problem stringlengths 10 5.15k | answer dict |
|---|---|
Given that the terminal side of angle $α$ passes through point $P(\frac{4}{5},-\frac{3}{5})$,
(1) Find the value of $\sin α$;
(2) Find the value of $\frac{\sin (\frac{π}{2}-α)}{\sin (α+π)}-\frac{\tan (α-π)}{\cos (3π-α)}$. | {
"answer": "\\frac{19}{48}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A clock currently shows the time $10:10$ . The obtuse angle between the hands measures $x$ degrees. What is the next time that the angle between the hands will be $x$ degrees? Round your answer to the nearest minute. | {
"answer": "11:15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the geometric sequence ${a_n}$, $a_3$ and $a_{15}$ are the roots of the equation $x^2 + 6x + 2 = 0$, calculate the value of $$\frac{a_{2}a_{16}}{a_{9}}.$$ | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there? | {
"answer": "115",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A = (2, 0) \) and \( B = (8, 6) \). Let \( P \) be a point on the circle \( x^2 + y^2 = 8x \). Find the smallest possible value of \( AP + BP \). | {
"answer": "6\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $(xOy)$, if the initial side of angle $\alpha$ is the non-negative semi-axis of $x$, and the terminal side is the ray $l$: $y=2x(x\leqslant 0)$.
(1) Find the value of $\tan \alpha$;
(2) Find the value of $\frac{\cos \left(\alpha-\pi\right)-2\cos \left( \frac{\pi}{2}+\alpha\right)}{\sin \left(\alpha- \frac{3\pi}{2}\right)-\sin \alpha}$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the length of the segment tangent from the point $(1,1)$ to the circle that passes through the points $(4,5),$ $(7,9),$ and $(6,14).$ | {
"answer": "5\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given complex numbers ${z_1}=2i$, ${z_2}=1-i$, where $i$ is the imaginary unit,
(1) Find the conjugate of the complex number $\frac{z_1}{z_2}$;
(2) In the complex plane, let points $Z_1$, $Z_2$ correspond to ${z_1}$, ${z_2}$ respectively, and $O$ be the origin. Form a parallelogram with $\overrightarrow{OZ_1}$, $\overrightarrow{OZ_2}$ as adjacent sides. Find the length of the diagonal of this parallelogram. | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of rectangle PRTV is divided into four rectangles, PQXW, QRSX, XSTU, and WXUV. Given that the area of PQXW is 9, the area of QRSX is 10, and the area of XSTU is 15, find the area of rectangle WXUV. | {
"answer": "\\frac{27}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ is a moving point on the parabola $y=\frac{1}{4}x^2$, determine the minimum value of the sum of the distance from point $P$ to the line $x+2y+4=0$ and the $x$-axis. | {
"answer": "\\frac{6\\sqrt{5}}{5}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ be the least common multiple of all integers from $10$ to $40$, inclusive. Let $Q$ be the least common multiple of $P$ and the integers $41, 42, 43, 44, 45, 46, 47, 48, 49,$ and $50$. What is the value of $\frac{Q}{P}?$
A) 1763
B) 82861
C) 9261
D) 14756 | {
"answer": "82861",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate \(\frac{2}{3} \cdot \frac{4}{7} \cdot \frac{5}{9} \cdot \frac{11}{13}\). | {
"answer": "\\frac{440}{2457}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
13. Given that $a$, $b$, $c$, are the lengths of the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$ respectively, with $a=2$, and $(2+b)(\sin A-\sin B)=(c-b)\sin C$, find the maximum area of $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $sinα-cosα=\frac{1}{5},0≤α≤π$, calculate $sin(2α-\frac{π}{4})$. | {
"answer": "\\frac{31\\sqrt{2}}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the integer $n$ such that $-150 < n < 150$ and $\tan n^\circ = \tan 286^\circ$. | {
"answer": "-74",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α∈\left( \frac{π}{2},π\right) $, and $\sin \left(π-α\right)+\cos \left(2π+α\right)= \frac{ \sqrt{2}}{3} $. Find the values of:
$(1)\sin {α} -\cos {α} .$
$(2)\tan {α} $. | {
"answer": "- \\frac{9+4 \\sqrt{2}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, points $D$ and $E$ lie on $\overline{BC}$ and $\overline{AC}$ respectively. Lines $\overline{AD}$ and $\overline{BE}$ intersect at point $T$ such that $AT/DT=2$ and $BT/ET=3$. Determine the ratio $CD/BD$. | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let point G be the centroid of triangle ABC. If $\angle A=120^\circ$ and $\overrightarrow {AB} \cdot \overrightarrow {AC}=-1$, find the minimum value of $|\overrightarrow {AG}|$. | {
"answer": "\\frac{\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a chess board, with the numbers $1$ through $64$ placed in the squares as in the diagram below.
\[\begin{tabular}{| c | c | c | c | c | c | c | c |}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8
\hline
9 & 10 & 11 & 12 & 13 & 14 & 15 & 16
\hline
17 & 18 & 19 & 20 & 21 & 22 & 23 & 24
\hline
25 & 26 & 27 & 28 & 29 & 30 & 31 & 32
\hline
33 & 34 & 35 & 36 & 37 & 38 & 39 & 40
\hline
41 & 42 & 43 & 44 & 45 & 46 & 47 & 48
\hline
49 & 50 & 51 & 52 & 53 & 54 & 55 & 56
\hline
57 & 58 & 59 & 60 & 61 & 62 & 63 & 64
\hline
\end{tabular}\]
Assume we have an infinite supply of knights. We place knights in the chess board squares such that no two knights attack one another and compute the sum of the numbers of the cells on which the knights are placed. What is the maximum sum that we can attain?
Note. For any $2\times3$ or $3\times2$ rectangle that has the knight in its corner square, the knight can attack the square in the opposite corner. | {
"answer": "1056",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$), its left and right vertices are $A$ and $B$, respectively. The right focus is $F$, and the line $l$ passing through point $F$ and perpendicular to the $x$-axis intersects the hyperbola at points $M$ and $N$. $P$ is a point on line $l$. When $\angle APB$ is maximized, point $P$ is exactly at $M$ (or $N$). Determine the eccentricity of the hyperbola. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the set of values of the parameter \(a\) for which the sum of the cubes of the roots of the equation \(x^{2}-a x+a+2=0\) is equal to -8. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x, y, z \) be complex numbers such that
\[
xy + 5y = -25, \\
yz + 5z = -25, \\
zx + 5x = -25.
\]
Find all possible values of \( xyz \). | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, $\angle Y = 90^\circ$, $YZ = 4$, and $XY = \sqrt{34}$. What is $\tan X$? | {
"answer": "\\frac{2\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence of numbers with only even digits in their decimal representation, determine the $2014^\text{th}$ number in the sequence. | {
"answer": "62048",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the volume of a cylinder formed by rotating a square with side length 10 centimeters about its horizontal line of symmetry. Express your answer in terms of $\pi$. | {
"answer": "250\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\left( r + \frac{1}{r} \right)^2 = 5,$ then find $r^3 + \frac{1}{r^3}.$ | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The left and right foci of a hyperbola are $F_{1}$ and $F_{2}$, respectively. A line passing through $F_{2}$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\triangle F_{1} A B$ is an equilateral triangle, what is the eccentricity of the hyperbola? | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(2,0)$, $(2,2)$, and $(0,2)$. What is the probability that $x^2 + y^2 < y$? | {
"answer": "\\frac{\\pi}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right triangle is inscribed in the ellipse given by the equation $x^2 + 9y^2 = 9$. One vertex of the triangle is at the point $(0,1)$, and one leg of the triangle is fully contained within the x-axis. Find the squared length of the hypotenuse of the inscribed right triangle, expressed as the ratio $\frac{m}{n}$ with $m$ and $n$ coprime integers, and give the value of $m+n$. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Chewbacca has 25 pieces of orange gum and 35 pieces of apple gum. Some of the pieces are in complete packs, while others are loose. Each complete pack has exactly $y$ pieces of gum. If Chewbacca loses two packs of orange gum, then the ratio of the number of pieces of orange gum he has to the number of pieces of apple gum will be exactly the same as if he instead finds 4 packs of apple gum. Find $y$. | {
"answer": "\\frac{15}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two diagonals of a regular decagon (a 10-sided polygon) are chosen. What is the probability that their intersection lies inside the decagon and forms a convex quadrilateral? | {
"answer": "\\dfrac{42}{119}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mr. Ambulando is at the intersection of $5^{\text{th}}$ and $\text{A St}$ , and needs to walk to the intersection of $1^{\text{st}}$ and $\text{F St}$ . There's an accident at the intersection of $4^{\text{th}}$ and $\text{B St}$ , which he'd like to avoid.
[center]<see attached>[/center]
Given that Mr. Ambulando wants to walk the shortest distance possible, how many different routes through downtown can he take? | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\overset{⇀}{m}=(2,1)$, $\overset{⇀}{n}=(\sin θ,\cos θ)$, where $θ∈(0, \dfrac{π}{2})$ is the inclination angle of line $l$ passing through point $A(1,4)$, if $\overset{⇀}{m}· \overset{⇀}{n}$ is at its maximum when line $l$ is tangent to the circle $(x+1)^{2}+(y-2)^{2}={r}^{2}(r > 0)$, then $r=$ . | {
"answer": "\\dfrac{2 \\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the convex quadrilateral $ABCD$ angle $\angle{BAD}=90$ , $\angle{BAC}=2\cdot\angle{BDC}$ and $\angle{DBA}+\angle{DCB}=180$ . Then find the angle $\angle{DBA}$ | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ΔABC$, $BC=a$, $AC=b$, where $a$ and $b$ are the two roots of the equation $x^2-2\sqrt{3}x+2=0$, and $2\cos(A+B)=1$.
(1) Find the angle $C$;
(2) Find the length of $AB$. | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $E$ and $F$ lie on $\overline{GH}$. The length of $\overline{GE}$ is $3$ times the length of $\overline{EH}$, and the length of $\overline{GF}$ is $5$ times the length of $\overline{FH}$. Determine the length of $\overline{EF}$ as a fraction of the length of $\overline{GH}$.
A) $\frac{1}{10}$
B) $\frac{1}{12}$
C) $\frac{1}{6}$
D) $\frac{1}{8}$ | {
"answer": "\\frac{1}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volume of the parallelepiped determined by vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is 6. Find the volume of the parallelepiped determined by $\mathbf{a} + 2\mathbf{b},$ $\mathbf{b} + \mathbf{c},$ and $2\mathbf{c} - 5\mathbf{a}.$ | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a finite sequence \(P = \left(p_1, p_2, \cdots, p_n\right)\), the Cesaro sum (named after the mathematician Cesaro) is defined as \(\frac{1}{n}(S_1 + S_2 + \cdots + S_n)\), where \(S_k = p_1 + p_2 + \cdots + p_k\) for \(1 \leq k \leq n\). If a sequence \(\left(p_1, p_2, \cdots, p_{99}\right)\) of 99 terms has a Cesaro sum of 1000, then the Cesaro sum of the 100-term sequence \(\left(1, p_1, p_2, \cdots, p_{99}\right)\) is ( ). | {
"answer": "991",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,150 \pi).$ | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square with a side length of 2 rotates around one of its sides, which is the axis of rotation. What is the volume of the cylinder obtained from this rotation? | {
"answer": "8\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Convert the quadratic equation $3x=x^{2}-2$ into general form and determine the coefficients of the quadratic term, linear term, and constant term. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Consider the function $f(x) = |x - \frac{5}{2}| + |x - a|$, where $x \in \mathbb{R}$. If the inequality $f(x) \geq a$ holds true for all $x \in \mathbb{R}$, find the maximum value of the real number $a$.
(2) Given positive numbers $x$, $y$, and $z$ satisfying $x + 2y + 3z = 1$, find the minimum value of $\frac{3}{x} + \frac{2}{y} + \frac{1}{z}$. | {
"answer": "16 + 8\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( N \) be the smallest positive integer whose digits have a product of 2000. The sum of the digits of \( N \) is | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the fourth-largest divisor of $1,234,560,000$. | {
"answer": "154,320,000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 = 16$ and $(x-5)^2 + 4y^2 = 64$?
A) $\frac{5\sqrt{110}}{6}$
B) $\frac{5\sqrt{119}}{6}$
C) $\frac{10\sqrt{119}}{6}$
D) $\frac{5\sqrt{125}}{6}$ | {
"answer": "\\frac{5\\sqrt{119}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x^3y = k$ for a positive constant $k$, find the percentage decrease in $y$ when $x$ increases by $20\%$. | {
"answer": "42.13\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The minimum distance from a point on the parabola $y=x^2$ to the line $2x-y-10=0$ is what? | {
"answer": "\\frac{9\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle C with its center C on the positive x-axis and a radius of 5, the chord intercepted by the line $x-y+3=0$ has a length of $2\sqrt{17}$.
(1) Find the equation of circle C;
(2) Suppose the line $ax-y+5=0$ intersects circle C at points A and B, find the range of the real number $a$;
(3) Under the condition of (2), is there a real number $a$ such that points A and B are symmetric about the line $l$ passing through point P(-2, 4)? If it exists, find the value of the real number $a$; if not, please explain why. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the geometric sequence $(-1, x, y, z, -2)$, find the value of $xyz$. | {
"answer": "-2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that points $P$ and $Q$ are moving points on the curve $y=xe^{-2x}$ and the line $y=x+2$ respectively, find the minimum distance between points $P$ and $Q$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $y= |x-1|+|2x-1|+|3x-1|+ |4x-1|+|5x-1|$ achieves its minimum value when the variable $x$ equals what value? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A quartic (4th degree) polynomial \( p(x) \) satisfies:
\[ p(n) = \frac{1}{n^2} \] for \( n = 1, 2, 3, 4, \) and \( 5 \). Find \( p(6) \). | {
"answer": "\\frac{1}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Juan wants to calculate the area of a large circular garden, whose actual diameter is 50 meters. Unfortunately, his measurement device has an accuracy error of up to 30%. Compute the largest possible percent error, in percent, in Juan’s computed area of the circle in square meters. | {
"answer": "69\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For natural numbers $m$ greater than or equal to 2, the decomposition of their cube powers can be represented as follows: $2^3 = 3 + 5$, $3^3 = 7 + 9 + 11$, $4^3 = 13 + 15 + 17 + 19$. Then,
(1) The smallest number in the decomposition of $8^3$ is;
(2) Following the above pattern, the $n$-th equation can be represented as $(n+1)^3 =$. | {
"answer": "57",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence of positive terms $\{a\_n\}$, where $a\_2=6$, and $\frac{1}{a\_1+1}$, $\frac{1}{a\_2+2}$, $\frac{1}{a\_3+3}$ form an arithmetic sequence, find the minimum value of $a\_1a\_3$. | {
"answer": "19+8\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Masha and the bear ate a basket of raspberries and 40 pies, starting and finishing at the same time. Initially, Masha ate raspberries while the bear ate pies, and then (at some moment) they switched. The bear ate both raspberries and pies 3 times faster than Masha. How many pies did Masha eat if they ate an equal amount of raspberries? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x \cdot \log_{27} 64 = 1$, then $4^x + 4^{-x} =$ \_\_\_\_\_\_\_\_. | {
"answer": "\\dfrac{10}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \\(\alpha\\) be an acute angle, and \\(\cos (\alpha+ \frac {\pi}{6})= \frac {3}{5}\\).
\\((1)\\) Find the value of \\(\cos (\alpha- \frac {\pi}{3})\\);
\\((2)\\) Find the value of \\(\cos (2\alpha- \frac {\pi}{6})\\). | {
"answer": "\\frac {24}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain district's education department wants to send 5 staff members to 3 schools for earthquake safety education. Each school must receive at least 1 person and no more than 2 people. How many different arrangements are possible? (Answer with a number) | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all functions \( f: \mathbb{Q} \rightarrow \{-1, 1\} \) such that for all distinct \( x, y \in \mathbb{Q} \) satisfying \( xy = 1 \) or \( x + y \in \{0, 1\} \), we have \( f(x) f(y) = -1 \).
Intermediate question: Let \( f \) be a function having the above property and such that \( f(0) = 1 \). What is \( f\left(\frac{42}{17}\right) \) ? | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle ABC$ have side lengths $AB = 12$, $AC = 16$, and $BC = 20$. Inside $\angle BAC$, two circles are positioned, each tangent to rays $\overline{AB}$ and $\overline{AC}$, and the segment $\overline{BC}$. Compute the distance between the centers of these two circles. | {
"answer": "20\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The conference has 12 teams. Each team plays every other team twice and an additional 6 games against non-conference opponents. Calculate the total number of games in a season involving the conference teams. | {
"answer": "204",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Bonnie constructs a frame for a cube using 12 pieces of wire that are each eight inches long. Meanwhile, Roark uses 2-inch-long pieces of wire to create a collection of unit cube frames that are not connected. The total volume of Roark's cubes is the same as the volume of Bonnie’s cube. What is the ratio of the total length of Bonnie's wire to the total length of Roark's wire? | {
"answer": "\\frac{1}{128}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 128 players in a single elimination tennis tournament, where exactly 32 players receive a bye in the first round. Calculate the total number of matches played in this tournament. | {
"answer": "126",
"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 $\cos A= \frac{ \sqrt{6}}{3}$.
(1) Find $\tan 2A$;
(2) If $\cos B= \frac{ 2\sqrt{2}}{3}, c=2\sqrt{2}$, find the area of $\triangle ABC$. | {
"answer": "\\frac{2\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2000} \left| x_{k} - x_{k+1} \right| = 2001 \). Let \( y_{k} = \frac{1}{k} \left( x_{1} + x_{2} + \cdots + x_{k} \right) \) for \( k = 1, 2, \cdots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} | y_{k} - y_{k+1} | \). (2001 Shanghai Mathematics Competition) | {
"answer": "2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A grocer sets up a pyramid display for a store promotion, where the topmost row has three cans and each succeeding row below has three more cans than the row immediately above it. If the grocer uses 225 cans in total for this display, how many rows are there in the display? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the domain of the function $f(x)$ is $R$, $f(2x+2)$ is an even function, $f(x+1)$ is an odd function, and when $x\in [0,1]$, $f(x)=ax+b$. If $f(4)=1$, find the value of $\sum_{i=1}^3f(i+\frac{1}{2})$. | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $\frac{{m^2-4m+4}}{{m-1}}÷(\frac{3}{{m-1}}-m-1)$, where $m=\sqrt{3}-2$. | {
"answer": "\\frac{-3+4\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a circle, a chord of length 10 cm is drawn. A tangent to the circle is drawn through one end of the chord, and a secant parallel to the tangent is drawn through the other end. The internal segment of the secant is 12 cm. Find the radius of the circle. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can the number 1500 be represented as a product of three natural numbers? Variations where the factors are the same but differ in order are considered identical. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle has an area of $A$. The midpoints of each side of the rectangle are connected to form a new, smaller rectangle inside the original. What is the ratio of the area of the smaller rectangle to the area of the original rectangle? Express your answer as a common fraction. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equation: $(2x+1)^2=3$. | {
"answer": "\\frac{-1-\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos (x- \frac {π}{4})-\sin (x- \frac {π}{4}).$
(I) Determine the evenness or oddness of the function $f(x)$ and provide a proof;
(II) If $θ$ is an angle in the first quadrant and $f(θ+ \frac {π}{3})= \frac { \sqrt {2}}{3}$, find the value of $\cos (2θ+ \frac {π}{6})$. | {
"answer": "\\frac {4 \\sqrt {2}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle has a radius of six inches. Two parallel chords $CD$ and $EF$ are drawn such that the distance from the center of the circle to chord $CD$ is three inches, and the distance to chord $EF$ is four inches. How long is chord $CD$ and $EF$? Express your answer in the simplest radical form. | {
"answer": "4\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the sum of the series:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))) \] | {
"answer": "510",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The polar equation of circle C is $\rho = 2\cos(θ + \frac{π}{4})$. The parametric equation of line l is
$$
\begin{cases}
x= \sqrt{2}t \\
y= \sqrt{2}t+4\sqrt{2}
\end{cases}
$$
(where t is the parameter). A tangent is drawn from a point P on line l to circle C at the point A, find the minimum value of the length of the tangent PA. | {
"answer": "2\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the prime factorization of $1007021035035021007001$ . (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$ where $p_1,\ldots,p_k$ are distinct prime numbers and $e_1,\ldots,e_k$ are positive integers.) | {
"answer": "7^7 \\times 11^7 \\times 13^7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | {
"answer": "\\sqrt{55}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the graphs of $y=\sin (\frac{1}{2}x-\frac{\pi }{6})$, determine the horizontal shift required to obtain the graph of $y=\sin \frac{1}{2}x$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Kristen has to clear snow from a driveway that is 30 feet long and 3 feet wide. If the snow is initially 8 inches deep, and compacting the snow reduces its volume by 10%, how much snow (in cubic feet) must Kristen move? | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the unique positive integer $m$ such that
\[1 \cdot 2^1 + 2 \cdot 2^2 + 3 \cdot 2^3 + \dots + m \cdot 2^m = 2^{m + 8}.\] | {
"answer": "129",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a game called "Twenty-Four Points" with the following rules: Choose four natural numbers between $1$ and $13$, and perform addition, subtraction, multiplication, and division operations on these four numbers (each number can be used only once and parentheses can be used) to make the result equal to $24$. Given four rational numbers: $-6$, $3$, $4$, $10$, please follow the rules of the "Twenty-Four Points" game to write an expression: ______, such that the result is equal to $24$. | {
"answer": "3 \\times \\left(10 - 6 + 4\\right)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two places, A and B, are 12 km apart. Cars A and B depart from place A towards place B one after the other at a constant speed. Car A takes 15 minutes to travel from A to B, while Car B takes 10 minutes. If Car B departs 2 minutes later than Car A:
1. Write the functions representing the distance traveled by Cars A and B with respect to Car A's travel time.
2. When do Cars A and B meet on the way? How far are they from place A at that time? | {
"answer": "7.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $A$ and $C$ lie on a circle centered at $P$, which is inside $\triangle ABC$ such that $\overline{AP}$ is perpendicular to $\overline{BC}$ and $\triangle ABC$ is equilateral. The circle intersects $\overline{BP}$ at $D$. If $\angle BAP = 45^\circ$, what is $\frac{BD}{BP}$?
A) $\frac{2}{3}$
B) $\frac{2 - \sqrt{2}}{2}$
C) $\frac{1 - \sqrt{2}}{2}$
D) $\frac{\sqrt{2}}{2}$ | {
"answer": "\\frac{2 - \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \dfrac {4^{x}+1}{4^{x- \frac {1}{2}}+1}$, find the sum of $f( \dfrac {1}{2014})+f( \dfrac {2}{2014})+…+f( \dfrac {2013}{2014})$. | {
"answer": "\\dfrac {6039}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\alpha \in (0, \frac{\pi}{2})$, and $\tan 2\alpha = \frac{\cos \alpha}{2-\sin \alpha}$, then find the value of $\tan \alpha$. | {
"answer": "\\frac{\\sqrt{15}}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Simplify: $\dfrac{\sin(\pi -\alpha)\cos(\pi +\alpha)\sin(\dfrac{\pi}{2}+\alpha)}{\sin(-\alpha)\sin(\dfrac{3\pi}{2}+\alpha)}$.
(2) Given $\alpha \in (\dfrac{\pi}{2}, \pi)$, and $\sin(\pi -\alpha) + \cos \alpha = \dfrac{7}{13}$, find $\tan \alpha$. | {
"answer": "-\\dfrac{12}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, there is a curve $C_{1}: x+y=4$, and another curve $C_{2}$ defined by the parametric equations $\begin{cases} x=1+\cos \theta, \\ y=\sin \theta \end{cases}$ (with $\theta$ as the parameter). A polar coordinate system is established with the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.
$(1)$ Find the polar equations of curves $C_{1}$ and $C_{2}$.
$(2)$ If a ray $l: \theta=\alpha (\rho > 0)$ intersects $C_{1}$ and $C_{2}$ at points $A$ and $B$ respectively, find the maximum value of $\dfrac{|OB|}{|OA|}$. | {
"answer": "\\dfrac{1}{4}(\\sqrt{2}+1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the lateral surface of a cone, when unrolled, forms a semicircle with a radius of $2\sqrt{3}$, and the vertex and the base circle of the cone are on the surface of a sphere O, calculate the volume of sphere O. | {
"answer": "\\frac{32}{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)$ defined on $\mathbb{R}$ and satisfying the condition $f(x+2) = 3f(x)$, when $x \in [0, 2]$, $f(x) = x^2 - 2x$. Find the minimum value of $f(x)$ when $x \in [-4, -2]$. | {
"answer": "-\\frac{1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The circumference of a circle has 50 numbers written on it, each of which is either +1 or -1. We want to find the product of these numbers. What is the minimum number of questions needed to determine this product if we can ask about the product of three consecutive numbers at a time? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos(2x+\varphi), |\varphi| \leqslant \frac{\pi}{2}$, if $f\left( \frac{8\pi}{3}-x \right)=-f(x)$, determine the horizontal shift required to obtain the graph of $y=\sin 2x$ from the graph of $y=f(x)$. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the lines $l_{1}$: $ax+y+3=0$ and $l_{2}$: $2x+\left(a-1\right)y+a+1=0$ are parallel, find the value of $a$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify the expression $\frac{5^5 + 5^3 + 5}{5^4 - 2\cdot5^2 + 5}$. | {
"answer": "\\frac{651}{116}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain sequence, the first term is \(a_1 = 1010\) and the second term is \(a_2 = 1011\). The values of the remaining terms are chosen so that \(a_n + a_{n+1} + a_{n+2} = 2n\) for all \(n \geq 1\). Determine \(a_{1000}\). | {
"answer": "1676",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can \(a, b, c\), and \(d\) be chosen from the set \(\{0,1,2, \ldots, 9\}\) so that \(a<b<c<d\) and \(a+b+c+d\) is a multiple of three? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other $2$ players and each pair plays until a winner is decided (i.e. in the event of a tie, they play again). What is the probability that each player wins exactly once? | {
"answer": "1/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of all odd terms in the first 10 terms of a geometric sequence is $85 \frac{1}{4}$, and the sum of all even terms is $170 \frac{1}{2}$, find the value of $S=a_{3}+a_{6}+a_{9}+a_{12}$. | {
"answer": "585",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system xOy, there is a line l₁: x = 2, and a curve C: {x = 2cosϕ, y = 2 + 2sinϕ} (where ϕ is a parameter). With O as the pole and the non-negative half-axis of the x-axis as the polar axis, establish a polar coordinate system. The polar coordinates of point M are $(3, \frac {π}{6})$.
1. Find the polar coordinate equations of the line l₁ and the curve C.
2. In the polar coordinate system, it is known that the ray ${l}_{2}：\;θ=α(0 < α < \frac{π}{2})$ intersects l₁ and C at points A and B respectively, and $|OA|\cdot |OB|=8 \sqrt {3}$. Find the area of △MOB. | {
"answer": "\\frac {3 \\sqrt {3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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