problem stringlengths 10 5.15k | answer dict |
|---|---|
In right triangle $DEF$, where $DE=15$, $DF=9$, and $EF=12$ units, what is the distance from $D$ to the midpoint of segment $EF$? | {
"answer": "7.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\binom{23}{5}=33649$, $\binom{23}{6}=42504$, and $\binom{23}{7}=33649$, find $\binom{25}{7}$. | {
"answer": "152306",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation \((7+4 \sqrt{3}) x^{2}+(2+\sqrt{3}) x-2=0\), calculate the difference between the larger root and the smaller root. | {
"answer": "6 - 3 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A standard domino game consists of 28 tiles. Each tile is made up of two integers ranging from 0 to 6, inclusive. All possible combinations $(a, b)$, where $a \leq b$, are listed exactly once. Note that tile $(4,2)$ is listed as tile $(2,4)$, because $2 \leq 4$. Excluding the piece $(0,0)$, for each of the other 27 tiles $(a, b)$, where $a \leq b$, we write the fraction $\frac{a}{b}$ on a board.
a) How many distinct values are written in the form of fractions on the board? (Note that the fractions $\frac{1}{2}$ and $\frac{2}{4}$ have the same value and should be counted only once.)
b) What is the sum of the distinct values found in the previous item? | {
"answer": "\\frac{13}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
<u>Set 4</u> **p10.** Eve has nine letter tiles: three $C$ ’s, three $M$ ’s, and three $W$ ’s. If she arranges them in a random order, what is the probability that the string “ $CMWMC$ ” appears somewhere in the arrangement?**p11.** Bethany’s Batteries sells two kinds of batteries: $C$ batteries for $\$ 4 $ per package, and $ D $ batteries for $ \ $7$ per package. After a busy day, Bethany looks at her ledger and sees that every customer that day spent exactly $\$ 2021 $, and no two of them purchased the same quantities of both types of battery. Bethany also notes that if any other customer had come, at least one of these two conditions would’ve had to fail. How many packages of batteries did Bethany sell?**p12.** A deck of cards consists of $ 30 $ cards labeled with the integers $ 1 $ to $ 30 $, inclusive. The cards numbered $ 1 $ through $ 15 $ are purple, and the cards numbered $ 16 $ through $ 30 $ are green. Lilith has an expansion pack to the deck that contains six indistinguishable copies of a green card labeled with the number $ 32$. Lilith wants to pick from the expanded deck a hand of two cards such that at least one card is green. Find the number of distinguishable hands Lilith can make with this deck.
PS. You should use hide for answers. | {
"answer": "\\frac{1}{28}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
George now has an unfair eight-sided die. The probabilities of rolling each number from 1 to 5 are each $\frac{1}{15}$, the probability of rolling a 6 or a 7 is $\frac{1}{6}$ each, and the probability of rolling an 8 is $\frac{1}{5}$. What is the expected value of the number shown when this die is rolled? Express your answer as a decimal. | {
"answer": "4.7667",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system $xOy$, the parameter equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{{\sqrt{2}}}{2}t\\ y=\frac{{\sqrt{2}}}{2}t\end{array}\right.$ (where $t$ is the parameter). Taking the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, establishing a polar coordinate system with the same unit length, the polar coordinate equation of the curve $C$ is $ρ=2\sqrt{2}\sin({θ+\frac{π}{4}})$.
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;
$(2)$ Let point $P(4,3)$, the intersection points of line $l$ and curve $C$ be $A$ and $B$, find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$. | {
"answer": "\\frac{{5\\sqrt{2}}}{{11}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cone is sliced into five pieces by planes parallel to its base. All of these pieces have the same height. What is the ratio of the volume of the third-largest piece to the volume of the largest piece? | {
"answer": "\\frac{19}{61}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regular octagonal pyramid $\allowbreak PABCDEFGH$ has the octagon $ABCDEFGH$ as its base. Each side of the octagon has length 5. Pyramid $PABCDEFGH$ has an additional feature where triangle $PAD$ is an equilateral triangle with side length 10. Calculate the volume of the pyramid. | {
"answer": "\\frac{250\\sqrt{3}(1 + \\sqrt{2})}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a rectangular grid comprising 5 rows and 4 columns of squares, how many different rectangles can be traced using the lines in the grid? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $C\_1$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and the hyperbola $C\_2$: $x^{2}- \frac{y^{2}}{4}=1$ share a common focus. One of the asymptotes of $C\_2$ intersects with the circle having the major axis of $C\_1$ as its diameter at points $A$ and $B$. If $C\_1$ precisely trisects the line segment $AB$, then the length of the minor axis of the ellipse $C\_1$ is _____. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\triangle ABC$ with $AC=1$, $\angle ABC= \frac{2\pi}{3}$, $\angle BAC=x$, let $f(x)= \overrightarrow{AB} \cdot \overrightarrow{BC}$.
$(1)$ Find the analytical expression of $f(x)$ and indicate its domain;
$(2)$ Let $g(x)=6mf(x)+1$ $(m < 0)$, if the range of $g(x)$ is $\left[- \frac{3}{2},1\right)$, find the value of the real number $m$. | {
"answer": "- \\frac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle $C: (x-1)^{2} + (y-2)^{2} = 25$ and a line $l: (2m+1)x + (m+1)y - 7m-4 = 0$, where $m \in \mathbb{R}$. Find the minimum value of the chord length $|AB|$ cut by line $l$ on circle $C$. | {
"answer": "4\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression $2000 \times 1995 \times 0.1995 - 10$. | {
"answer": "0.2 \\times 1995^2 - 10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the complex plane, let $A$ be the set of solutions to $z^3 - 27 = 0$ and let $B$ be the set of solutions to $z^3 - 9z^2 - 27z + 243 = 0,$ find the distance between the point in $A$ closest to the origin and the point in $B$ closest to the origin. | {
"answer": "3(\\sqrt{3} - 1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the vertex of angle α is at the origin of the coordinate system, its initial side coincides with the non-negative half-axis of the x-axis, and its terminal side passes through the point (-√3,2), find the value of tan(α - π/6). | {
"answer": "-3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line $10x + 8y = 80$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
A) $\frac{18\sqrt{41} + 40}{\sqrt{41}}$
B) $\frac{360}{17}$
C) $\frac{107}{5}$
D) $\frac{43}{2}$
E) $\frac{281}{13}$ | {
"answer": "\\frac{18\\sqrt{41} + 40}{\\sqrt{41}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their spot to the spectator (there were no draws). It turned out that Egor participated in 13 games, and Nikita in 27 games. How many games did Innokentiy play? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In right triangle $DEF$ with $\angle D = 90^\circ$, we have $DE = 8$ and $EF = 17$. Find $\cos F$. | {
"answer": "\\frac{8}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that the distance between any two of the given $n(n=2,3,4,5)$ points in the plane is at least 1. What is the minimum value that the diameter of this system of points can have? | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given: $A=2a^{2}-5ab+3b$, $B=4a^{2}+6ab+8a$.
$(1)$ Simplify: $2A-B$;
$(2)$ If $a=-1$, $b=2$, find the value of $2A-B$;
$(3)$ If the value of the algebraic expression $2A-B$ is independent of $a$, find the value of $b$. | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to $\angle A$, $\angle B$, $\angle C$ respectively. It is known that $a$, $b$, $c$ are in geometric progression, and $a^{2}-c^{2}=ac-bc$,
(1) Find the measure of $\angle A$;
(2) Find the value of $\frac{b\sin B}{c}$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Arrange 3 boys and 4 girls in a row. Calculate the number of different arrangements under the following conditions:
(1) Person A and Person B must stand at the two ends;
(2) All boys must stand together;
(3) No two boys stand next to each other;
(4) Exactly one person stands between Person A and Person B. | {
"answer": "1200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pyramid is intersected by a plane parallel to its base, dividing its lateral surface into two parts of equal area. In what ratio does this plane divide the lateral edges of the pyramid? | {
"answer": "\\frac{1}{\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $a=2$, $b=3$, $\cos C=\frac{1}{3}$. The radius of the circumcircle is ______. | {
"answer": "\\frac{9 \\sqrt{2}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate plane $(xOy)$, the focus of the parabola $y^{2}=2x$ is $F$. Let $M$ be a moving point on the parabola, then the maximum value of $\frac{MO}{MF}$ is _______. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer $m$ such that the following is true?
*Given $\it m$ integers between $\it1$ and $\it{2023},$ inclusive, there must exist two of them $\it a, b$ such that $1 < \frac ab \le 2.$* | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
After shifting the graph of the function $y=\sin^2x-\cos^2x$ to the right by $m$ units, the resulting graph is symmetric to the graph of $y=k\sin x\cos x$ ($k>0$) with respect to the point $\left( \frac{\pi}{3}, 0 \right)$. Find the minimum positive value of $k+m$. | {
"answer": "2+ \\frac{5\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circular cylindrical post has a circumference of 6 feet and a height of 18 feet. A string is wrapped around the post which spirals evenly from the bottom to the top, looping around the post exactly six times. What is the length of the string, in feet? | {
"answer": "18\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The total in-store price for a laptop is $299.99. A radio advertisement offers the same laptop for five easy payments of $55.98 and a one-time shipping and handling charge of $12.99. Calculate the amount of money saved by purchasing the laptop from the radio advertiser. | {
"answer": "710",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. Determine the area of their overlapping interiors. Express your answer in expanded form in terms of $\pi$. | {
"answer": "\\frac{9}{2}\\pi - 9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(\cos x) = -f'(\frac{1}{2})\cos x + \sqrt{3}\sin^2 x$, find the value of $f(\frac{1}{2})$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α$ is an angle in the second quadrant and $\sin α= \frac {3}{5}$, find $\sin 2α$. | {
"answer": "- \\frac{24}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a trapezoid, the two non parallel sides and a base have length $1$ , while the other base and both the diagonals have length $a$ . Find the value of $a$ . | {
"answer": "\\frac{\\sqrt{5} + 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin \left(x+ \frac {\pi}{3}\right)= \frac {1}{3}$, then the value of $\sin \left( \frac {5\pi}{3}-x\right)-\cos \left(2x- \frac {\pi}{3}\right)$ is \_\_\_\_\_\_. | {
"answer": "\\frac {4}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express $0.000 000 04$ in scientific notation. | {
"answer": "4 \\times 10^{-8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A taxi driver passes through six traffic checkpoints on the way from the restaurant to the train station. Assuming that the events of encountering a red light at each checkpoint are independent of each other and the probability is $\frac{1}{3}$ at each checkpoint, calculate the probability that the driver has passed two checkpoints before encountering a red light. | {
"answer": "\\frac{4}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{a_n\}$ with the general term formula $a_n = -n^2 + 12n - 32$, determine the maximum value of $S_n - S_m$ for any $m, n \in \mathbb{N^*}$ and $m < n$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos(\frac{1}{2}x-\frac{π}{3})$, the graph is shifted to the right by $φ(0＜φ＜\frac{π}{2})$ units to obtain the graph of the function $g(x)$, and $g(x)+g(-x)=0$. Determine the value of $g(2φ+\frac{π}{6})$. | {
"answer": "\\frac{\\sqrt{6}+\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \sqrt {2}\cos (x+ \frac {\pi}{4})$, after translating the graph of $f(x)$ by the vector $\overrightarrow{v}=(m,0)(m > 0)$, the resulting graph exactly matches the function $y=f′(x)$. The minimum value of $m$ is \_\_\_\_\_\_. | {
"answer": "\\frac {3\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Circle $\Gamma$ has diameter $\overline{AB}$ with $AB = 6$ . Point $C$ is constructed on line $AB$ so that $AB = BC$ and $A \neq C$ . Let $D$ be on $\Gamma$ so that $\overleftrightarrow{CD}$ is tangent to $\Gamma$ . Compute the distance from line $\overleftrightarrow{AD}$ to the circumcenter of $\triangle ADC$ .
*Proposed by Justin Hsieh* | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ a,b,c,d$ be rational numbers with $ a>0$ . If for every integer $ n\ge 0$ , the number $ an^{3} \plus{}bn^{2} \plus{}cn\plus{}d$ is also integer, then the minimal value of $ a$ will be | {
"answer": "$\\frac{1}{6}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=e^{x}\cos x-x$.
(Ⅰ) Find the equation of the tangent line to the curve $y=f(x)$ at the point $(0,f(0))$;
(Ⅱ) Find the maximum and minimum values of the function $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$. | {
"answer": "-\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Place four balls numbered 1, 2, 3, and 4 into three boxes labeled A, B, and C.
(1) If none of the boxes are empty and ball number 3 must be in box B, how many different arrangements are there?
(2) If ball number 1 cannot be in box A and ball number 2 cannot be in box B, how many different arrangements are there? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle $C: (x-1)^{2} + (y-2)^{2} = 25$ and a line $l: mx-y-3m+1=0$ intersect at points $A$ and $B$. Find the minimum value of $|AB|$. | {
"answer": "4\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ and $y$ be positive real numbers. Find the minimum value of
\[x^2 + y^2 + \frac{4}{(x + y)^2}.\] | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $20$-quart radiator initially contains a mixture of $18$ quarts of water and $2$ quarts of antifreeze. Six quarts of the mixture are removed and replaced with pure antifreeze liquid. This process is repeated three more times. Calculate the fractional part of the final mixture that is water.
**A)** $\frac{10.512}{20}$
**B)** $\frac{1}{3}$
**C)** $\frac{7.42}{20}$
**D)** $\frac{4.322}{20}$
**E)** $\frac{10}{20}$ | {
"answer": "\\frac{4.322}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $y^2 = 4x$ whose directrix intersects the x-axis at point $P$, draw line $l$ through point $P$ with the slope $k (k > 0)$, intersecting the parabola at points $A$ and $B$. Let $F$ be the focus of the parabola. If $|FB| = 2|FA|$, then calculate the length of segment $AB$. | {
"answer": "\\frac{\\sqrt{17}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$(1)$ Calculate: $2^{-1}+|\sqrt{6}-3|+2\sqrt{3}\sin 45^{\circ}-\left(-2\right)^{2023}\cdot (\frac{1}{2})^{2023}$.
$(2)$ Simplify and then evaluate: $\left(\frac{3}{a+1}-a+1\right) \div \frac{{{a}^{2}}-4}{{{a}^{2}}+2a+1}$, where $a$ takes a suitable value from $-1$, $2$, $3$ for evaluation. | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\theta \in (0^\circ, 360^\circ)$ and the terminal side of angle $\theta$ is symmetric to the terminal side of the $660^\circ$ angle with respect to the x-axis, and point $P(x, y)$ is on the terminal side of angle $\theta$ (not the origin), find the value of $$\frac {xy}{x^{2}+y^{2}}.$$ | {
"answer": "\\frac {\\sqrt {3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $\sin (C-A)=1$, $\sin B= \frac{1}{3}$.
(I) Find the value of $\sin A$;
(II) Given $b= \sqrt{6}$, find the area of $\triangle ABC$. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $A$, $B$, $C$ are the three internal angles of $\triangle ABC$, and their respective opposite sides are $a$, $b$, $c$, and $2\cos ^{2} \frac {A}{2}+\cos A=0$.
(1) Find the value of angle $A$;
(2) If $a=2 \sqrt {3},b+c=4$, find the area of $\triangle ABC$. | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of the infinite product $(3^{1/4})(9^{1/16})(27^{1/64})(81^{1/256}) \dotsm$ plus 2, the result in the form of "$\sqrt[a]{b}$ plus $c$". | {
"answer": "\\sqrt[9]{81} + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(0 \le x_0 < 1\), let
\[x_n = \left\{ \begin{array}{ll}
3x_{n-1} & \text{if } 3x_{n-1} < 1 \\
3x_{n-1} - 1 & \text{if } 1 \le 3x_{n-1} < 2 \\
3x_{n-1} - 2 & \text{if } 3x_{n-1} \ge 2
\end{array}\right.\]
for all integers \(n > 0\), determine the number of values of \(x_0\) for which \(x_0 = x_6\). | {
"answer": "729",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a, b, c \) be real numbers such that \( 9a^2 + 4b^2 + 25c^2 = 1 \). Find the maximum value of
\[ 3a + 4b + 5c. \] | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function f(x) = $\frac{1}{3}$x^3^ + $\frac{1−a}{2}$x^2^ - ax - a, x ∈ R, where a > 0.
(1) Find the monotonic intervals of the function f(x);
(2) If the function f(x) has exactly two zeros in the interval (-3, 0), find the range of values for a;
(3) When a = 1, let the maximum value of the function f(x) on the interval [t, t+3] be M(t), and the minimum value be m(t). Define g(t) = M(t) - m(t), find the minimum value of the function g(t) on the interval [-4, -1]. | {
"answer": "\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a complex number $z$ that satisfies the following two conditions:
① $1 < z + \frac{2}{z} \leqslant 4$.
② The real part and the imaginary part of $z$ are both integers, and the corresponding point in the complex plane is located in the fourth quadrant.
(I) Find the complex number $z$;
(II) Calculate $|\overline{z} + \frac{2 - i}{2 + i}|$. | {
"answer": "\\frac{\\sqrt{65}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest composite number that has no prime factors less than 20. | {
"answer": "667",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f\left(x\right)=x-{e}^{-x}$, if the line $y=mx+n$ is a tangent line to the curve $y=f\left(x\right)$, find the minimum value of $m+n$. | {
"answer": "1-\\dfrac{1}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of the distances from one vertex of a rectangle with sides of lengths $3$ and $5$ to the midpoints of each of the sides of the rectangle.
A) $11.2$
B) $12.4$
C) $13.1$
D) $14.5$
E) $15.2$ | {
"answer": "13.1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $\frac{{\cos190°(1+\sqrt{3}\tan10°)}}{{\sin290°\sqrt{1-\cos40°}}}=\_\_\_\_\_\_$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ S $ be the set of all sides and diagonals of a regular hexagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? | {
"answer": "\\frac{17}{35}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $y=f(x)$ is symmetric around the line $y=x$. After moving it left by one unit, the graph is still symmetric around the line $y=x$. If $f(1)=0$, then $f(2011)=$ __(A)__. | {
"answer": "-2010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The stem and leaf plot shows the heights, in inches, of the players on the Westvale High School boys' basketball team. Calculate the mean height of the players on the team. (Note: $6|2$ represents 62 inches.)
Height of the Players on the Basketball Team (inches)
$5|7$
$6|2\;4\;4\;5\;7\;8$
$7|0\;1\;2\;2\;3\;4\;5\;5$ | {
"answer": "68.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest prime divisor of $36^2 + 49^2$. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the distinct digits \( a, b, \) and \( c \), Araceli wrote the number \( abc \), and Luana wrote the numbers \( ab, bc, \) and \( ca \). Find the digits \( a, b, \) and \( c \), knowing that the sum of the numbers written by Luana is equal to the number written by Araceli. | {
"answer": "198",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is now 3:00:00 PM, as read on a 12-hour digital clock. In 189 hours, 58 minutes, and 52 seconds, the time will be $X:Y:Z$ on the clock. What is the value of $X + Y + Z$? | {
"answer": "122",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\ln x - ax^2 + 3$,
(1) Discuss the monotonicity of the function $y=f(x)$;
(2) If there exist real numbers $m, n \in [1, 5]$ such that $f(m)=f(n)$ holds when $n-m \geq 2$, find the maximum value of the real number $a$. | {
"answer": "\\frac{\\ln 3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let point $P$ be the intersection point in the first quadrant of the hyperbola $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ and the circle $x^{2}+y^{2}=a^{2}+b^{2}$. Let $F\_1$ and $F\_2$ be the left and right foci of the hyperbola, respectively, and $|PF\_1| = 2|PF\_2|$. Find the eccentricity of the hyperbola. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is
$$
\begin{cases}
x=2- \frac { \sqrt {2}}{2}t \\
y=1+ \frac { \sqrt {2}}{2}t
\end{cases}
(t \text{ is the parameter}).
$$
In the polar coordinate system (using the same unit length as the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis), the equation of circle $C$ is $\rho=4\cos \theta$.
- (I) Find the Cartesian coordinate equation of circle $C$;
- (II) Suppose circle $C$ intersects line $l$ at points $A$ and $B$. If the coordinates of point $P$ are $(2,1)$, find $|PA|+|PB|$. | {
"answer": "\\sqrt {14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α$ is an angle in the second quadrant, let point $P(x, \sqrt {5})$ be a point on the terminal side of $α$, and $\cos α= \frac { \sqrt {2}}{4}x$. Find the value of $4\cos (α+ \frac {π}{2})-3\tan α$. | {
"answer": "\\sqrt {15}- \\sqrt {10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$ , $b$ and $c$ . | {
"answer": "719",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin \alpha + \cos \alpha = \frac{\sqrt{2}}{3}$, and $0 < \alpha < \pi$, find $\tan\left(\alpha - \frac{\pi}{4}\right) = \_\_\_\_\_\_\_\_\_\_.$ | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine how many prime dates occurred in 2008, a leap year. A "prime date" is when both the month and the day are prime numbers. | {
"answer": "53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A kite is inscribed in a circle with center $O$ and radius $60$ . The diagonals of the kite meet at a point $P$ , and $OP$ is an integer. The minimum possible area of the kite can be expressed in the form $a\sqrt{b}$ , where $a$ and $b$ are positive integers and $b$ is squarefree. Find $a+b$ . | {
"answer": "239",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $a_{1} < a_{2}< \cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.
*Ray Li* | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Since 2021, the "Study Strong Country" app has launched a "Four-Person Match" answer module. The rules are as follows: Users need to answer two rounds of questions in the "Four-Person Match". At the beginning of each round, the system will automatically match 3 people to answer questions with the user. At the end of each round, the four participants will be ranked first, second, third, and fourth based on their performance. In the first round, the first place earns 3 points, the second and third places earn 2 points each, and the fourth place earns 1 point. In the second round, the first place earns 2 points, and the rest earn 1 point each. The sum of the scores from the two rounds is the total score of the user in the "Four-Person Match". Assuming that the user has an equal chance of getting first, second, third, or fourth place in the first round; if the user gets first place in the first round, the probability of getting first place in the second round is 1/5, and if the user does not get first place in the first round, the probability of getting first place in the second round is 1/3.
$(1)$ Let the user's score in the first round be $X$, find the probability distribution of $X$;
$(2)$ Find the expected value of the user's total score in the "Four-Person Match". | {
"answer": "3.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin \alpha - \cos \alpha = \frac{1}{5}$, and $0 \leqslant \alpha \leqslant \pi$, find the value of $\sin (2\alpha - \frac{\pi}{4})$ = $\_\_\_\_\_\_\_\_$. | {
"answer": "\\frac{31\\sqrt{2}}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white and white becomes black. We call this operation a *step*. If $C$ is the original coloring, let $S(C)$ be the least number of steps required to make all the unit squares black. Find with proof the greatest possible value of $S(C)$ . | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There exist constants $a_1, a_2, a_3, a_4, a_5, a_6, a_7$ such that
\[
\cos^7 \theta = a_1 \cos \theta + a_2 \cos 2 \theta + a_3 \cos 3 \theta + a_4 \cos 4 \theta + a_5 \cos 5 \theta + a_6 \cos 6 \theta + a_7 \cos 7 \theta
\]
for all angles $\theta.$ Find $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 + a_6^2 + a_7^2.$ | {
"answer": "\\frac{1716}{4096}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=\sin (\pi x+\varphi)-2\cos (\pi x+\varphi)$ $(0 < \varphi < \pi)$, its graph is symmetric about the line $x=1$. Find $\sin 2\varphi$. | {
"answer": "- \\frac {4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40, m] = 120$ and $\mathop{\text{lcm}}[m, 45] = 180$, what is $m$? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A and B start from points A and B simultaneously, moving towards each other and meet at point C. If A starts 2 minutes earlier, then their meeting point is 42 meters away from point C. Given that A's speed is \( a \) meters per minute, B's speed is \( b \) meters per minute, where \( a \) and \( b \) are integers, \( a > b \), and \( b \) is not a factor of \( a \). What is the value of \( a \)? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the volumes of the solids obtained by rotating the regions bounded by the graphs of the functions about the y-axis.
$$
y=x^{3}, \quad y=x
$$ | {
"answer": "\\frac{4\\pi}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the vectors $\overrightarrow{m}=(x,y)$ and $\overrightarrow{n}=(x-y)$, let $P$ be a moving point on the curve $\overrightarrow{m}\cdot \overrightarrow{n}=1 (x > 0)$. If the distance from point $P$ to the line $x-y+1=0$ is always greater than $\lambda$, find the maximum value of the real number $\lambda$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A semicircular sheet of iron with a radius of 6 is rolled into the lateral surface of a cone. The volume of this cone is \_\_\_\_\_\_. | {
"answer": "9\\sqrt{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many whole numbers between $200$ and $500$ contain the digit $3$? | {
"answer": "138",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\alpha$ and $\beta \in \left(0,\pi \right)$, where $\tan \alpha$ and $\tan \beta$ are two roots of the equation ${x^2}+3\sqrt{3}x+4=0$, find the value of $\alpha +\beta$. | {
"answer": "\\frac{4\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \( x \) and \( y \) satisfy the equation:
$$
\sqrt{x y}+\sqrt{(1-x)(1-y)}=\sqrt{7 x(1-y)}+\frac{\sqrt{y(1-x)}}{\sqrt{7}}
$$
Find the maximum value of the expression \( x + 7y \). Justify your answer. | {
"answer": "57/8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right tetrahedron \(ABCD\), \(DA\), \(DB\), and \(DC\) are mutually perpendicular. Let \(S\) and \(R\) represent its surface area and the radius of the circumscribed sphere, respectively. What is the maximum value of \(\frac{S}{R^2}\)? | {
"answer": "\\frac{2}{3}(3+\\sqrt{3})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
According to census statistics, the probability of a woman of childbearing age giving birth to a boy or a girl is equal. If a second child is allowed, calculate the probability that a woman of childbearing age will have two girls. | {
"answer": "\\frac {1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain set of integers is assigned to the letters of the alphabet such that $H=10$. The value of a word is the sum of its assigned letter values. Given that $THIS=50$, $HIT=35$ and $SIT=40$, find the value of $I$. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given three lines $l_1$: $4x+y-4=0$, $l_2$: $mx+y=0$, $l_3$: $x-my-4=0$ that do not intersect at the same point:
(1) When these three lines cannot form a triangle, find the value of the real number $m$.
(2) When $l_3$ is perpendicular to both $l_1$ and $l_2$, find the distance between the two foot points. | {
"answer": "\\frac{4\\sqrt{17}}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Julia invested a certain amount of money in two types of assets: real estate and mutual funds. The total amount she invested was $\$200,000$. If she invested 6 times as much in real estate as she did in mutual funds, what was her total investment in real estate? | {
"answer": "171,428.58",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, let $a$, $b$, and $c$ be the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $\cos B = \frac{4}{5}$ and $b = 2$.
1. Find the value of $a$ when $A = \frac{\pi}{6}$.
2. Find the value of $a + c$ when the area of $\triangle ABC$ is $3$. | {
"answer": "2\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest possible value of the expression $$\frac{(2a+3b)^2 + (b-c)^2 + (2c-a)^2}{b^2}$$ where \( b > a > c \) are real numbers, and \( b \neq 0 \). | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the base-$10$ logarithms of the divisors of $6^n$ is $540$. What is $n$?
A) 9
B) 10
C) 11
D) 12
E) 13 | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Below is pictured a regular seven-pointed star. Find the measure of angle \( a \) in radians. | {
"answer": "\\frac{5\\pi}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ is on the line $y=2x+1$, and point $Q$ is on the curve $y=x+\ln x$, determine the minimum distance between points $P$ and $Q$. | {
"answer": "\\frac{2\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with an eccentricity of $\frac{\sqrt{3}}{2}$ and a length of the minor axis of $4$. <br/>$(1)$ Find the equation of the ellipse; <br/>$(2)$ A chord passing through $P(2,1)$ divides $P$ in half. Find the equation of the line containing this chord and the length of the chord. | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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