problem stringlengths 10 5.15k | answer dict |
|---|---|
A pyramid with a square base has all edges of 1 unit in length. What is the radius of the sphere that can be inscribed in the pyramid? | {
"answer": "\\frac{\\sqrt{6} - \\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The real function $g$ has the property that, whenever $x,$ $y,$ $m$ are positive integers such that $x + y = 3^m,$ the equation
\[g(x) + g(y) = 2m^2\]holds. What is $g(2187)$? | {
"answer": "98",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The year 2009 has the property that rearranging its digits never results in a smaller four-digit number (numbers do not start with zero). In which year will this property first repeat? | {
"answer": "2022",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In acute triangle $\triangle ABC$, $b=2$, $B=\frac{\pi }{3}$, $\sin 2A+\sin (A-C)-\sin B=0$, find the area of $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A weightless pulley has a rope with masses of 3 kg and 6 kg. Neglecting friction, find the force exerted by the pulley on its axis. Consider the acceleration due to gravity to be $10 \, \mathrm{m/s}^2$. Give the answer in newtons, rounding it to the nearest whole number if necessary. | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the limit of the function:
\[
\lim _{x \rightarrow 1}\left(\frac{x+1}{2 x}\right)^{\frac{\ln (x+2)}{\ln (2-x)}}
\] | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of triangle $PQR$ are in the ratio of $3:4:5$. Segment $QS$ is the angle bisector drawn to the shortest side, dividing it into segments $PS$ and $SR$. What is the length, in inches, of the longer subsegment of side $PR$ if the length of side $PR$ is $15$ inches? Express your answer as a common fraction. | {
"answer": "\\frac{60}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), the left and right foci coincide with the symmetric points about the two asymptotes, respectively. Then, the eccentricity of the hyperbola is __________. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the quadrilateral $ABCD$, it is known that $\angle BAC = \angle CAD = 60^\circ$ and $AB + AD = AC$. It is also given that $\angle ACD = 23^\circ$. What is the measure of angle $ABC$ in degrees? | {
"answer": "83",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Place 5 balls, numbered 1, 2, 3, 4, 5, into three different boxes, with two boxes each containing 2 balls and the other box containing 1 ball. How many different arrangements are there? (Answer with a number). | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of digits in the value of $2^{15} \times 5^{10} \times 3^2$. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(1990 = 2^{\alpha_{1}} + 2^{\alpha_{2}} + \cdots + 2^{\alpha_{n}}\), where \(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n}\) are distinct non-negative integers. Find \(\alpha_{1} + \alpha_{2} + \cdots + \alpha_{n}\). | {
"answer": "43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A 7' × 11' table sits in the corner of a square room. The table is to be rotated so that the side formerly 7' now lies along what was previously the end side of the longer dimension. Determine the smallest integer value of the side S of the room needed to accommodate this move. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $P$ is on the ellipse $\frac{x^{2}}{16}+ \frac{y^{2}}{9}=1$. The maximum and minimum distances from point $P$ to the line $3x-4y=24$ are $\_\_\_\_\_\_$. | {
"answer": "\\frac{12(2- \\sqrt{2})}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four scores belong to Alex and the other three to Morgan: 78, 82, 90, 95, 98, 102, 105. Alex's mean score is 91.5. What is Morgan's mean score? | {
"answer": "94.67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the inequality
$$
\log _{x^{2}+y^{2}}(x+y) \geqslant 1
$$
find the maximum value of \( y \) among all \( x \) and \( y \) that satisfy the inequality. | {
"answer": "\\frac{1}{2} + \\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A random variable \(X\) is given by the probability density function \(f(x) = \frac{1}{2} \sin x\) within the interval \((0, \pi)\); outside this interval, \(f(x) = 0\). Find the variance of the function \(Y = \varphi(X) = X^2\) using the probability density function \(g(y)\). | {
"answer": "\\frac{\\pi^4 - 16\\pi^2 + 80}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number whose digits sum up to 47. | {
"answer": "299999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two regular tetrahedra are inscribed in a cube in such a way that four vertices of the cube serve as the vertices of one tetrahedron, and the remaining four vertices of the cube serve as the vertices of the other. What fraction of the volume of the cube is occupied by the volume of the intersection of these tetrahedra? | {
"answer": "1/6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{1, 2, \cdots, 2005\} \). If any \( n \) pairwise coprime numbers in \( S \) always include at least one prime number, find the minimum value of \( n \). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If Yen has a 5 × 7 index card and reduces the length of the shorter side by 1 inch, the area becomes 24 square inches. Determine the area of the card if instead she reduces the length of the longer side by 2 inches. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Corners are sliced off from a cube of side length 2 so that all its six faces each become regular octagons. Find the total volume of the removed tetrahedra.
A) $\frac{80 - 56\sqrt{2}}{3}$
B) $\frac{80 - 48\sqrt{2}}{3}$
C) $\frac{72 - 48\sqrt{2}}{3}$
D) $\frac{60 - 42\sqrt{2}}{3}$ | {
"answer": "\\frac{80 - 56\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volume of a hemisphere is $\frac{500}{3}\pi$. What is the total surface area of the hemisphere including its base? Express your answer in terms of $\pi$. | {
"answer": "3\\pi \\times 250^{2/3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given $a+c=8$, $\cos B= \frac{1}{4}$.
(1) If $\overrightarrow{BA}\cdot \overrightarrow{BC}=4$, find the value of $b$;
(2) If $\sin A= \frac{\sqrt{6}}{4}$, find the value of $\sin C$. | {
"answer": "\\frac{3\\sqrt{6}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A telephone station serves 400 subscribers. For each subscriber, the probability of calling the station within an hour is 0.01. Find the probabilities of the following events: "within an hour, 5 subscribers will call the station"; "within an hour, no more than 4 subscribers will call the station"; "within an hour, at least 3 subscribers will call the station". | {
"answer": "0.7619",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right circular cone with a base radius of \(1 \, \text{cm}\) and a slant height of \(3 \, \text{cm}\), point \(P\) is on the circumference of the base. Determine the shortest distance from the vertex \(V\) of the cone to the shortest path from \(P\) back to \(P\). | {
"answer": "1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the two real roots of the equation (lgx)<sup>2</sup>\-lgx+lg2•lg5=0 with respect to x are m and n, then 2<sup>m+n</sup>\=\_\_\_\_\_\_. | {
"answer": "128",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terminal side of angle $\alpha$ passes through point $P(\sqrt{3}, m)$ ($m \neq 0$), and $\cos \alpha = \frac{m}{6}$, find the value of $\sin \alpha$ ___. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right, where each integer is squared. For example, $b_3 = 149$ (since $1^2 = 1$, $2^2 = 4$, $3^2 = 9$), and $b_5 = 1491625$. For $1 \le k \le 100$, determine how many $b_k$ are divisible by 4. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the $x O y$ coordinate plane, there is a Chinese chess "knight" at the origin $(0,0)$. The "knight" needs to be moved to the point $P(1991,1991)$ using the movement rules of the chess piece. What is the minimum number of moves required? | {
"answer": "1328",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle has dimensions 12 by 15, and a circle centered at one of its vertices has a radius of 15. What is the area of the union of the regions enclosed by the rectangle and the circle? Express your answer in terms of \( \pi \). | {
"answer": "180 + 168.75\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the 150th term of the sequence that consists of all those positive integers which are either powers of 3 or sums of distinct powers of 3. | {
"answer": "2280",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagonals AC and CE of the regular hexagon ABCDEF are divided by inner points M and N respectively, so that AM/AC = CN/CE = r. Determine r if B, M, and N are collinear. | {
"answer": "\\frac{1}{\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the minimum length of the second longest side of a triangle with an area of one unit? | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set
\[ S=\{1, 2, \cdots, 12\}, \quad A=\{a_{1}, a_{2}, a_{3}\} \]
where \( a_{1} < a_{2} < a_{3}, \quad a_{3} - a_{2} \leq 5, \quad A \subseteq S \). Find the number of sets \( A \) that satisfy these conditions. | {
"answer": "185",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the following infinite product: $3^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 27^{\frac{1}{27}} \cdot 81^{\frac{1}{81}} \dotsm.$ | {
"answer": "3^{\\frac{3}{4}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\frac{1}{3}x^{3}+ax^{2}+bx-\frac{2}{3}$, the equation of the tangent line at $x=2$ is $x+y-2=0$.
(I) Find the values of the real numbers $a$ and $b$.
(II) Find the extreme values of the function $f(x)$. | {
"answer": "-\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f(x) = \lg \frac{1 + x}{1 - x} \), if \( f\left(\frac{y + z}{1 + y z}\right) = 1 \) and \( f\left(\frac{y - z}{1 - y z}\right) = 2 \), where \( -1 < y, z < 1 \), find the value of \( f(y) \cdot f(z) \). | {
"answer": "-3/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The shortest distance from a point on the curve $f(x) = \ln(2x-1)$ to the line $2x - y + 3 = 0$ is what? | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of the product \(x^{2} y^{2} z^{2} u\) given the condition that \(x, y, z, u \geq 0\) and:
\[ 2x + xy + z + yz u = 1 \] | {
"answer": "1/512",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( M \) is the midpoint of side \( BC \) of the triangle \( ABC \), where \( AB = 17 \), \( AC = 30 \), and \( BC = 19 \). A circle is constructed with diameter \( AB \). A point \( X \) is chosen arbitrarily on this circle. What is the minimum possible length of the segment \( MX \)? | {
"answer": "6.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) with three points \(P\), \(Q\), and \(R\) on it, where \(P\) and \(Q\) are symmetric with respect to the origin. Find the maximum value of \(|RP| + |RQ|\). | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a parallelogram with $\angle ABC=135^\circ$, $AB=14$ and $BC=8$. Extend $\overline{CD}$ through $D$ to $E$ so that $DE=3$. If $\overline{BE}$ intersects $\overline{AD}$ at $F$, then find the length of segment $FD$.
A) $\frac{6}{17}$
B) $\frac{18}{17}$
C) $\frac{24}{17}$
D) $\frac{30}{17}$ | {
"answer": "\\frac{24}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a \star b = ab + a + b \) for all integers \( a \) and \( b \). Evaluate \( 1 \star (2 \star (3 \star (4 \star \ldots (99 \star 100) \ldots))) \). | {
"answer": "101! - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A basketball team consists of 12 players, including two pairs of twins, Alex and Brian, and Chloe and Diana. In how many ways can we choose a team of 5 players if no pair of twins can both be in the team lineup simultaneously? | {
"answer": "560",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $y=ax^{2}+bx+c$ ($a\neq 0$) with its axis of symmetry to the left of the $y$-axis, where $a$, $b$, $c \in \{-3,-2,-1,0,1,2,3\}$, let the random variable $X$ be the value of "$|a-b|$". Then, the expected value $EX$ is \_\_\_\_\_\_. | {
"answer": "\\dfrac {8}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f(x) = \cos x + \log_2 x \) for \( x > 0 \), if the positive real number \( a \) satisfies \( f(a) = f(2a) \), then find the value of \( f(2a) - f(4a) \). | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a positive number $x$, define $f(x)=\frac{2x}{x+1}$. Calculate: $f(\frac{1}{101})+f(\frac{1}{100})+f(\frac{1}{99})+\ldots +f(\frac{1}{3})+f(\frac{1}{2})+f(1)+f(2)+f(3)+\ldots +f(99)+f(100)+f(101)$. | {
"answer": "201",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x \neq y \), and the two sequences \( x, a_{1}, a_{2}, a_{3}, y \) and \( b_{1}, x, b_{2}, b_{3}, y, b_{4} \) are both arithmetic sequences. Then \(\frac{b_{4}-b_{3}}{a_{2}-a_{1}}\) equals $\qquad$. | {
"answer": "2.6666666666666665",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence \( x_{n} \), satisfying \( (n+1) x_{n+1}=x_{n}+n \), and \( x_{1}=2 \), find \( x_{2009} \). | {
"answer": "\\frac{2009! + 1}{2009!}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Factorize the expression $27x^6 - 512y^6$ and find the sum of all integer coefficients in its factorized form. | {
"answer": "92",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a peculiar four-digit number (with the first digit not being 0). It is a perfect square, and the sum of its digits is also a perfect square. Dividing this four-digit number by the sum of its digits results in yet another perfect square. Additionally, the number of divisors of this number equals the sum of its digits, which is a perfect square. If all the digits of this four-digit number are distinct, what is this four-digit number? | {
"answer": "2601",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given four points \( O, A, B, C \) on a plane, with \( OA=4 \), \( OB=3 \), \( OC=2 \), and \( \overrightarrow{OB} \cdot \overrightarrow{OC}=3 \), find the maximum area of triangle \( ABC \). | {
"answer": "2 \\sqrt{7} + \\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of triangle ABC, whose vertices have coordinates A(0,0), B(1424233,2848467), C(1424234,2848469). Round the answer to two decimal places. | {
"answer": "0.50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One of the angles in a triangle is $120^{\circ}$, and the lengths of the sides form an arithmetic progression. Find the ratio of the lengths of the sides of the triangle. | {
"answer": "3 : 5 : 7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(2,4)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is not above the $x$-axis? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A straight one-way city street has 8 consecutive traffic lights. Every light remains green for 1.5 minutes, yellow for 3 seconds, and red for 1.5 minutes. The lights are synchronized so that each light turns red 10 seconds after the preceding one turns red. What is the longest interval of time, in seconds, during which all 8 lights are green? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the maximum and minimum values of the function $y=2\sin \left( \frac{\pi x}{6}- \frac{\pi}{3}\right)$ where $(0\leqslant x\leqslant 9)$ is to be determined. | {
"answer": "2-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the complex numbers \( z_1 \) and \( z_2 \) such that \( \left| z_1 + z_2 \right| = 20 \) and \( \left| z_1^2 + z_2^2 \right| = 16 \), find the minimum value of \( \left| z_1^3 + z_2^3 \right| \). | {
"answer": "3520",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the limits:
1) \(\lim_{x \to 3}\left(\frac{1}{x-3}-\frac{6}{x^2-9}\right)\)
2) \(\lim_{x \to \infty}\left(\sqrt{x^2 + 1}-x\right)\)
3) \(\lim_{n \to \infty} 2^n \sin \frac{x}{2^n}\)
4) \(\lim_{x \to 1}(1-x) \tan \frac{\pi x}{2}\). | {
"answer": "\\frac{2}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To encourage residents to conserve water, a city charges residents for domestic water use in a tiered pricing system. The table below shows partial information on the tiered pricing for domestic water use for residents in the city, each with their own water meter:
| Water Sales Price | Sewage Treatment Price |
|-------------------|------------------------|
| Monthly Water Usage per Household | Unit Price: yuan/ton | Unit Price: yuan/ton |
| 17 tons or less | $a$ | $0.80$ |
| More than 17 tons but not more than 30 tons | $b$ | $0.80$ |
| More than 30 tons | $6.00$ | $0.80$ |
(Notes: 1. The amount of sewage generated by each household is equal to the amount of tap water used by that household; 2. Water bill = tap water cost + sewage treatment fee)
It is known that in April 2020, the Wang family used 15 tons of water and paid 45 yuan, and in May, they used 25 tons of water and paid 91 yuan.
(1) Find the values of $a$ and $b$;
(2) If the Wang family paid 150 yuan for water in June, how many tons of water did they use that month? | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of $453_6$, $436_6$, and $42_6$ in base 6. | {
"answer": "1415_6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many pairs of two-digit positive integers have a difference of 50? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of $3^{12} \cdot 3^3$ and express it as some integer raised to the third power. | {
"answer": "243",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the finals of a beauty contest among giraffes, there were two finalists: the Tall one and the Spotted one. There are 135 voters divided into 5 districts, each district is divided into 9 precincts, and each precinct has 3 voters. Voters in each precinct choose the winner by majority vote; in a district, the giraffe that wins in the majority of precincts is the winner; finally, the giraffe that wins in the majority of districts is declared the winner of the final. The Tall giraffe won. What is the minimum number of voters who could have voted for the Tall giraffe? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When two fair dice are thrown, the numbers obtained are $a$ and $b$, respectively. Express the probability that the slope $k$ of the line $bx+ay=1$ is greater than or equal to $-\dfrac{2}{5}$. | {
"answer": "\\dfrac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola C: x² = 2py (p > 0), draw a line l: y = 6x + 8, which intersects the parabola C at points A and B. Point O is the origin, and $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$. A moving circle P has its center on the parabola C and passes through a fixed point D(0, 4). If the moving circle P intersects the x-axis at points E and F, and |DE| < |DF|, find the minimum value of $\frac{|DE|}{|DF|}$. | {
"answer": "\\sqrt{2} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The real numbers \( x, y, z \) satisfy \( x + y + z = 2 \) and \( xy + yz + zx = 1 \). Find the maximum possible value of \( x - y \). | {
"answer": "\\frac{2 \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Digital clocks display hours and minutes (for example, 16:15). While practicing arithmetic, Buratino finds the sum of the digits on the clock $(1+6+1+5=13)$. Write down such a time of day when the sum of the digits on the clock will be the greatest. | {
"answer": "19:59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given three points $A$, $B$, $C$ on a straight line in the Cartesian coordinate system, satisfying $\overrightarrow{OA}=(-3,m+1)$, $\overrightarrow{OB}=(n,3)$, $\overrightarrow{OC}=(7,4)$, and $\overrightarrow{OA} \perp \overrightarrow{OB}$, where $O$ is the origin.
$(1)$ Find the values of the real numbers $m$, $n$;
$(2)$ Let $G$ be the centroid of $\triangle AOC$, and $\overrightarrow{OG}= \frac{2}{3} \overrightarrow{OB}$, find the value of $\cos \angle AOC$. | {
"answer": "-\\frac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( x \) is a multiple of \( 7200 \), what is the greatest common divisor of \( f(x)=(5x+6)(8x+3)(11x+9)(4x+12) \) and \( x \)? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the sequence \(\left\{a_{n}\right\}\), it is known that \(a_{1}=1\) and \(a_{n+1}>a_{n}\), and that \(a_{n+1}^{2}+a_{n}^{2}+1=2\left(a_{n+1}+a_{n}+2 a_{n+1} a_{n}\right)\). Find \(\lim \limits_{n \rightarrow \infty} \frac{S_{n}}{n a_{n}}\). | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify $\sqrt{18} \times \sqrt{32} \times \sqrt{2}$. | {
"answer": "24\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 19 weights with values $1, 2, 3, \ldots, 19$ grams: nine iron, nine bronze, and one gold. It is known that the total weight of all the iron weights is 90 grams more than the total weight of the bronze weights. Find the weight of the gold weight. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $A, B, C, D, E, F$ are the vertices of a regular hexagon with a side length of 2, and a parabola passes through the points $A, B, C, D$, find the distance from the focus of the parabola to its directrix. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, with angles A, B, and C, the sides opposite these angles are labeled as $a, b,$ and $c$ respectively. Given the vectors $\overrightarrow{m}=\left( \frac{a}{2}, \frac{c}{2} \right)$ and $\overrightarrow{n}=(\cos C, \cos A)$, it is also known that $\overrightarrow{n} \cdot \overrightarrow{m} = b\cos B$.
1. Determine the value of angle B;
2. If $\cos \left( \frac{A-C}{2} \right) = \sqrt{3}\sin A$ and $|\overrightarrow{m}|=\sqrt{5}$, find the area of $\triangle ABC$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $D$ be the circle with equation $x^2 - 10y - 7 = -y^2 - 8x + 4$. Find the center $(a, b)$ and radius $r$ of $D$, and determine the value of $a + b + r$. | {
"answer": "1 + 2\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(\lg 2 = 0.30103\), calculate the number of digits in \( M = 1 + 10^4 + \frac{10^4 (10^4 - 1)}{1 \cdot 2} + \frac{10^4 (10^4-1)(10^4-2)}{1 \cdot 2 \cdot 3} + \cdots + \frac{10^4 (10^4 - 1)}{1 \cdot 2} + 10^4 + 1\). | {
"answer": "3011",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify \(\left(\cos 42^{\circ}+\cos 102^{\circ}+\cos 114^{\circ}+\cos 174^{\circ}\right)^{2}\) into a rational number. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line passing through the point $(0,-2)$ intersects the parabola $y^{2}=16x$ at two points $A(x_1,y_1)$ and $B(x_2,y_2)$, with $y_1^2-y_2^2=1$. Calculate the area of the triangle $\triangle OAB$, where $O$ is the origin. | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are very many symmetrical dice. They are thrown simultaneously. With a certain probability \( p > 0 \), it is possible to get a sum of 2022 points. What is the smallest sum of points that can fall with the same probability \( p \)? | {
"answer": "337",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( OP \) be the diameter of the circle \( \Omega \), and \( \omega \) be a circle with center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). The chord \( OB \) of the circle \( \Omega \) intersects the second circle at point \( A \). Find the length of segment \( AB \) if \( BD \cdot BC = 5 \). | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest 6-digit palindrome in base 2, that can be expressed as a 4-digit palindrome in a different base. Provide your response in base 2. | {
"answer": "100001_2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many triangles are there with all sides being integers and the longest side being 11? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=4\sin x\sin \left(x+ \frac {\pi}{3}\right)-1$.
$(1)$ Calculate the value of $f\left( \frac {5\pi}{6}\right)$:
$(2)$ Let $A$ be the smallest angle in $\triangle ABC$, and $f(A)= \frac {8}{5}$, find the value of $f\left(A+ \frac {\pi}{4}\right)$. | {
"answer": "\\frac {6}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find \( k \) such that, for all \( n \), the following expression is a perfect square:
$$
4 n^{2} + k n + 9
$$ | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x) = \sqrt{-x^2 + 5x + 6}$.
$(1)$ Find the domain of $f(x)$.
$(2)$ Determine the intervals where $f(x)$ is increasing or decreasing.
$(3)$ Find the maximum and minimum values of $f(x)$ on the interval $[1,5]$. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x^{2} + y^{2} - 2x - 2y + 1 = 0 \) where \( x, y \in \mathbb{R} \), find the minimum value of \( F(x, y) = \frac{x + 1}{y} \). | {
"answer": "3/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express $0.3\overline{206}$ as a common fraction. | {
"answer": "\\frac{5057}{9990}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the following four propositions:
(1) Two lines parallel to the same plane are parallel to each other.
(2) Two lines perpendicular to the same line are parallel to each other.
(3) Through a point outside a known plane, there exists exactly one plane parallel to the given plane.
(4) Through a line outside a known plane, a plane can always be constructed parallel to the given plane.
The sequence numbers of the true propositions are ____. (List all the numbers of the true propositions.) | {
"answer": "(3)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
150 people were surveyed and asked: "Do you think teal is more green or blue?" Of them, 90 believe teal is "more green," and 50 believe it's "more blue." Additionally, 40 believe it's both "more green" and "more blue." Another 20 think teal is neither "more green" nor "more blue."
How many of those 150 people believe that teal is "more blue"? | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $y^{2}=2px(p > 0)$ and the hyperbola $\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a > 0,b > 0)$ have the same focus $F$, and point $A$ is an intersection point of the two curves, and $AF$ is perpendicular to the x-axis, calculate the eccentricity of the hyperbola. | {
"answer": "\\sqrt {2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Gill leaves Lille at 09:00, the train travels the first 27 km at 96 km/h and then stops at Lens for 3 minutes before traveling the final 29 km to Lillers at 96 km/h, calculate the arrival time at Lillers. | {
"answer": "09:38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among five numbers, if we take the average of any four numbers and add the remaining number, the sums will be 74, 80, 98, 116, and 128, respectively. By how much is the smallest number less than the largest number among these five numbers? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(\alpha, \beta \in \left(\frac{3\pi}{4}, \pi \right)\), \(\cos (\alpha + \beta) = \frac{4}{5}\), and \(\sin \left(\alpha - \frac{\pi}{4}\right) = \frac{12}{13}\), find \(\cos \left(\beta + \frac{\pi}{4}\right)\). | {
"answer": "-\\frac{56}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the curve $C$ is given by the parametric equations $ \begin{cases} x=3\cos \alpha \\ y=\sin \alpha \end{cases} $ (where $\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive x-axis as the polar axis, the polar equation of line $l$ is $\rho\sin (\theta- \frac {\pi}{4})= \sqrt {2}$.
$(1)$ Find the implicit equation for $C$ and the inclination angle of $l$.
$(2)$ Let point $P(0,2)$ be given, and line $l$ intersects curve $C$ at points $A$ and $B$. Find the $|PA|+|PB|$. | {
"answer": "\\frac {18 \\sqrt {2}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The $10\times15$ rectangle $EFGH$ is cut into two congruent pentagons, which are repositioned to form a square. Determine the length $z$ of one side of the pentagons that aligns with one side of the square.
A) $5\sqrt{2}$
B) $5\sqrt{3}$
C) $10\sqrt{2}$
D) $10\sqrt{3}$ | {
"answer": "5\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a particular game, each of $4$ players rolls a standard $8$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again continuing until one player wins. Hugo is one of the players. What is the probability that Hugo's first roll was a $7$, given that he won the game?
A) $\frac{25}{256}$
B) $\frac{27}{128}$
C) $\frac{13}{64}$
D) $\frac{17}{96}$ | {
"answer": "\\frac{27}{128}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set \( S = \left\{ z \mid |z - 7 - 8i| = |z_1^4 + 1 - 2z_1^2| ; z, z_1 \in \mathbb{C}, |z_1| = 1 \right\} \), find the area of the region corresponding to \( S \) in the complex plane. | {
"answer": "16\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the book "Nine Chapters on the Mathematical Art," a tetrahedron with all four faces being right-angled triangles is called a "biēnào." Given that tetrahedron $ABCD$ is a "biēnào," $AB\bot $ plane $BCD$, $BC\bot CD$, and $AB=\frac{1}{2}BC=\frac{1}{3}CD$. If the volume of this tetrahedron is $1$, then the surface area of its circumscribed sphere is ______. | {
"answer": "14\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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