problem stringlengths 10 5.15k | answer dict |
|---|---|
If two different properties are randomly selected from the five types of properties (metal, wood, water, fire, and earth) where metal overcomes wood, wood overcomes earth, earth overcomes water, water overcomes fire, and fire overcomes metal, determine the probability that the two selected properties do not overcome each other. | {
"answer": "\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(\alpha) = \frac {\sin(\pi-\alpha)\cos(2\pi-\alpha)\tan(-\alpha+\pi)}{-\tan(-\alpha -\pi )\cos( \frac {\pi}{2}-\alpha )}$:
1. Simplify $f(\alpha)$.
2. If $\alpha$ is an angle in the third quadrant and $\cos(\alpha- \frac {3\pi}{2}) = \frac {1}{5}$, find the value of $f(\alpha)$. | {
"answer": "\\frac {2\\sqrt{6}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of $\sin 68^{\circ} \sin 67^{\circ} - \sin 23^{\circ} \cos 68^{\circ}$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A four-meter gas pipe has rusted in two places. Determine the probability that all three resulting pieces can be used as connections to gas stoves, given that regulations require a stove to be at least 1 meter away from the main gas pipeline. | {
"answer": "1/8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The point $P$ $(4,5)$ is reflected over the $y$-axis to $Q$. Then $Q$ is reflected over the line $y=-x$ to $R$. What is the area of triangle $PQR$? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle \(ABCD\), \(E\) and \(F\) are chosen on \(\overline{AB}\) and \(\overline{CD}\), respectively, so that \(AEFD\) is a square. If \(\frac{AB}{BE} = \frac{BE}{BC}\), determine the value of \(\frac{AB}{BC}\). | {
"answer": "\\frac{3 + \\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Lucas makes a batch of lemonade using 200 grams of lemon juice, 100 grams of sugar, and 300 grams of water. If there are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar, and water has no calories, determine the total number of calories in 200 grams of this lemonade. | {
"answer": "145",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x, y, z, u, v \in \mathbf{R}_{+} \). Determine the maximum value of \( f = \frac{xy + yz + zu + uv}{2x^2 + y^2 + 2z^2 + u^2 + 2v^2} \). | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sin(\omega x+\varphi)$, where $(\omega > 0, |\varphi| < \frac{\pi}{2})$, the graph passes through the point $B(0,-1)$, and is monotonically increasing on the interval $\left(\frac{\pi}{18}, \frac{\pi}{3}\right)$. Additionally, the graph of $f(x)$ coincides with its original graph after being shifted to the left by $\pi$ units. If $x_{1}, x_{2} \in \left(-\frac{17\pi}{12}, -\frac{2\pi}{3}\right)$ and $x_{1} \neq x_{2}$, and $f(x_{1}) = f(x_{2})$, calculate $f(x_{1}+x_{2})$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the angle of inclination of the tangent line to the curve $y=\frac{1}{3}x^3-5$ at the point $(1,-\frac{3}{2})$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin \alpha$ is a root of the equation $5x^{2}-7x-6=0$, find:
$(1)$ The value of $\frac {\cos (2\pi-\alpha)\cos (\pi+\alpha)\tan ^{2}(2\pi-\alpha)}{\cos ( \frac {\pi}{2}+\alpha)\sin (2\pi-\alpha)\cot ^{2}(\pi-\alpha)}$.
$(2)$ In $\triangle ABC$, $\sin A+ \cos A= \frac { \sqrt {2}}{2}$, $AC=2$, $AB=3$, find the value of $\tan A$. | {
"answer": "-2- \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider an arithmetic sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$. Given that $a_1=9$, $a_2$ is an integer, and $S_n \leq S_5$, find the sum of the first 9 terms of the sequence $\{\frac{1}{a_n a_{n+1}}\}$. | {
"answer": "-\\frac{1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cone with a base radius of $1$ and a height of $\sqrt{3}$, both the apex of the cone and the base circle are on the surface of a sphere $O$, calculate the surface area of this sphere. | {
"answer": "\\frac{16\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence \(\left\{a_{n}\right\}\) with the general term
\[ a_{n} = n^{4} + 6n^{3} + 11n^{2} + 6n, \]
find the sum of the first 12 terms \( S_{12} \). | {
"answer": "104832",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $f(x), g(x), h(x), p(x)$ are linear equations where $f(x) = x + 1$, $g(x) = -x + 5$, $h(x) = 4$, and $p(x) = 1$. Define new functions $j(x)$ and $k(x)$ as follows:
$$j(x) = \max\{f(x), g(x), h(x), p(x)\},$$
$$k(x)= \min\{f(x), g(x), h(x), p(x)\}.$$
Find the length squared, $\ell^2$, of the graph of $y=k(x)$ from $x = -4$ to $x = 4$. | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a \) and \( b \) be positive real numbers. Given that \(\frac{1}{a} + \frac{1}{b} \leq 2\sqrt{2}\) and \((a - b)^2 = 4(ab)^3\), find \(\log_a b\). | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\alpha$ and $\beta$ be real numbers. Find the minimum value of
\[(3 \cos \alpha + 4 \sin \beta - 10)^2 + (3 \sin \alpha + 4 \cos \beta - 12)^2.\] | {
"answer": "(\\sqrt{244} - 7)^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $2006$ students and $14$ teachers in a school. Each student knows at least one teacher (knowing is a symmetric relation). Suppose that, for each pair of a student and a teacher who know each other, the ratio of the number of the students whom the teacher knows to that of the teachers whom the student knows is at least $t.$ Find the maximum possible value of $t.$ | {
"answer": "143",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the total distance on a Cartesian coordinate plane starting at $(2, -3)$, going to $(8, 9)$, and then moving to $(3, 2)?$ | {
"answer": "6\\sqrt{5} + \\sqrt{74}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
John's flight departed from Chicago at 3:15 PM and landed in Denver at 4:50 PM. Considering that Denver is one hour behind Chicago, if his flight took $h$ hours and $m$ minutes, with $0 < m < 60$, calculate the value of $h + m$. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person rolls a die twice, obtaining the numbers $m$ and $n$, which are used as the coefficients of a quadratic equation $x^2 + mx + n = 0$. The probability that the equation has real roots is ______. | {
"answer": "\\dfrac{19}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From three red balls numbered $1$, $2$, $3$ and two white balls numbered $2$, $3$, find the probability that two balls drawn at random have different numbers and colors. | {
"answer": "\\dfrac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle{ABC}, AB=13, \angle{A}=45^\circ$, and $\angle{C}=30^\circ$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\perp{BC}$, $\angle{BAD}=\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such that $PN\perp{BC}$. Find $AP^2$ expressed as a reduced fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and determine $m+n$. | {
"answer": "171",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terminal side of angle $\alpha$ passes through point $P(m, 2\sqrt{2})$, $\sin{\alpha} = \frac{2\sqrt{2}}{3}$, and $\alpha$ is in the second quadrant.
(1) Find the value of $m$;
(2) If $\tan{\beta} = \sqrt{2}$, find the value of $\frac{\sin{\alpha}\cos{\beta} + 3\sin({\frac{\pi}{2} + \alpha})\sin{\beta}}{\cos{(\pi + \alpha)}\cos{(-\beta)} - 3\sin{\alpha}\sin{\beta}}$. | {
"answer": "\\frac{\\sqrt{2}}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The first term of a sequence is \( a_{1} = 1 \), and each subsequent term is defined by
\[ a_{n+1} = 1 + \frac{n}{a_{n}}, \quad n = 1, 2, 3, \ldots \]
Does the following limit exist? If it exists, determine it.
\[ \lim_{n \rightarrow \infty} \left(a_{n} - \sqrt{n}\right) \] | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube with a side length of 1 meter was cut into smaller cubes with a side length of 1 centimeter and arranged in a straight line. What is the length of the resulting line? | {
"answer": "10000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Use the angle addition formula for cosine to simplify the expression $\cos 54^{\circ}\cos 24^{\circ}+2\sin 12^{\circ}\cos 12^{\circ}\sin 126^{\circ}$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of the circle inscribed in a right triangle if the projections of the legs onto the hypotenuse are 9 meters and 16 meters, respectively. | {
"answer": "25 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For which smallest \( n \) do there exist \( n \) numbers in the interval \( (-1, 1) \) such that their sum is 0, and the sum of their squares is 36? | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\(ABCD\) is a square-based pyramid with base \(ABCD\) and apex \(E\). Point \(E\) is directly above point \(A\), with \(AE = 1024\) units and \(AB = 640\) units. The pyramid is sliced into two parts by a horizontal plane parallel to the base \(ABCD\), at a height \(h\) above the base. The portion of the pyramid above the plane forms a new smaller pyramid. For how many integer values of \(h\) does the volume of this new pyramid become an integer? | {
"answer": "85",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S_1, S_2, \ldots, S_{10} \) be the first ten terms of an arithmetic progression (A.P.) consisting of positive integers. If \( S_1 + S_2 + \ldots + S_{10} = 55 \) and \( \left(S_{10} - S_{8}\right) + \left(S_{9} - S_{7}\right) + \ldots + \left(S_{3} - S_{1}\right) = d \), find \( d \). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The hyperbola C: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) has one focus that is also the focus of the parabola $y^{2} = 4x$. Line $l$ is an asymptote of $C$ and intersects the circle $(x-1)^{2} + y^{2} = a^{2}$ at points $A$ and $B$. If $|AB| = b$, determine the eccentricity of the hyperbola $C$. | {
"answer": "\\dfrac{3\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Currently, 7 students are to be assigned to participate in 5 sports events, with the conditions that students A and B cannot participate in the same event, each event must have at least one participant, and each student can only participate in one event. How many different ways can these conditions be satisfied? (Answer with a number) | {
"answer": "15000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ a,\ b$ be the real numbers such that $ 0\leq a\leq b\leq 1$ . Find the minimum value of $ \int_0^1 |(x\minus{}a)(x\minus{}b)|\ dx$ . | {
"answer": "1/12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find a positive integer whose last digit is 2, and if this digit is moved from the end to the beginning, the new number is twice the original number. | {
"answer": "105263157894736842",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A classroom is paved with cubic bricks that have an edge length of 0.3 meters, requiring 600 bricks. If changed to cubic bricks with an edge length of 0.5 meters, how many bricks are needed? (Solve using proportions.) | {
"answer": "216",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are three saline solutions with concentrations of 5%, 8%, and 9%, labeled A, B, and C, weighing 60g, 60g, and 47g respectively. We need to prepare 100g of a saline solution with a concentration of 7%. What is the maximum and minimum amount of solution A (5% concentration) that can be used? Please write down the sum of these two numbers as the answer. | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
15 balls numbered 1 through 15 are placed in a bin. Joe produces a list of four numbers by performing the following sequence four times: he chooses a ball, records the number, and places the ball back in the bin. Finally, Joe chooses to make a unique list by selecting 3 numbers from these 4, and forgetting the order in which they were drawn. How many different lists are possible? | {
"answer": "202500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α, β ∈ (0, \frac{π}{2})$, and $\frac{\sin β}{\sin α} = \cos(α + β)$,
(1) If $α = \frac{π}{6}$, then $\tan β =$ _______;
(2) The maximum value of $\tan β$ is _______. | {
"answer": "\\frac{\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two arithmetic sequences $ \{ a_n \} $ and $ \{ b_n \} $ with the sum of the first $ n $ terms denoted as $ S_n $ and $ T_n $ respectively, and satisfying $ \frac {S_{n}}{T_{n}} = \frac {7n+2}{n+3} $, find the ratio $ \frac {a_{5}}{b_{5}} $. | {
"answer": "\\frac {65}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. If $2a_n + (-1)^n \cdot a_n = 2^n + (-1)^n \cdot 2^n$ ($n \in \mathbb{N}^*$), then $S_{10}=$ \_\_\_\_\_\_. | {
"answer": "\\dfrac{2728}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The positive integers \( x \) and \( y \), for which \( \gcd(x, y) = 3 \), are the coordinates of the vertex of a square centered at the origin with an area of \( 20 \cdot \operatorname{lcm}(x, y) \). Find the perimeter of the square. | {
"answer": "24\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Comprehensive exploration: When two algebraic expressions containing square roots are multiplied together and the product does not contain square roots, we call these two expressions rationalizing factors of each other. For example, $\sqrt{2}+1$ and $\sqrt{2}-1$, $2\sqrt{3}+3\sqrt{5}$ and $2\sqrt{3}-3\sqrt{5}$ are all rationalizing factors of each other. When performing calculations involving square roots, using rationalizing factors can eliminate square roots in the denominator. For example: $\frac{1}{\sqrt{2}+1}=\frac{1\times(\sqrt{2}-1)}{(\sqrt{2}+1)(\sqrt{2}-1)}=\sqrt{2}-1$; $\frac{1}{\sqrt{3}+\sqrt{2}}=\frac{1\times(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}=\sqrt{3}-\sqrt{2}$. Based on the above information, answer the following questions:
$(1)$ $\sqrt{2023}-\sqrt{2022}$ and ______ are rationalizing factors of each other;
$(2)$ Please guess $\frac{1}{\sqrt{n+1}+\sqrt{n}}=\_\_\_\_\_\_$; ($n$ is a positive integer)
$(3)$ $\sqrt{2023}-\sqrt{2022}$ ______ $\sqrt{2022}-\sqrt{2021}$ (fill in "$>$", "$<$", or "$=$");
$(4)$ Calculate: $(\frac{1}{\sqrt{3}+1}+\frac{1}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{7}+\sqrt{5}}+\ldots +\frac{1}{\sqrt{2023}+\sqrt{2021}})\times (\sqrt{2023}+1)$. | {
"answer": "1011",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence of 2020 natural numbers is written in a row. Each of them, starting from the third number, is divisible by the previous one and by the sum of the two preceding ones.
What is the smallest possible value for the last number in the sequence? | {
"answer": "2019!",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Integers from 1 to 100 are placed in a row in some order. Let us call a number *large-right*, if it is greater than each number to the right of it; let us call a number *large-left*, is it is greater than each number to the left of it. It appears that in the row there are exactly $k$ large-right numbers and exactly $k$ large-left numbers. Find the maximal possible value of $k$ . | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, where $AB = 6$ and $AC = 10$. Let $M$ be a point on $BC$ such that $BM : MC = 2:3$. If $AM = 5$, what is the length of $BC$?
A) $7\sqrt{2.2}$
B) $5\sqrt{6.1}$
C) $10\sqrt{3.05}$
D) $15 - 3\sqrt{6.1}$ | {
"answer": "5\\sqrt{6.1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two balls, a red one and a green one, are randomly and independently tossed into bins numbered with positive integers. For each ball, the probability that it is tossed into bin $k$ is given by $p_k = \frac{1}{k(k+1)}$ for $k = 1, 2, 3, ...$. Calculate the probability that the red ball is tossed into an odd-numbered bin and the green ball into an even-numbered bin.
A) $\frac{1}{6}$
B) $\frac{1}{5}$
C) $\frac{1}{4}$
D) $\frac{1}{3}$
E) $\frac{1}{2}$ | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sphere, a cylinder with a square axial section, and a cone. The cylinder and cone have identical bases, and their heights equal the diameter of the sphere. How do the volumes of the cylinder, sphere, and cone compare to each other? | {
"answer": "3 : 2 : 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The organizers of a ping-pong tournament have only one table. They call two participants to play, who have not yet played against each other. If after the game the losing participant suffers their second defeat, they are eliminated from the tournament (since there are no ties in tennis). After 29 games, it turned out that all participants were eliminated except for two. How many participants were there in the tournament? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In \( \triangle ABC \), if \( |\overrightarrow{AB}| = 2 \), \( |\overrightarrow{BC}| = 3 \), and \( |\overrightarrow{CA}| = 4 \), find the value of \( \overrightarrow{AB} \cdot \overrightarrow{BC} + \overrightarrow{BC} \cdot \overrightarrow{CA} + \overrightarrow{CA} \cdot \overrightarrow{AB} \). | {
"answer": "-\\frac{29}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a computer keyboard, the key for the digit 1 is not working. For example, if you try to type the number 1231234, only the number 23234 will actually print.
Sasha tried to type an 8-digit number, but only 202020 was printed. How many 8-digit numbers satisfy this condition? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the ceiling is 3 meters above the floor, the light bulb is 15 centimeters below the ceiling, Alice is 1.6 meters tall and can reach 50 centimeters above the top of her head, and a 5 centimeter thick book is placed on top of a stool to reach the light bulb, find the height of the stool in centimeters. | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the three interior angles $A$, $B$, $C$ of $\triangle ABC$ form an arithmetic sequence, and the side $b$ opposite to angle $B$ equals $\sqrt{3}$, and the function $f(x)=2 \sqrt{3}\sin ^{2}x+2\sin x\cos x- \sqrt{3}$ reaches its maximum value at $x=A$, then the area of $\triangle ABC$ is __________. | {
"answer": "\\frac{3+ \\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the number of parcels received by a person in the months from January to May are $1$, $3$, $2$, $2$, $2$ respectively, find the variance ($s^{2}=$ ___) of these $5$ numbers. | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $sin({α+\frac{π}{4}})=\frac{{12}}{{13}}$, and $\frac{π}{4}＜α＜\frac{{3π}}{4}$, find the value of $\cos \alpha$____. | {
"answer": "\\frac{7\\sqrt{2}}{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Ms. Johnson's class, each student averages two days absent out of thirty school days. What is the probability that out of any three students chosen at random, exactly two students will be absent and one will be present on a Monday, given that on Mondays the absence rate increases by 10%? Express your answer as a percent rounded to the nearest tenth. | {
"answer": "1.5\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular decagon $ABCDEFGHIJ$, points $K$, $L$, $M$, $N$, $O$, $P$, $Q$, $R$, and $S$ are selected on the sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DE}$, $\overline{EF}$, $\overline{FG}$, $\overline{GH}$, $\overline{HI}$, and $\overline{IJ}$ respectively. Each of these points divides their respective sides into segments with a ratio of 3:1, starting from the vertices $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, $I$, $J$. Lines connected from alternate points as $KMOQSU$ create a new hexagon inside the decagon. What is the ratio of the area of hexagon $KMOQSU$ to the area of decagon $ABCDEFGHIJ$?
A) $\frac{1}{4}$
B) $\frac{3\sqrt{3}}{40}$
C) $\frac{9\sqrt{3}}{40}$
D) $\frac{1}{10}$ | {
"answer": "\\frac{3\\sqrt{3}}{40}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The minimum value of the function \( y = |\cos x| + |\cos 2x| \) (for \( x \in \mathbf{R} \)) is ______. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When numbers are represented in "base fourteen", the digit before it becomes full is fourteen. If in "base fourteen", the fourteen digits are sequentially noted as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ten, J, Q, K, then convert the three-digit number JQK in "base fourteen" into a "binary" number and determine the number of digits. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a rectangular table of size \( x \) cm \(\times 80\) cm, identical sheets of paper of size 5 cm \(\times 8\) cm are placed. The first sheet is placed in the bottom left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet is placed in the top right corner. What is the length \( x \) in centimeters? | {
"answer": "77",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The hypotenuse of a right triangle measures $9$ inches, and one angle is $30^{\circ}$. What is the number of square inches in the area of the triangle? | {
"answer": "10.125\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer \( n \) such that
\[
\sqrt{5 n}-\sqrt{5 n-4}<0.01
\] | {
"answer": "8001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 500 machines, each using 6 parts of the same type. These parts must be replaced by the end of the week if they fail. Out of all new parts, 10% fail by the end of the first week, 30% fail by the end of the second week, and the remaining 60% fail by the end of the third week. No part lasts beyond the third week. How many new parts need to be replaced by the end of the third week? | {
"answer": "3000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five people are crowding into a booth against a wall at a noisy restaurant. If at most three can fit on one side, how many seating arrangements accommodate them all? | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos(\pi+\theta) = -\frac{1}{2}$, find the value of $\tan(\theta - 9\pi)$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cut a cube into two cuboids. If the ratio of their surface areas is 1:2, what is the ratio of their volumes? | {
"answer": "1:5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle \(ABC\) with the midpoints of sides \(BC\), \(AC\), and \(AB\) denoted as \(D\), \(E\), and \(F\) respectively, it is known that the medians \(AD\) and \(BE\) are perpendicular to each other, with lengths \(\overline{AD} = 18\) and \(\overline{BE} = 13.5\). Calculate the length of the third median \(CF\) of this triangle. | {
"answer": "22.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( m > n \geqslant 1 \), find the smallest value of \( m + n \) such that
\[ 1000 \mid 1978^{m} - 1978^{n} . \ | {
"answer": "106",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A dog and a cat simultaneously grab a sausage from different ends with their teeth. If the dog bites off its piece and runs away, the cat will get 300 grams more than the dog. If the cat bites off its piece and runs away, the dog will get 500 grams more than the cat. How much sausage is left if both bite off their pieces and run away? | {
"answer": "400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\alpha$ be an acute angle. If $\sin\left(\alpha - \frac{\pi}{4}\right) = \frac{1}{3}$, then $\cos2\alpha = \_\_\_\_\_\_$. | {
"answer": "-\\frac{4\\sqrt{2}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many 9-digit numbers, divisible by 2, can be formed by permuting the digits of the number 231157152? | {
"answer": "3360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
You are given the numbers 1, 2, 3, 4, 5, 6, 7, 8 to be placed at the eight vertices of a cube, such that the sum of any three numbers on each face of the cube is at least 10. Find the minimum possible sum of the four numbers on any face. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
12 soccer teams participate in a round-robin tournament. Each pair of teams plays a match, with the winning team earning 2 points and the losing team earning 0 points. If the match results in a draw, each team earns 1 point. Three referees recorded the total points for all teams and obtained three different results: 3,086; 2,018; and 1,238. They calculated the averages of each pair of these numbers and found that the averages were still different. Then, they calculated the average of the initial three numbers. If one of these seven numbers is correct, what is the number of teams \( n \)? | {
"answer": "47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regular octagon \( CH I L D R E N \) has area 1. Determine the area of quadrilateral \( L I N E \). | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a hyperbola $mx^{2}+ny^{2}=1$ has a focus that is the same as the focus of the parabola $y=\frac{1}{8}{x^2}$ and an eccentricity of $2$, calculate the distance from the focus of the parabola to one of the asymptotes of the hyperbola. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find four distinct positive integers $a, b, c, d$ less than $15$ which are invertible modulo $15$. Calculate the remainder when $(abc + abd + acd + bcd)(abcd)^{-1}$ is divided by $15$. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a psychiatric hospital, there is a chief doctor and many madmen. During the week, each madman bit someone once a day (possibly themselves). At the end of the week, it was found that each patient has two bites, and the chief doctor has one hundred bites. How many madmen are there in the hospital? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given sets \( A = \{ x \mid 5x - a \leq 0 \} \) and \( B = \{ x \mid 6x - b > 0 \} \), where \( a, b \in \mathbf{N} \), and \( A \cap B \cap \mathbf{N} = \{ 2, 3, 4 \} \), determine the number of integer pairs \((a, b)\). | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In tetrahedron \( ABCD \), the dihedral angle between face \( ABC \) and face \( BCD \) is \( 60^\circ \). Vertex \( A \)'s projection onto face \( BCD \) is point \( H \), which is the orthocenter of \( \triangle BCD \). Point \( G \) is the centroid of \( \triangle ABC \). Given that \( AH = 4 \) and \( AB = AC \), find \( GH \). | {
"answer": "\\frac{4\\sqrt{21}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Train 109 T departs from Beijing at 19:33 and arrives in Shanghai the next day at 10:26; train 1461 departs from Beijing at 11:58 and arrives in Shanghai the next day at 8:01. How many minutes are the running times of these two trains different? | {
"answer": "310",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence is formed by arranging the following arrays into a row: $$( \frac {1}{1}), ( \frac {2}{1}, \frac {1}{2}), ( \frac {3}{1}, \frac {2}{2}, \frac {1}{3}), ( \frac {4}{1}, \frac {3}{2}, \frac {2}{3}, \frac {1}{4}), ( \frac {5}{1}, \frac {4}{2}, \frac {3}{3}, \frac {2}{4}, \frac {1}{5}),…$$ If we remove the parentheses from the arrays, we get a sequence: $$\frac {1}{1}, \frac {2}{1}, \frac {1}{2}, \frac {3}{1}, \frac {2}{2}, \frac {1}{3}, \frac {4}{1}, \frac {3}{2}, \frac {2}{3}, \frac {1}{4}, \frac {5}{1}, \frac {4}{2}, \frac {3}{3}, \frac {2}{4}, \frac {1}{5},…$$ A student observed that $$\frac {63×64}{2}=2016,$$ based on this, the 2012th term of the sequence is ______. | {
"answer": "\\frac {5}{59}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of a triangle if it is known that its medians \(CM\) and \(BN\) are 6 and 4.5 respectively, and \(\angle BKM = 45^\circ\), where \(K\) is the point of intersection of the medians. | {
"answer": "9\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(1,-1)$ and $\overrightarrow{b}=(2,-1)$, the projection of $\overrightarrow{a}+\overrightarrow{b}$ in the direction of $\overrightarrow{a}$ is ______. | {
"answer": "\\frac{5\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the greatest integer less than or equal to \[\frac{5^{50} + 3^{50}}{5^{45} + 3^{45}}.\] | {
"answer": "3124",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a line and a point at a distance of $1 \mathrm{~cm}$ from the line. What is the volume of a cube where the point is one of the vertices and the line is one of the body diagonals? | {
"answer": "\\frac{3 \\sqrt{6}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $b_1, b_2, \ldots$ is geometric with $b_1=b$ and common ratio $s,$ where $b$ and $s$ are positive integers. Given that $\log_4 b_1+\log_4 b_2+\cdots+\log_4 b_{10} = 2010,$ find the number of possible ordered pairs $(b,s).$ | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle is divided into two segments by a chord equal to the side of a regular inscribed triangle. Determine the ratio of the areas of these segments. | {
"answer": "\\frac{4\\pi - 3\\sqrt{3}}{8\\pi + 3\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of the geometric series $3 - \left(\frac{3}{4}\right) + \left(\frac{3}{4}\right)^2 - \left(\frac{3}{4}\right)^3 + \dots$. Express your answer as a common fraction. | {
"answer": "\\frac{12}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin \alpha + \cos \alpha = -\frac{3}{\sqrt{5}}$, and $|\sin \alpha| > |\cos \alpha|$, find the value of $\tan \frac{\alpha}{2}$. | {
"answer": "-\\frac{\\sqrt{5} + 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A function $f(x)$ defined on $R$ satisfies $f(x+1) = 2f(x)$. When $x \in (-1,0]$, $f(x) = x^{3}$. Find $f(\frac{21}{2})$. | {
"answer": "-256",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The cafeteria is tiled with the same floor tiles, and it takes 630 tiles to cover an area with 18 square decimeters of tiles. How many tiles will it take if we switch to square tiles with a side length of 6 decimeters? | {
"answer": "315",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the volume of the solid formed by rotating around the $O Y$ axis the curvilinear trapezoid which is bounded by the hyperbola $x y=2$ and the lines $y_{1}=1, y_{2}=4$, and $y_{3}=0$. | {
"answer": "3\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\( x_{1} = 2001 \). When \( n > 1, x_{n} = \frac{n}{x_{n-1}} \). Given that \( x_{1} x_{2} x_{3} \ldots x_{10} = a \), find the value of \( a \). | {
"answer": "3840",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The edge of the cube \(E F G H E_1 F_1 G_1 H_1\) is equal to 2. Points \(A\) and \(B\) are taken on the edges \(E H\) and \(H H_1\) such that \(\frac{E A}{A H} = 2\) and \(\frac{B H}{B H_1} = \frac{1}{2}\). A plane is drawn through the points \(A\), \(B\), and \(G_1\). Find the distance from point \(E\) to this plane. | {
"answer": "2 \\sqrt{\\frac{2}{11}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain quadratic polynomial is known to have the following properties: its leading coefficient is equal to one, it has integer roots, and its graph (parabola) intersects the line \( y = 2017 \) at two points with integer coordinates. Can the ordinate of the vertex of the parabola be uniquely determined based on this information? | {
"answer": "-1016064",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ten distinct natural numbers are such that the product of any 5 of them is even, and the sum of all 10 numbers is odd. What is their smallest possible sum? | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the line $y=kx+t$ is a tangent line to the curve $y=e^x+2$ and also a tangent line to the curve $y=e^{x+1}$, find the value of $t$. | {
"answer": "4-2\\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 8 keys numbered 1 to 8 and 8 boxes numbered 1 to 8. Each key can only open the box with the same number. All keys are placed in these boxes and locked up so that each box contains one key. How many different ways are there to place the keys in the boxes such that at least two boxes have to be opened to unlock all the boxes? (Assume the keys are not duplicated and a box can either be opened with its corresponding key or broken open with no other means.) | {
"answer": "35280",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For arbitrary real numbers \(a\) and \(b\) (\(a \neq 0\)), find the minimum value of the expression \(\frac{1}{a^{2}} + 2a^{2} + 3b^{2} + 4ab\). | {
"answer": "\\sqrt{\\frac{8}{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with its center on side $AB$ of triangle $ABC$ touches the other two sides. Find the area of the circle if $a = 13$ cm, $b = 14$ cm, and $c = 15$ cm, where $a$, $b$, and $c$ are the lengths of the sides of the triangle. | {
"answer": "\\frac{3136}{81} \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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