speed of 320 meters per minute. On the way, due to a bike malfunction, he pushes the bike and continues walking for 5 minutes to a location 1800 meters from $B$ to repair the bike. Fifteen minutes later, he resumes riding towards $B$ at 1.5 times his original cycling speed. Upon reaching $B$, he is 17 minutes later than the estimated time. What is Li Shuang's speed while pushing the bike in meters per minute? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $\left(\sqrt[4]{(\sqrt{5})^5}\right)^2$. | {
"answer": "5 \\sqrt[4]{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the vertices of a rectangle are $A(0,0)$, $B(2,0)$, $C(2,1)$, and $D(0,1)$. A particle starts from the midpoint $P_{0}$ of $AB$ and moves in a direction forming an angle $\theta$ with $AB$, reaching a point $P_{1}$ on $BC$. The particle then sequentially reflects to points $P_{2}$ on $CD$, $P_{3}$ on $DA$, and $P_{4}$ on $AB$, with the reflection angle equal to the incidence angle. If $P_{4}$ coincides with $P_{0}$, find $\tan \theta$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two real numbers \( p > 1 \) and \( q > 1 \) such that \( \frac{1}{p} + \frac{1}{q} = 1 \) and \( pq = 9 \), what is \( q \)? | {
"answer": "\\frac{9 + 3\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Given $\frac{\sin\alpha + 3\cos\alpha}{3\cos\alpha - \sin\alpha} = 5$, find the value of $\sin^2\alpha - \sin\alpha\cos\alpha$.
(2) Given a point $P(-4, 3)$ on the terminal side of angle $\alpha$, determine the value of $\frac{\cos\left(\frac{\pi}{2} + \alpha\right)\sin\left(-\pi - \alpha\right)}{\cos\left(\frac{11\pi}{2} - \alpha\right)\sin\left(\frac{9\pi}{2} + \alpha\right)}$. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the volume of an octahedron which has an inscribed sphere of radius 1. | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p$, $q$, $r$, $s$, and $t$ be distinct integers such that $(8-p)(8-q)(8-r)(8-s)(8-t) = -120$. Calculate the sum $p+q+r+s+t$. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and it is given that $(a+b)(\sin A-\sin B)=c(\sin C-\sin B)$.
$(1)$ Find $A$.
$(2)$ If $a=4$, find the maximum value of the area $S$ of $\triangle ABC$. | {
"answer": "4 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$BL$ is the angle bisector of triangle $ABC$. Find its area, given that $|AL| = 2$, $|BL| = 3\sqrt{10}$, and $|CL| = 3$. | {
"answer": "\\frac{15 \\sqrt{15}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the function \( f(x)=\cos 4x + 6\cos 3x + 17\cos 2x + 30\cos x \) for \( x \in \mathbb{R} \). | {
"answer": "-18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The first term of a sequence is 2, the second term is 3, and each subsequent term is formed such that each term is 1 less than the product of its two neighbors. What is the sum of the first 1095 terms of the sequence? | {
"answer": "1971",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 1000 lights and 1000 switches. Each switch simultaneously controls all lights whose numbers are multiples of the switch's number. Initially, all lights are on. Now, if switches numbered 2, 3, and 5 are pulled, how many lights will remain on? | {
"answer": "499",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( S = x^{2} + y^{2} - 2(x + y) \), where \( x \) and \( y \) satisfy \( \log_{2} x + \log_{2} y = 1 \), find the minimum value of \( S \). | {
"answer": "4 - 4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the polynomial \[ p(x) = x^4 - 3x^3 - 9x^2 + 27x - 8, \] where $x$ is a positive number such that $x^2 - 3x - 9 = 0$. | {
"answer": "\\frac{65 + 81\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Appending three digits at the end of 2007, one obtains an integer \(N\) of seven digits. In order to get \(N\) to be the minimal number which is divisible by 3, 5, and 7 simultaneously, what are the three digits that one would append? | {
"answer": "075",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a clock workshop, there are several digital clocks (more than one), displaying time in a 12-hour format (the number of hours on the clock screen ranges from 1 to 12). All clocks run at the same speed but show completely different times: the number of hours on the screen of any two different clocks is different, and the number of minutes as well.
One day, the master added up the number of hours on the screens of all available clocks, then added up the number of minutes on the screens of all available clocks, and remembered the two resulting numbers. After some time, he did the same thing again and found that both the total number of hours and the total number of minutes had decreased by 1. What is the maximum number of digital clocks that could be in the workshop? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum of $x^{2} y^{2} z$ under the condition that $x, y, z \geq 0$ and $2 x + 3 x y^{2} + 2 z = 36$. | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The military kitchen needs 1000 jin of rice and 200 jin of millet for dinner. Upon arriving at the rice store, the quartermaster finds a promotion: "Rice is 1 yuan per jin, with 1 jin of millet given for every 10 jin purchased (fractions of 10 jins do not count); Millet is 2 yuan per jin, with 2 jins of rice given for every 5 jin purchased (fractions of 5 jins do not count)." How much money does the quartermaster need to spend to buy enough rice and millet for dinner? | {
"answer": "1200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At 8:00 AM, Xiao Cheng and Xiao Chen set off from locations A and B respectively, heading towards each other. They meet on the way at 9:40 AM. Xiao Cheng says: "If I had walked 10 km more per hour, we would have met 10 minutes earlier." Xiao Chen says: "If I had set off half an hour earlier, we would have met 20 minutes earlier." If both of their statements are correct, how far apart are locations A and B? (Answer in kilometers). | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of \( x + y \), given that \( x^2 + y^2 - 3y - 1 = 0 \). | {
"answer": "\\frac{\\sqrt{26}+3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the 10 numbers $0, 1, 2, \cdots, 9$, select 3 such that their sum is an even number not less than 10. How many different ways are there to make such a selection? | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a, b \in R^{+}$ and $a + b = 1$, find the supremum of $- \frac{1}{2a} - \frac{2}{b}$. | {
"answer": "-\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the final of the giraffe beauty contest, two giraffes, Tall and Spotted, reached the finals. There are 135 voters divided into 5 districts, with each district divided into 9 precincts, and each precinct having 3 voters. The voters in each precinct choose the winner by majority vote; in a district, the giraffe that wins in the majority of precincts wins the district; finally, the giraffe that wins in the majority of the districts is declared the winner of the final. The giraffe Tall won. What is the minimum number of voters who could have voted for Tall? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the set $\{2, 7, 12, 17, 22, 27, 32\}$. Calculate the number of different integers that can be expressed as the sum of three distinct members of this set. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set \( P = \{1, 2, \ldots, 2014\} \) and \( A \subseteq P \). If the difference between any two numbers in the set \( A \) is not a multiple of 99, and the sum of any two numbers in the set \( A \) is also not a multiple of 99, then the set \( A \) can contain at most how many elements? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Snow White entered a room with a round table surrounded by 30 chairs. Some of the chairs were occupied by dwarfs. It turned out that Snow White couldn't sit in a way such that no one was sitting 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. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $a=2$ and $\cos C=-\frac{1}{4}$.
1. If $b=3$, find the value of $c$.
2. If $c=2\sqrt{6}$, find the value of $\sin B$. | {
"answer": "\\frac{\\sqrt{10}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Divide a square into 25 smaller squares, where 24 of these smaller squares are unit squares, and the remaining piece can also be divided into squares with a side length of 1. Find the area of the original square. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
30 students from five courses created 40 problems for the olympiad, with students from the same course creating the same number of problems, and students from different courses creating different numbers of problems. How many students created exactly one problem? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $-{1^{2022}}+{({\frac{1}{3}})^{-2}}+|{\sqrt{3}-2}|$. | {
"answer": "10-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a polynomial of degree 2022 with integer coefficients and a leading coefficient of 1, what is the maximum number of roots it can have within the interval \( (0,1) \)? | {
"answer": "2021",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the number $2016 * * * * 02 *$, you need to replace each of the 5 asterisks with any of the digits $0, 2, 4, 6, 7, 8$ (digits may repeat) so that the resulting 11-digit number is divisible by 6. How many ways can this be done? | {
"answer": "2160",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a $3 \times 4$ grid, you need to place 4 crosses so that there is at least one cross in each row and each column. How many ways are there to do this? | {
"answer": "36",
"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, and $4b\sin A = \sqrt{7}a$.
(1) Find the value of $\sin B$;
(2) If $a$, $b$, and $c$ form an arithmetic sequence with a positive common difference, find the value of $\cos A - \cos C$. | {
"answer": "\\frac{\\sqrt{7}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many three-digit numbers are there in which each digit is greater than the digit to its right? | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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