problem stringlengths 10 5.15k | answer dict |
|---|---|
Louise is designing a custom dress and needs to provide her hip size in millimeters. If there are $12$ inches in a foot and $305$ millimeters in a foot, and Louise's hip size is $42$ inches, what size should she specify in millimeters? | {
"answer": "1067.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of the shape enclosed by the curve $y=x^2$ (where $x>0$), the tangent line at point A(2, 4), and the x-axis. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $A=\{x|ax^{2}+bx+c\leqslant 0\left(a \lt b\right)\}$ has one and only one element, then the minimum value of $M=\frac{{a+3b+4c}}{{b-a}}$ is ______. | {
"answer": "2\\sqrt{5} + 5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a biased coin with probabilities of $\frac{3}{4}$ for heads and $\frac{1}{4}$ for tails, determine the difference between the probability of winning Game A, which involves 4 coin tosses and at least three heads, and the probability of winning Game B, which involves 5 coin tosses with the first two tosses and the ... | {
"answer": "\\frac{89}{256}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $a$, $b$, $c$ are the opposite sides of the internal angles $A$, $B$, $C$, respectively, and $\sin ^{2}A+\sin A\sin C+\sin ^{2}C+\cos ^{2}B=1$.
$(1)$ Find the measure of angle $B$;
$(2)$ If $a=5$, $b=7$, find $\sin C$. | {
"answer": "\\frac{3\\sqrt{3}}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the acute triangle \(KLM\), \(V\) is the intersection of its heights, and \(X\) is the foot of the height onto side \(KL\). The angle bisector of angle \(XVL\) is parallel to side \(LM\), and the angle \(MKL\) measures \(70^\circ\).
What are the measures of the angles \(KLM\) and \(KML\)? | {
"answer": "55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xoy$, the parametric equation of line $l$ is $\begin{cases}x= \frac{ \sqrt{2}}{2}t \\ y=3+ \frac{ \sqrt{2}}{2}t\end{cases} (t$ is the parameter$)$, in the polar coordinate system with $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$... | {
"answer": "\\frac{2 \\sqrt{5}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Take a standard set of 28 dominoes and put back double 3, double 4, double 5, and double 6, as they will not be needed. Arrange the remaining dominoes to form 3 square frames, as shown in the image, so that the sum of the points along each side is equal. In the given example, these sums are equal to 15. If this is one ... | {
"answer": "15",
"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 $\cos A= \frac{4}{5}$.
(1) Find the value of $\sin ^{2} \frac{B+C}{2}+\cos 2A$;
(2) If $b=2$, the area of $\triangle ABC$ is $S=3$, find $a$. | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("flipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it). Thus, Student 3 will clo... | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A supermarket purchases two types of goods, $A$ and $B$. Buying 4 items of type $A$ costs $10$ yuan less than buying 5 items of type $B$. Buying 20 items of type $A$ and 10 items of type $B$ costs a total of $160$ yuan.
$(1)$ Find the cost price per item of goods $A$ and $B$ respectively.
$(2)$ If the store purchas... | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many $n=2,3,4,\ldots,109,110$ is the base-$n$ number $432143_n$ a multiple of $11$? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $| \overrightarrow{a}|=2$, $\overrightarrow{e}$ is a unit vector, and the angle between $\overrightarrow{a}$ and $\overrightarrow{e}$ is $\dfrac {\pi}{3}$, find the projection of $\overrightarrow{a}+ \overrightarrow{e}$ on $\overrightarrow{a}- \overrightarrow{e}$. | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is now 3:15:20 PM, as read on a 12-hour digital clock. In 305 hours, 45 minutes, and 56 seconds, the time will be $X:Y:Z$. What is the value of $X + Y + Z$? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the Horner's method (also known as Qin Jiushao's algorithm), calculate the value of the polynomial \\(f(x)=12+35x-8x^{2}+79x^{3}+6x^{4}+5x^{5}+3x^{6}\\) when \\(x=-4\\), and determine the value of \\(V_{3}\\). | {
"answer": "-57",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sequence $\{a_n\}$ is a geometric sequence, and the sequence $\{b_n\}$ is an arithmetic sequence. If $a_1-a_6-a_{11}=-3\sqrt{3}$ and $b_1+b_6+b_{11}=7\pi$, then the value of $\tan \frac{b_3+b_9}{1-a_4-a_3}$ is ______. | {
"answer": "-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. It is given that $(2c-a)\cos B = b\cos A$.
1. Find angle $B$.
2. If $b=6$ and $c=2a$, find the area of $\triangle ABC$. | {
"answer": "6\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\alpha$ , $\beta$ , and $\gamma$ are the roots of $x^3 - x - 1 = 0$ , compute $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$ . | {
"answer": "-7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a triangular pyramid $P-ABC$, $PC \perp$ plane $ABC$, $\angle CAB=90^{\circ}$, $PC=3$, $AC=4$, $AB=5$, find the surface area of the circumscribed sphere of the triangular pyramid. | {
"answer": "50\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest value of $n$ less than 50,000 for which the expression $3(n-3)^2 - 4n + 28$ is a multiple of 7? | {
"answer": "49999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=(\sin x+\cos x)^{2}+2\cos ^{2}x-2$.
$(1)$ Find the smallest positive period and the intervals of monotonic increase for the function $f(x)$;
$(2)$ When $x\in\left[ \frac {\pi}{4}, \frac {3\pi}{4}\right]$, find the maximum and minimum values of the function $f(x)$. | {
"answer": "- \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( x^{4} + ax^{2} + bx + c = 0 \) has roots 1, 2, and 3 (one root is repeated), find \( a + c \).
(17th Annual American High School Mathematics Examination, 1966) | {
"answer": "-61",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a set of paired data $(18,24)$, $(13,34)$, $(10,38)$, $(-1,m)$, the regression equation for these data is $y=-2x+59.5$. Find the correlation coefficient $r=$______(rounded to $0.001$). | {
"answer": "-0.998",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $PQR$ has vertices $P = (4,0)$, $Q = (0,4)$, and $R$, where $R$ is on the line $x + y = 8$ and also on the line $y = 2x$. Find the area of $\triangle PQR$.
A) $\frac{4}{3}$
B) $\frac{6}{3}$
C) $\frac{8}{3}$
D) $\frac{10}{3}$
E) $\frac{12}{3}$ | {
"answer": "\\frac{8}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify and evaluate:
(1) Calculate the value of $\frac {1}{\log_{4}6}+6^{\log_{6} \sqrt {3}-1}-2\log_{6} \frac {1}{3}$;
(2) Given $\tan\alpha=2$ and $\sin\alpha+\cos\alpha < 0$, find the value of $\frac {\tan(\pi-\alpha)\cdot \sin(-\alpha+ \frac {3\pi}{2})}{\cos(\pi +\alpha )\cdot \sin(-\pi -\alpha )}$. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cylinder has a radius of 2 inches and a height of 3 inches. What is the radius of a sphere that has the same volume as this cylinder? | {
"answer": "\\sqrt[3]{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The constant term in the expansion of $(x^2-2)\left(x-\frac{2}{\sqrt{x}}\right)^{6}$ is ______. | {
"answer": "-480",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,$ $b,$ $c,$ $d$ be real numbers such that $a + b + c + d = 10$ and
\[ab + ac + ad + bc + bd + cd = 20.\] Find the largest possible value of $d$. | {
"answer": "\\frac{5 + 5\\sqrt{21}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the graph of \( y = g(x) \), with \( 1 \) unit between grid lines, where \( g(x) = \frac{(x-4)(x-2)(x)(x+2)(x+4)(x+6)}{720} - 2.5 \), defined only on the shown domain.
Determine the sum of all integers \( c \) for which the equation \( g(x) = c \) has exactly \( 4 \) solutions.
[asy]
size(150);
real f(real ... | {
"answer": "-5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The fraction of the area of rectangle P Q R S that is shaded must be calculated. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify: $$\sqrt[3]{5488000}$$ | {
"answer": "176.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Laura typically performs her routine using two 10-pound dumbbells for 30 repetitions. If she decides to use two 8-pound dumbbells instead, how many repetitions must she complete to lift the same total weight as her usual routine? | {
"answer": "37.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, medians $XM$ and $YN$ intersect at $Q$, $QN=3$, $QM=4$, and $MN=5$. What is the area of $XMYN$? | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$ , point $D$ lies on side $AC$ such that $\angle ABD=\angle C$ . Point $E$ lies on side $AB$ such that $BE=DE$ . $M$ is the midpoint of segment $CD$ . Point $H$ is the foot of the perpendicular from $A$ to $DE$ . Given $AH=2-\sqrt{3}$ and $AB=1$ , find the size of $\angle AME$ . | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many triangles can be formed using the vertices of a regular hexacontagon (a 60-sided polygon), avoiding the use of any three consecutive vertices in forming these triangles? | {
"answer": "34160",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $f: x \rightarrow \sqrt{x}$ is a function from set $A$ to set $B$.
1. If $A=[0,9]$, then the range of the function $f(x)$ is ________.
2. If $B={1,2}$, then $A \cap B =$ ________. | {
"answer": "{1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = |\overrightarrow{b}| = 1$ and $|3\overrightarrow{a} - 2\overrightarrow{b}| = \sqrt{7}$,
(I) Find the magnitude of the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$;
(II) Find the value of $|3\overrightarrow{... | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle and $k$ be a positive number such that altitudes $AD$, $BE$, and $CF$ are extended past $A$, $B$, and $C$ to points $A'$, $B'$, and $C'$ respectively, where $AA' = kBC$, $BB' = kAC$, and $CC' = kAB$. Suppose further that $A''$ is a point such that the line segment $AA''$ is a rotation of line se... | {
"answer": "\\frac{1}{\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $\frac{7}{4} \times \frac{8}{14} \times \frac{14}{8} \times \frac{16}{40} \times \frac{35}{20} \times \frac{18}{45} \times \frac{49}{28} \times \frac{32}{64}$ | {
"answer": "\\frac{49}{200}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Lil writes one of the letters \( \text{P}, \text{Q}, \text{R}, \text{S} \) in each cell of a \( 2 \times 4 \) table. She does this in such a way that, in each row and in each \( 2 \times 2 \) square, all four letters appear. In how many ways can she do this? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \(a, b, c, d\) belong to the interval \([-4.5, 4.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the system of inequalities about $x$ is $\left\{{\begin{array}{l}{-2({x-2})-x＜2}\\{\frac{{k-x}}{2}≥-\frac{1}{2}+x}\end{array}}\right.$ has at most $2$ integer solutions, and the solution to the one-variable linear equation about $y$ is $3\left(y-1\right)-2\left(y-k\right)=7$, determine the sum of all integers $k$ th... | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is $2\frac{1}{4}$ divided by $\frac{3}{5}$? | {
"answer": "3 \\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equation of curve $C_1$ as $$\begin{cases} x=2t-1 \\ y=-4t-2 \end{cases}$$ (where $t$ is the parameter), and establishing a polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C_2$ is $$\rho= \frac {2}{1-\cos\thet... | {
"answer": "\\frac {3 \\sqrt {5}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F\_1$ and $F\_2$ are the two foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$, and $P$ is a point on the hyperbola such that $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2} = 0$ and $|\overrightarrow{PF\_1}| \cdot |\overrightarrow{PF\_2}| = 2ac (c$ is the semi-focal distance$)$... | {
"answer": "\\frac{\\sqrt{5} + 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse C: $\frac{x^{2}}{3}+y^{2}=1$ with left focus and right focus as $F_{1}$ and $F_{2}$ respectively. The line $y=x+m$ intersects C at points A and B. If the area of $\triangle F_{1}AB$ is twice the area of $\triangle F_{2}AB$, find $m$. | {
"answer": "-\\frac{\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $\frac{1}{{{x^2}+2x+1}}\cdot (1+\frac{3}{x-1})\div \frac{x+2}{{{x^2}-1}$, where $x=2\sqrt{5}-1$. | {
"answer": "\\frac{\\sqrt{5}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From six balls numbered respectively $1$, $2$, $3$, $4$, $5$, $6$, which are all of the same size, three balls are randomly drawn. Find the probability that exactly two of the balls have consecutive numbers. | {
"answer": "\\dfrac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a finite sequence \(P = \left(p_{1}, p_{2}, \cdots, p_{n}\right)\), the Caesar sum (named after a mathematician Caesar) is defined as \(\frac{s_{1}+s_{2}+\cdots+s_{n}}{n}\), where \(s_{k} = p_{1} + p_{2} + \cdots + p_{k}\) for \(1 \leq k \leq n\). If a sequence of 99 terms \(\left(p_{1}, p_{2}, \cdots, p_{99}\right... | {
"answer": "991",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In writing the integers from 20 through 99 inclusive, how many times is the digit 7 written? | {
"answer": "18",
"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. Given that $a > b$, $a=5$, $c=6$, and $\sin B=\dfrac{3}{5}$.
1. Find the values of $b$ and $\sin A$.
2. Find the value of $\sin (2A+\dfrac{\pi}{4})$. | {
"answer": "\\dfrac {7 \\sqrt {2}}{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The set of vectors $\mathbf{u}$ such that
\[\mathbf{u} \cdot \mathbf{u} = \mathbf{u} \cdot \begin{pmatrix} 8 \\ -28 \\ 12 \end{pmatrix}\]forms a solid in space. Find the volume of this solid. | {
"answer": "\\frac{4}{3} \\pi \\cdot 248^{3/2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of the larger base of a truncated pyramid is $T$, the area of the smaller base is $t$, and the height is $m$. The volume is $V = kTm$, where $\frac{1}{3} \leq k \leq 1$. What is the proportionality ratio $\lambda(<1)$ between the two bases? (The proportionality ratio is the ratio of corresponding distances in ... | {
"answer": "\\frac{2 - \\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the geometric sequence $\{a_{n}\}$, $a_{3}$ and $a_{7}$ are two distinct extreme points of the function $f\left(x\right)=\frac{1}{3}x^{3}+4x^{2}+9x-1$. Find $a_{5}$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sample of size 66 with a frequency distribution as follows: $(11.5, 15.5]$: $2$, $(15.5, 19.5]$: $4$, $(19.5, 23.5]$: $9$, $(23.5, 27.5]$: $18$, $(27.5, 31.5]$: $11$, $(31.5, 35.5]$: $12$, $[35.5, 39.5)$: $7$, $[39.5, 43.5)$: $3$, estimate the probability that the data falls in [31.5, 43.5). | {
"answer": "\\frac{1}{3}",
"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 they satisfy the equation $$\frac {2c-b}{a} = \frac {\cos{B}}{\cos{A}}$$. If $a = 2\sqrt {5}$, find the maximum value of $b + c$. | {
"answer": "4\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\frac{1}{2}c\sin B=(c-a\cos B)\sin C$.
$(1)$ Find angle $A$;
$(2)$ If $D$ is a point on side $AB$ such that $AD=2DB$, $AC=2$, and $BC=\sqrt{7}$, find the area of triangle $\triangle ACD$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Mr. Fox's class, there are seven more girls than boys, and the total number of students is 35. What is the ratio of the number of girls to the number of boys in his class?
**A)** $2 : 3$
**B)** $3 : 2$
**C)** $4 : 3$
**D)** $5 : 3$
**E)** $7 : 4$ | {
"answer": "3 : 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During the New Year's Day holiday of 2018, 8 high school students from four different classes, each with 2 students, plan to carpool for a trip. Classes are denoted as (1), (2), (3), and (4). They will be divided between two cars, Car A and Car B, each with a capacity for 4 students (seating positions within the same c... | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $y^{2}=4x$ and the circle $E:(x-4)^{2}+y^{2}=12$. Let $O$ be the origin. The line passing through the center of circle $E$ intersects the circle at points $A$ and $B$. The lines $OA$ and $OB$ intersect the parabola at points $P$ and $Q$ (points $P$ and $Q$ are not coincident with point $O$). Let $S_{... | {
"answer": "\\frac{9}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A whole number, $M$, is chosen so that $\frac{M}{4}$ is strictly between 8 and 9. What is the value of $M$? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An electronic flea lands on a point $k$ on the number line. In the first step, it jumps 1 unit to the left to $k_1$, in the second step it jumps 2 units to the right to $k_2$, in the third step it jumps 3 units to the left to $k_3$, in the fourth step it jumps 4 units to the right to $k_4$, and so on. Following this pa... | {
"answer": "-30.06",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum possible volume of a cylinder inscribed in a cone with a height of 27 and a base radius of 9. | {
"answer": "324\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p$, $q$, $r$, $s$, and $t$ be positive integers such that $p+q+r+s+t=2022$. Let $N$ be the largest of the sum $p+q$, $q+r$, $r+s$, and $s+t$. What is the smallest possible value of $N$? | {
"answer": "506",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 2$, and $\overrightarrow{a} \cdot \overrightarrow{b} = -\sqrt{3}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest sum of distances from an arbitrary point on the plane to the vertices of a unit square.
A problem of the shortest connection for four points. Four points: \(A, B, C,\) and \(D\) are the vertices of a square with side length 1. How should these points be connected by roads to ensure that it is possibl... | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest number, all of whose digits are either 5, 3, or 1, and whose digits add up to $15$? | {
"answer": "555",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles, each with a radius of 12 cm, overlap such that each circle passes through the center of the other. Determine the length, in cm, of the common chord formed by these circles. Express your answer in simplest radical form. | {
"answer": "12\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the integer is a 4-digit positive number with four different digits, the leading digit is not zero, the integer is a multiple of 5, 7 is the largest digit, and the first and last digits of the integer are the same, calculate the number of such integers. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ m\equal{}\left(abab\right)$ and $ n\equal{}\left(cdcd\right)$ be four-digit numbers in decimal system. If $ m\plus{}n$ is a perfect square, find the largest value of $ a\cdot b\cdot c\cdot d$. | {
"answer": "600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line $l_{1}$: $x+my+6=0$ is parallel to the line $l_{2}$: $(m-2)x+3y+2m=0$. Find the value of $m$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\overrightarrow{OA}=(-2,4)$, $\overrightarrow{OB}=(-a,2)$, $\overrightarrow{OC}=(b,0)$, $a > 0$, $b > 0$, and $O$ is the coordinate origin. If points $A$, $B$, and $C$ are collinear, find the minimum value of $\frac{1}{a}+\frac{1}{b}$. | {
"answer": "\\frac{3 + 2\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The World Cup football tournament is held in Brazil, and the host team Brazil is in group A. In the group stage, the team plays a total of 3 matches. The rules stipulate that winning one match scores 3 points, drawing one match scores 1 point, and losing one match scores 0 points. If the probability of Brazil winning, ... | {
"answer": "0.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)$ be a function defined on $\mathbb{R}$ with a minimum positive period of $3\pi$, and its expression in the interval $(-\pi,2\pi]$ is $f(x)= \begin{cases} \sin x & (0\leqslant x\leqslant 2\pi) \\ \cos x & (-\pi < x < 0) \end{cases}$. Evaluate the expression $f(- \frac {308\pi}{3})+f( \frac {601\pi}{6})$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The median of the numbers 3, 7, x, 14, 20 is equal to the mean of those five numbers. Calculate the sum of all real numbers \( x \) for which this is true. | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A club has between 150 and 250 members. Every month, all the members meet up for a group activity that requires the members to be divided into seven distinct groups. If one member is unable to attend, the remaining members can still be evenly divided into the seven groups. Calculate the sum of all possible numbers of m... | {
"answer": "2807",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $\{a\_n\}$ satisfies $a\_1=1$, $a\_2=3$, and $a_{n+2}=|a_{n+1}|-a_{n}$, where $n∈N^{*}$. Let $S_{n}$ denote the sum of the first $n$ terms of the sequence. Find $S_{100}$. | {
"answer": "89",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle of radius $3$ is cut into six congruent arcs. These arcs are then rearranged symmetrically to form a hexagonal star as illustrated below. Determine the ratio of the area of the hexagonal star to the area of the original circle.
A) $\frac{4.5}{\pi}$
B) $\frac{4.5\sqrt{2}}{\pi}$
C) $\frac{4.5\sqrt{3}}{\pi}$
D) $... | {
"answer": "\\frac{4.5\\sqrt{3}}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a \) and \( b \) be nonnegative real numbers such that
\[
\sin (ax + b) = \sin 17x
\]
for all integers \( x \). Find the smallest possible value of \( a \). | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $3\vec{a} + 4\vec{b} + 5\vec{c} = 0$ and $|\vec{a}| = |\vec{b}| = |\vec{c}| = 1$, calculate $\vec{b} \cdot (\vec{a} + \vec{c})$. | {
"answer": "-\\dfrac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, each of the five boxes is to contain a number. Each number in a shaded box must be the average of the number in the box to the left of it and the number in the box to the right of it. Given the numbers:
| 8 | | | 26 | $x$ |
What is the value of $x$ ? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere intersects the $xy$-plane in a circle centered at $(3,5,0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0,5,-8),$ with radius $r.$ Find $r.$ | {
"answer": "\\sqrt{59}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A merchant's cumulative sales from January to May reached 38.6 million yuan. It is predicted that the sales in June will be 5 million yuan, the sales in July will increase by x% compared to June, and the sales in August will increase by x% compared to July. The total sales in September and October are equal to the tota... | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \sin(\omega x + \varphi)$ ($\omega > 0$, $|\varphi| < \frac{\pi}{2}$) has a minimum positive period of $\pi$, and its graph is translated to the right by $\frac{\pi}{6}$ units to obtain the graph of the function $g(x) = \sin(\omega x)$, determine the value of $\varphi$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Use the Horner's method to calculate the value of the polynomial $f(x) = 2x^5 + 5x^3 - x^2 + 9x + 1$ when $x = 3$. What is the value of $v_3$ in the third step? | {
"answer": "68",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a > 0, b > 0$ and $a+b=1$, find the minimum value of $\frac{1}{a} + \frac{2}{b}$. | {
"answer": "3 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $A$, $B$, and $C$ are the three interior angles of $\triangle ABC$, then the minimum value of $$\frac {4}{A}+ \frac {1}{B+C}$$ is \_\_\_\_\_\_. | {
"answer": "\\frac {9}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve for $x$ if $\frac{2}{x+3} + \frac{3x}{x+3} - \frac{4}{x+3} = 4$. | {
"answer": "-14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the $xy$-plane, consider a T-shaped region bounded by horizontal and vertical segments with vertices at $(0,0), (0,4), (4,4), (4,2), (7,2), (7,0)$. What is the slope of the line through the origin that divides the area of this region exactly in half?
A) $\frac{1}{4}$
B) $\frac{1}{3}$
C) $\frac{1}{2}$
D) $\frac{2}{3}... | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equations of curve $C_1$ are $$\begin{cases} x=2\cos\theta \\ y= \sqrt {3}\sin\theta\end{cases}$$ (where $\theta$ is the parameter), with the origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, establishing a polar coordinate system with the same unit length, th... | {
"answer": "2 \\sqrt {7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x$, $y$, $z \in \mathbb{R}$, if $-1$, $x$, $y$, $z$, $-3$ form a geometric sequence, calculate the value of $xyz$. | {
"answer": "-3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of students in Carlos' graduating class is more than 100 and fewer than 200 and is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. How many students are in Carlos' graduating class? | {
"answer": "182",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Selina takes a sheet of paper and cuts it into 10 pieces. She then takes one of these pieces and cuts it into 10 smaller pieces. She then takes another piece and cuts it into 10 smaller pieces and finally cuts one of the smaller pieces into 10 tiny pieces. How many pieces of paper has the original sheet been cut into?
... | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line segment connecting the focus F of the parabola $y^2=4x$ and the point M(0,1) intersects the parabola at point A. Let O be the origin, then the area of △OAM is _____. | {
"answer": "\\frac {3}{2} - \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest integer less than $\log_2 \frac{3}{2} + \log_2 \frac{6}{3} + \cdots + \log_2 \frac{3030}{3029}$? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)=3x^{2}+2x+1$, if $\int_{-1}^{1}f(x)\,dx=2f(a)$, then $a=$ ______. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, all rows, columns, and diagonals have the sum 12. Find the sum of the four corner numbers. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S = \{1, 22, 333, \dots , 999999999\}$ . For how many pairs of integers $(a, b)$ where $a, b \in S$ and $a < b$ is it the case that $a$ divides $b$ ? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The random variable $X$ follows a normal distribution $N(1,4)$. Given that $P(X \geqslant 2) = 0.2$, calculate the probability that $0 \leqslant X \leqslant 1$. | {
"answer": "0.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f(x)=\sin (ωx+φ)(ω > 0,0 < φ < π)$ has a distance of $\frac {π}{2}$ between adjacent symmetry axes, and the function $y=f(x+ \frac {π}{2})$ is an even function.
1. Find the analytical expression of $f(x)$.
2. If $α$ is an acute angle, and $f(\frac {α}{2}+ \frac {π}{12})= \frac {3}{5}$, find the... | {
"answer": "\\frac {24+7 \\sqrt {3}}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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