problem stringlengths 10 5.15k | answer dict |
|---|---|
Calculate the volume of a tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$, and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.
$A_{1}(-2, 0, -4)$
$A_{2}(-1, 7, 1)$
$A_{3}(4, -8, -4)$
$A_{4}(1, -4, 6)$ | {
"answer": "5\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $2\sin ^{2} \frac{A}{2}= \sqrt{3}\sin A$, $\sin (B-C)=2\cos B\sin C$, find the value of $\frac{AC}{AB}$ . | {
"answer": "\\frac{1+\\sqrt{13}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a finite sequence \( B = (b_1, b_2, \dots, b_{50}) \) of numbers, the Cesaro sum is defined as
\[
\frac{S_1 + \cdots + S_{50}}{50},
\]
where \( S_k = b_1 + \cdots + b_k \) and \( 1 \leq k \leq 50 \).
If the Cesaro sum of the 50-term sequence \( (b_1, \dots, b_{50}) \) is 500, what is the Cesaro sum of the 51-term sequence \( (2, b_1, \dots, b_{50}) \)? | {
"answer": "492",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles trapezoid, the longer base \(AB\) is 24 units, the shorter base \(CD\) is 12 units, and each of the non-parallel sides has a length of 13 units. What is the length of the diagonal \(AC\)? | {
"answer": "\\sqrt{457}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, if $\cos B= \frac {4}{5}$, $a=5$, and the area of $\triangle ABC$ is $12$, find the value of $\frac {a+c}{\sin A+\sin C}$. | {
"answer": "\\frac {25}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An eight-sided die numbered from 1 to 8 is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The prime factorization of 1386 is $2 \times 3 \times 3 \times 7 \times 11$. How many ordered pairs of positive integers $(x,y)$ satisfy the equation $xy = 1386$, and both $x$ and $y$ are even? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \sqrt {3}\sin x\cdot\cos x- \frac {1}{2}\cos 2x$ $(x\in\mathbb{R})$.
$(1)$ Find the minimum value and the smallest positive period of the function $f(x)$.
$(2)$ Let $\triangle ABC$ have internal angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively, and $f(C)=1$, $B=30^{\circ}$, $c=2 \sqrt {3}$. Find the area of $\triangle ABC$. | {
"answer": "2 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles triangle \(ABC\) with base \(AC\) equal to 37, the exterior angle at vertex \(B\) is \(60^\circ\). Find the distance from vertex \(C\) to line \(AB\). | {
"answer": "18.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( p \) and \( q \) are positive integers such that \( p + q > 2017 \), \( 0 < p < q \leq 2017 \), and \((p, q) = 1\), find the sum of all fractions of the form \(\frac{1}{pq}\). | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Convert the following angles between degrees and radians: ① -15°; ② $\frac {7\pi }{12}$
(2) Given that the terminal side of angle $\alpha$ passes through point P(2sin30°, -2cos30°), find the sine, cosine, and tangent values of angle $\alpha$. | {
"answer": "-\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Frederik wants to make a special seven-digit password. Each digit of his password occurs exactly as many times as its digit value. The digits with equal values always occur consecutively, e.g., 4444333 or 1666666. How many possible passwords can he make?
A) 6
B) 7
C) 10
D) 12
E) 13 | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average age of 40 sixth-graders is 12. The average age of 30 of their teachers is 45. What is the average age of all these sixth-graders and their teachers? | {
"answer": "26.14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular quadrilateral pyramid $S-ABCD$ with lateral edge length of $4$ and $\angle ASB = 30^\circ$, points $E$, $F$, and $G$ are taken on lateral edges $SB$, $SC$, and $SD$ respectively. Find the minimum value of the perimeter of the spatial quadrilateral $AEFG$. | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fix a sequence $ a_1,a_2,a_3,... $ of integers satisfying the following condition:for all prime numbers $ p $ and all positive integers $ k $ , we have $ a_{pk+1}=pa_k-3a_p+13 $ .Determine all possible values of $ a_{2013} $ . | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that our number system has a base of eight, determine the fifteenth number in the sequence. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many natural numbers between 200 and 400 are divisible by 8? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from the vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(-2, -1, -1) \)
\( A_{2}(0, 3, 2) \)
\( A_{3}(3, 1, -4) \)
\( A_{4}(-4, 7, 3) \) | {
"answer": "\\frac{140}{\\sqrt{1021}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 11 of the number 1, 22 of the number 2, 33 of the number 3, and 44 of the number 4 on the blackboard. The following operation is performed: each time, three different numbers are erased, and the fourth number, which is not erased, is written 2 extra times. For example, if 1 of 1, 1 of 2, and 1 of 3 are erased, then 2 more of 4 are written. After several operations, there are only 3 numbers left on the blackboard, and no further operations can be performed. What is the product of the last three remaining numbers? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a cyclic quadrilateral with sides $AB$, $BC$, $CD$, and $DA$. The side lengths are distinct integers less than $10$ and satisfy $BC + CD = AB + DA$. Find the largest possible value of the diagonal $BD$.
A) $\sqrt{93}$
B) $\sqrt{\frac{187}{2}}$
C) $\sqrt{\frac{191}{2}}$
D) $\sqrt{100}$ | {
"answer": "\\sqrt{\\frac{191}{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two non-collinear vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}|=|\overrightarrow{b}|$, and $\overrightarrow{a}\perp(\overrightarrow{a}-2\overrightarrow{b})$, the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\ln x+ \frac {1}{2}ax^{2}-x-m$ ($m\in\mathbb{Z}$).
(I) If $f(x)$ is an increasing function, find the range of values for $a$;
(II) If $a < 0$, and $f(x) < 0$ always holds, find the minimum value of $m$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle ABC, angle C is a right angle, and CD is the altitude. Find the radius of the circle inscribed in triangle ABC if the radii of the circles inscribed in triangles ACD and BCD are 6 and 8, respectively. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average of five distinct natural numbers is 15, and the median is 18. What is the maximum possible value of the largest number among these five numbers? | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a 200-meter race, Sonic is 16 meters ahead of Dash when Sonic finishes the race. The next time they race, Sonic starts 2.5 times this lead distance behind Dash, who is at the starting line. Both runners run at the same constant speed as they did in the first race. Determine the distance Sonic is ahead of Dash when Sonic finishes the second race. | {
"answer": "19.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve tiles numbered $1$ through $12$ are turned face down. One tile is turned up at random, and a die with 8 faces numbered 1 to 8 is rolled. Determine the probability that the product of the numbers on the tile and the die will be a square. | {
"answer": "\\frac{7}{48}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a month with 31 days, where the number of the month is a product of two distinct primes (e.g., July, represented as 7). Determine how many days in July are relatively prime to the month number. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $A$ lies on the line $y=\frac{8}{15} x-6$, and point $B$ on the parabola $y=x^{2}$. What is the minimum length of the segment $AB$? | {
"answer": "1334/255",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Kanga labelled the vertices of a square-based pyramid using \(1, 2, 3, 4,\) and \(5\) once each. For each face, Kanga calculated the sum of the numbers on its vertices. Four of these sums equaled \(7, 8, 9,\) and \(10\). What is the sum for the fifth face? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive real numbers $a$, $b$, $c$, $d$ satisfying $a^{2}-ab+1=0$ and $c^{2}+d^{2}=1$, find the value of $ab$ when $\left(a-c\right)^{2}+\left(b-d\right)^{2}$ reaches its minimum. | {
"answer": "\\frac{\\sqrt{2}}{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of triangle \(ABC\) is 1. Let \(A_1\), \(B_1\), and \(C_1\) be the midpoints of the sides \(BC\), \(CA\), and \(AB\) respectively. Points \(K\), \(L\), and \(M\) are taken on segments \(AB_1\), \(CA_1\), and \(BC_1\) respectively. What is the minimum area of the common part of triangles \(KLM\) and \(A_1B_1C_1\)? | {
"answer": "1/8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a^{2}+c^{2}-b^{2}=ac$, $c=2$, and point $G$ satisfies $| \overrightarrow{BG}|= \frac { \sqrt {19}}{3}$ and $\overrightarrow{BG}= \frac {1}{3}( \overrightarrow{BA}+ \overrightarrow{BC})$, find the value of $\sin A$. | {
"answer": "\\frac {3 \\sqrt {21}}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine how many ordered pairs $(a, b)$, where $a$ is a positive real number and $b$ is an integer between $1$ and $210$, inclusive, satisfy the equation $(\log_b a)^{2023} = \log_b(a^{2023})$. | {
"answer": "630",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In parallelogram $ABCD$ , $AB = 10$ , and $AB = 2BC$ . Let $M$ be the midpoint of $CD$ , and suppose that $BM = 2AM$ . Compute $AM$ . | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Integers less than $4010$ but greater than $3000$ have the property that their units digit is the sum of the other digits and also the full number is divisible by 3. How many such integers exist? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point P is any point on the graph of the function $f(x) = 2\sqrt{2x}$, and a tangent line is drawn from point P to circle D: $x^2 + y^2 - 4x + 3 = 0$, with the points of tangency being A and B, find the minimum value of the area of quadrilateral PADB. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Lee can make 24 cookies with four cups of flour. If the ratio of flour to sugar needed is 2:1 and he has 3 cups of sugar available, how many cookies can he make? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)$ be a function defined on $\mathbb{R}$ with a period of $2$, and for any real number $x$, it always holds that $f(x)-f(-x)=0$. When $x \in [0,1]$, $f(x)=-\sqrt{1-x^{2}}$. Determine the number of zeros of the function $g(x)=f(x)-e^{x}+1$ in the interval $[-2018,2018]$. | {
"answer": "2018",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $w$ is a complex number such that $w^2 = 45-21i$. Find $|w|$. | {
"answer": "\\sqrt[4]{2466}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the vector $\overrightarrow {a} = (3\cos\alpha, 2)$ is parallel to the vector $\overrightarrow {b} = (3, 4\sin\alpha)$, find the value of the acute angle $\alpha$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. Given that $AB=25$ and $PQ = QR = 2.5$, calculate the perimeter of $\triangle APR$. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Of all positive integers between 10 and 100, what is the sum of the non-palindrome integers that take exactly eight steps to become palindromes? | {
"answer": "187",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a pot, there are 6 sesame-filled dumplings, 5 peanut-filled dumplings, and 4 red bean paste-filled dumplings. These three types of dumplings look exactly the same from the outside. If 4 dumplings are randomly scooped out, the probability that at least one dumpling of each type is scooped out is ______. | {
"answer": "\\dfrac{48}{91}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the complex numbers $z\_1=a^2-2-3ai$ and $z\_2=a+(a^2+2)i$, if $z\_1+z\_2$ is a purely imaginary number, determine the value of the real number $a$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Some bugs are sitting on squares of $10\times 10$ board. Each bug has a direction associated with it **(up, down, left, right)**. After 1 second, the bugs jump one square in **their associated**direction. When the bug reaches the edge of the board, the associated direction reverses (up becomes down, left becomes right, down becomes up, and right becomes left) and the bug moves in that direction. It is observed that it is **never** the case that two bugs are on same square. What is the maximum number of bugs possible on the board? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the ratio of the length of its horizontal side to the length of its vertical side. Find the smallest possible value of the sum of all the written numbers. | {
"answer": "5/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is \( \frac{1}{4} \) more than 32.5? | {
"answer": "32.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Una rolls 8 standard 6-sided dice simultaneously and calculates the product of the 8 numbers obtained. What is the probability that the product is divisible by 8?
A) $\frac{1}{4}$
B) $\frac{57}{64}$
C) $\frac{199}{256}$
D) $\frac{57}{256}$
E) $\frac{63}{64}$ | {
"answer": "\\frac{199}{256}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with a focal length of $2\sqrt{2}$, and passing through the point $A(\frac{3}{2}, -\frac{1}{2})$.
(1) Find the equation of the ellipse;
(2) Find the coordinates of a point $P$ on the ellipse $C$ such that its distance to the line $l$: $x+y+4=0$ is minimized, and find this minimum distance. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a function $B(m, n)$ by \[ B(m,n) = \left\{ \begin{aligned} &n+2& \text{ if } m = 0 \\ &B(m-1, 2) & \text{ if } m > 0 \text{ and } n = 0 \\ &B(m-1, B(m, n-1))&\text{ if } m > 0 \text{ and } n > 0. \end{aligned} \right.\]Compute $B(2, 2).$ | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a rational number $r$ , its *period* is the length of the smallest repeating block in its decimal expansion. for example, the number $r=0.123123123...$ has period $3$ . If $S$ denotes the set of all rational numbers of the form $r=\overline{abcdefgh}$ having period $8$ , find the sum of all elements in $S$ . | {
"answer": "50000000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $\triangle ABC$, $\triangle BCD$, and $\triangle CDE$ are right-angled at $B$, $C$, and $D$ respectively, with $\angle ACB=\angle BCD = \angle CDE = 45^\circ$, and $AB=15$. [asy]
pair A, B, C, D, E;
A=(0,15);
B=(0,0);
C=(10.6066,0);
D=(15,0);
E=(21.2132,0);
draw(A--B--C--D--E);
draw(B--C);
draw(C--D);
label("A", A, N);
label("B", B, W);
label("C", C, S);
label("D", D, S);
label("E", E, S);
[/asy] Find the length of $CD$. | {
"answer": "\\frac{15\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the quadrilateral $ABCD$ , $AB = BC = CD$ and $\angle BMC = 90^\circ$ , where $M$ is the midpoint of $AD$ . Determine the acute angle between the lines $AC$ and $BD$ . | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a trapezoidal field where it's planted uniformly with wheat. The trapezoid has the following measurements: side $AB$ is 150 m, base $AD$ (the longest side) is 300 m, and the other base $BC$ is 150 m. The angle at $A$ is $75^\circ$, and the angle at $B$ is $105^\circ$. At harvest time, all the wheat is collected at the point nearest to the trapezoid's perimeter. What fraction of the crop is brought to the longest side $AD$? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | {
"answer": "0.298",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC, AB = 10, BC = 8, CA = 7$ and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. Calculate the length of $PC$. | {
"answer": "\\frac{56}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the lateral side \( CD \) of the trapezoid \( ABCD \) (\( AD \parallel BC \)), a point \( M \) is marked. A perpendicular \( AH \) is dropped from vertex \( A \) to segment \( BM \). It turns out that \( AD = HD \). Find the length of segment \( AD \), given that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \). | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{1, 2, \cdots, 2005\} \). If in any set of \( n \) pairwise coprime numbers in \( S \) there is at least one prime number, find the minimum value of \( n \). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $10$ divides the number $1\cdot2^1+2\cdot2^2+3\cdot2^3+\dots+n\cdot2^n$ , what is the least integer $n\geq 2012$ ? | {
"answer": "2014",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x) = \log_a x$, its inverse function is $g(x)$.
(1) Solve the equation for $x$: $f(x-1) = f(a-x) - f(5-x)$;
(2) Let $F(x) = (2m-1)g(x) + \left( \frac{1}{m} - \frac{1}{2} \right)g(-x)$. If $F(x)$ has a minimum value, find the expression for $h(m)$;
(3) Find the maximum value of $h(m)$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When \( x^{2} \) is added to the quadratic polynomial \( f(x) \), its maximum value increases by \( \frac{27}{2} \), and when \( 4x^{2} \) is subtracted from it, its maximum value decreases by 9. How will the maximum value of \( f(x) \) change if \( 2x^{2} \) is subtracted from it? | {
"answer": "\\frac{27}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}+\overrightarrow{b}|+2\overrightarrow{a}\cdot\overrightarrow{b}=0$, determine the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be a fixed integer, $n \geqslant 2$.
1. Determine the smallest constant $c$ such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_i x_j (x_i^2 + x_j^2) \leqslant c \left(\sum_{i=1}^n x_i \right)^4
$$
holds for all non-negative real numbers $x_1, x_2, \cdots, x_n$.
2. For this constant $c$, determine the necessary and sufficient conditions for which equality holds. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\binom{18}{8}=31824$, $\binom{18}{9}=48620$, and $\binom{18}{10}=43758$, calculate $\binom{20}{10}$. | {
"answer": "172822",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=(x-a)^{2}+(2\ln x-2a)^{2}$, where $x > 0, a \in \mathbb{R}$, find the value of the real number $a$ such that there exists $x_{0}$ such that $f(x_{0}) \leqslant \frac{4}{5}$. | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 2012 distinct points in the plane, each of which is to be coloured using one of \( n \) colours so that the number of points of each colour are distinct. A set of \( n \) points is said to be multi-coloured if their colours are distinct. Determine \( n \) that maximizes the number of multi-coloured sets. | {
"answer": "61",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $|\cos\theta|= \frac {1}{5}$ and $\frac {5\pi}{2}<\theta<3\pi$, find the value of $\sin \frac {\theta}{2}$. | {
"answer": "-\\frac{\\sqrt{15}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many roots does the equation
$$
\overbrace{f(f(\ldots f}^{10 \text{ times }}(x) \ldots))+\frac{1}{2}=0
$$
where $f(x)=|x|-1$ have? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)$ be a function defined on $R$ such that $f(x+3) + f(x+1) = f(2) = 1$. Find $\sum_{k=1}^{2023} f(k) =$ ____. | {
"answer": "1012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sixteen 6-inch wide square posts are evenly spaced with 6 feet between them to enclose a square field. What is the outer perimeter, in feet, of the fence? | {
"answer": "106",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $\frac{b}{a}+\sin({A-B})=\sin C$. Find:<br/>
$(1)$ the value of angle $A$;<br/>
$(2)$ if $a=2$, find the maximum value of $\sqrt{2}b+2c$ and the area of triangle $\triangle ABC$. | {
"answer": "\\frac{12}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( y = \sqrt{2x^2 + 2} \) with its graph represented as curve \( G \), and the focus of curve \( G \) denoted as \( F \), two lines \( l_1 \) and \( l_2 \) pass through \( F \) and intersect curve \( G \) at points \( A, C \) and \( B, D \) respectively, such that \( \overrightarrow{AC} \cdot \overrightarrow{BD} = 0 \).
(1) Find the equation of curve \( G \) and the coordinates of its focus \( F \).
(2) Determine the minimum value of the area \( S \) of quadrilateral \( ABCD \). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $\tan B= \sqrt {3}$, $AB=3$, and the area of triangle $ABC$ is $\frac {3 \sqrt {3}}{2}$. Find the length of $AC$. | {
"answer": "\\sqrt {7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three people jointly start a business with a total investment of 143 million yuan. The ratio of the highest investment to the lowest investment is 5:3. What is the maximum and minimum amount the third person could invest in millions of yuan? | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the hexagonal pyramid $(P-ABCDEF)$, the base is a regular hexagon with side length $\sqrt{2}$, $PA=2$ and is perpendicular to the base. Find the volume of the circumscribed sphere of the hexagonal pyramid. | {
"answer": "4\\sqrt{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Gauss is a famous German mathematician, known as the "Prince of Mathematics". There are 110 achievements named after "Gauss". Let $x\in\mathbb{R}$, use $[x]$ to represent the largest integer not exceeding $x$, and use $\{x\}=x-[x]$ to represent the non-negative fractional part of $x$. Then, $y=[x]$ is called the Gauss function. It is known that the sequence $\{a_n\}$ satisfies: $$a_{1}= \sqrt {3}, a_{n+1}=[a_{n}]+ \frac {1}{\{a_{n}\}}, (n\in\mathbb{N}^{*})$$, then $a_{2017}=$ \_\_\_\_\_\_. | {
"answer": "3024+ \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of
\[\sqrt{x^2 + (1 + 2x)^2} + \sqrt{(x - 1)^2 + (x - 1)^2}\]over all real numbers $x.$ | {
"answer": "\\sqrt{2} \\times \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest $n$ for which there exists an $n$-gon that can be divided into a triangle, quadrilateral, ..., up to a 2006-gon? | {
"answer": "2006",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Person A arrives between 7:00 and 8:00, while person B arrives between 7:20 and 7:50. The one who arrives first waits for the other for 10 minutes, after which they leave. Calculate the probability that the two people will meet. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube has a square pyramid placed on one of its faces. Determine the sum of the combined number of edges, corners, and faces of this new shape. | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\pi$. | {
"answer": "\\frac{9\\pi - 18}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $0 < \alpha < \pi$, $\tan\alpha = -2$.
(1) Find the value of $\sin\left(\alpha + \frac{\pi}{6}\right)$;
(2) Calculate the value of $$\frac{2\cos\left(\frac{\pi}{2} + \alpha\right) - \cos(\pi - \alpha)}{\sin\left(\frac{\pi}{2} - \alpha\right) - 3\sin(\pi + \alpha)};$$
(3) Simplify $2\sin^2\alpha - \sin\alpha\cos\alpha + \cos^2\alpha$. | {
"answer": "\\frac{11}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $c$ such that $6x^2 + cx + 16$ equals the square of a binomial. | {
"answer": "8\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In diagram square $ABCD$, four triangles are removed resulting in rectangle $PQRS$. Two triangles at opposite corners ($SAP$ and $QCR$) are isosceles with each having area $120 \text{ m}^2$. The other two triangles ($SDR$ and $BPQ$) are right-angled at $D$ and $B$ respectively, each with area $80 \text{ m}^2$. What is the length of $PQ$, in meters? | {
"answer": "4\\sqrt{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A grocer creates a display of cans where the top row contains two cans and each subsequent lower row has three more cans than the row preceding it. If the total number of cans used in the display is 120, how many rows are there in the display? | {
"answer": "n = 9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the car's average miles-per-gallon for the entire trip given that the odometer readings are $34,500, 34,800, 35,250$, and the gas tank was filled with $8, 10, 15$ gallons of gasoline. | {
"answer": "22.7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)= \sqrt {3}\sin x\cos (x+ \dfrac {π}{6})+\cos x\sin (x+ \dfrac {π}{3})+ \sqrt {3}\cos ^{2}x- \dfrac { \sqrt {3}}{2}$.
(I) Find the range of $f(x)$ when $x\in(0, \dfrac {π}{2})$;
(II) Given $\dfrac {π}{12} < α < \dfrac {π}{3}$, $f(α)= \dfrac {6}{5}$, $- \dfrac {π}{6} < β < \dfrac {π}{12}$, $f(β)= \dfrac {10}{13}$, find $\cos (2α-2β)$. | {
"answer": "-\\dfrac{33}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangular lattice is formed with seven points arranged as follows: six points form a regular hexagon and one point is at the center. Each point is one unit away from its nearest neighbor. Determine how many equilateral triangles can be formed where all vertices are on this lattice. Assume the points are numbered 1 to 7, with 1 to 6 being the peripheral points and 7 being the center. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Evaluate: $\sin^2 120^\circ + \cos 180^\circ + \tan 45^\circ - \cos^2 (-330^\circ) + \sin (-210^\circ)$;
(2) Determine the monotonic intervals of the function $f(x) = \left(\frac{1}{3}\right)^{\sin x}$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Diagonals \( AC \) and \( BD \) of the cyclic quadrilateral \( ABCD \) are perpendicular and intersect at point \( M \). It is known that \( AM = 3 \), \( BM = 4 \), and \( CM = 6 \). Find \( CD \). | {
"answer": "10.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tom, John, and Lily each shot six arrows at a target. Arrows hitting anywhere within the same ring scored the same number of points. Tom scored 46 points and John scored 34 points. How many points did Lily score? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $m \in \mathbb{R}$. A moving line passing through a fixed point $A$ is given by $x+my=0$, and a line passing through a fixed point $B$ is given by $mx-y-m+3=0$. These two lines intersect at point $P(x, y)$. Find the maximum value of $|PA|+|PB|$. | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equation \[-2x^2 = \frac{4x + 2}{x + 2}.\] | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive three-digit integers are there in which each of the three digits is either prime or a perfect square? | {
"answer": "343",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}, BC = 14,$ and $CA = 22$ . Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$ . Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$ , respectively, at points $X$ and $Y$ . If $XY$ can be expressed as $a\sqrt{b} - c$ for positive integers $a,b,c$ with $c$ squarefree, find $a + b + c$ .
*Proposed by Andrew Wu* | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$ and $b$ be positive integers such that $a$ has $4$ factors and $b$ has $2a$ factors. If $b$ is divisible by $a$, what is the least possible value of $b$? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two ways of choosing six different numbers from the list \( 1,2,3,4,5,6,7,8,9 \) so that the product of the six numbers is a perfect square. Suppose that these two perfect squares are \( m^{2} \) and \( n^{2} \), with \( m \) and \( n \) positive integers and \( m \neq n \). What is the value of \( m+n \)? | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One fine summer day, François was looking for Béatrice in Cabourg. Where could she be? Perhaps on the beach (one chance in two) or on the tennis court (one chance in four), it could be that she is in the cafe (also one chance in four). If Béatrice is on the beach, which is large and crowded, François has a one in two chance of not finding her. If she is on one of the courts, there is another one in three chance of missing her, but if she went to the cafe, François will definitely find her: he knows which cafe Béatrice usually enjoys her ice cream. François visited all three possible meeting places but still did not find Béatrice.
What is the probability that Béatrice was on the beach, assuming she did not change locations while François was searching for her? | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.