problem stringlengths 10 5.15k | answer dict |
|---|---|
Let $x$ be a real number selected uniformly at random between 100 and 300. If $\lfloor \sqrt{x} \rfloor = 14$, find the probability that $\lfloor \sqrt{100x} \rfloor = 140$.
A) $\frac{281}{2900}$
B) $\frac{4}{29}$
C) $\frac{1}{10}$
D) $\frac{96}{625}$
E) $\frac{1}{100}$ | {
"answer": "\\frac{281}{2900}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is 2 5 that of triangle AOB, determine a : b.
[asy]
pair A = (0,3);
pair B = (0,0);
pair C = (3,0);
pair D = (0,1.5);
pair E = (0.35,0);
pair F = (1.2,1.8);
pair J = (0.17,0);
pair Y = (0.17,0.75);
pair Z = (1.6,0.2);
draw(A--B);
draw(B--C);
draw(C--A);
draw(D--F--Z--E--D);
draw(" $O$ ", B, dir(180));
draw(" $B$ ", A, dir(45));
draw(" $A$ ", C, dir(45));
draw(" $Q$ ", E, dir(45));
draw(" $P$ ", D, dir(45));
draw(" $R$ ", Z, dir(45));
draw(" $S$ ", F, dir(45));
draw(" $a$ ", Y, dir(210));
draw(" $b$ ", J, dir(100));
[/asy] | {
"answer": "2 : 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram below, $WXYZ$ is a trapezoid where $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$. Given $YZ = 15$, $\tan Z = 2$, and $\tan X = 2.5$, what is the length of $XY$? | {
"answer": "2\\sqrt{261}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the perimeter of the triangle that has area $3-\sqrt{3}$ and angles $45^\circ$ , $60^\circ$ , and $75^\circ$ . | {
"answer": "3\\sqrt{2} + 2\\sqrt{3} - \\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C$ are $ \begin{cases} x=2+2\cos θ \
y=2\sin θ\end{cases} $ for the parameter $(θ)$, and in the polar coordinate system $(rOθ)$ (with the same unit length as the rectangular coordinate system $(xOy)$, and the origin $O$ as the pole, and the positive semi-axis of $x$ as the polar axis), the equation of the line $l$ is $ρ\sin (θ+ \dfrac {π}{4})=2 \sqrt {2}$.
(I) Find the equation of the curve $C$ in the polar coordinate system;
(II) Find the length of the chord cut off by the line $l$ on the curve $C$. | {
"answer": "2 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with a radius of 3 is centered at the midpoint of one side of an equilateral triangle each side of which has a length of 9. Determine the difference between the area inside the circle but outside the triangle and the area inside the triangle but outside the circle. | {
"answer": "9\\pi - \\frac{81\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Semicircles of diameter 3 inches are lined up as shown. What is the area, in square inches, of the shaded region in an 18-inch length of this pattern? Express your answer in terms of \(\pi\). | {
"answer": "\\frac{27}{4}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the following expression:
$$2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2(1+2))))))))))$$ | {
"answer": "4094",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin (ωx+φ)(ω > 0,|φ|\leqslant \dfrac {π}{2})$, $y=f(x- \dfrac {π}{4})$ is an odd function, $x= \dfrac {π}{4}$ is the symmetric axis of the graph of $y=f(x)$, and $f(x)$ is monotonic in $(\dfrac {π}{14}, \dfrac {13π}{84})$, determine the maximum value of $ω$. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$ and $b$ are rational numbers, a new operation is defined as follows: $a$☼$b=a^{3}-2ab+4$. For example, $2$☼$5=2^{3}-2\times 2\times 5+4=-8$. Find $4$☼$\left(-9\right)=\_\_\_\_\_\_$. | {
"answer": "140",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p(x)=x^4-4x^3+2x^2+ax+b$ . Suppose that for every root $\lambda$ of $p$ , $\frac{1}{\lambda}$ is also a root of $p$ . Then $a+b=$ [list=1]
[*] -3
[*] -6
[*] -4
[*] -8
[/list] | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
John has 15 marbles of different colors, including two reds, two greens, and two blues. In how many ways can he choose 5 marbles, if exactly one of the chosen marbles must be red and one must be green? | {
"answer": "660",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An arithmetic sequence {a_n} has a sum of the first n terms as S_n, and S_6/S_3 = 4. Find the value of S_9/S_6. | {
"answer": "\\dfrac{9}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terminal side of angle $\alpha$ passes through the point $P(\sqrt{3}, m)$ ($m \neq 0$), and $\cos\alpha = \frac{m}{6}$, then $\sin\alpha = \_\_\_\_\_\_$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum possible product of three different numbers from the set $\{-9, -7, -2, 0, 4, 6, 8\}$, where the product contains exactly one negative number? | {
"answer": "-96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of six, standard, six-sided dice is rolled once. What is the probability that there is exactly one pair and one triplet (three dice showing the same value), and the remaining dice show different values? | {
"answer": "\\frac{25}{162}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a school trip, there are 8 students and a teacher. They want to take pictures in groups where each group consists of either 4 or 5 students. How many different group combinations can they make? | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{\begin{array}{l}{x=\frac{4}{1+{t}^{2}}},\\{y=\frac{4t}{1+{t}^{2}}}\end{array}\right.$ $(t\in R)$.<br/>$(Ⅰ)$ Find the rectangular coordinate equation of curve $C$;<br/>$(Ⅱ)$ Given that the parametric equations of line $l$ are $\left\{\begin{array}{l}{x=1+\sqrt{3}t,}\\{y=t}\end{array}\right.$ $(t\in R)$, point $M(1,0)$, and line $l$ intersects curve $C$ at points $A$ and $B$, find $\frac{1}{|MA|}+\frac{1}{|MB|}$. | {
"answer": "\\frac{\\sqrt{15}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Exhibit a $13$ -digit integer $N$ that is an integer multiple of $2^{13}$ and whose digits consist of only $8$ s and $9$ s. | {
"answer": "8888888888888",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 15 girls in a class of 27 students. The ratio of boys to girls in this class is: | {
"answer": "4:5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two positive integers differ by 8 and their product is 168. What is the larger integer? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The letter T is formed by placing two $3\:\text{inch}\!\times\!5\:\text{inch}$ rectangles to form a T shape. The vertical rectangle is placed in the middle of the horizontal one, overlapping it by $1.5$ inches on both sides. What is the perimeter of the new T, in inches? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $n$ such that $2^6 \cdot 3^3 \cdot n = 10!$. | {
"answer": "1050",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the phase shift of the graph of \( y = \cos(5x - \frac{\pi}{2}) \). | {
"answer": "\\frac{\\pi}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $BC=a$, $AC=b$, and $a$, $b$ are the roots of the equation $x^{2}-2 \sqrt{3}x+2=0$, $2\cos (A+B)=1$
$(1)$ Find the degree of angle $C$.
$(2)$ Find the length of $AB$. | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the interval \([-6, 6]\), an element \(x_0\) is arbitrarily chosen. If the slope of the tangent line to the parabola \(y = x^2\) at \(x = x_0\) has an angle of inclination \(\alpha\), find the probability that \(\alpha \in \left[ \frac{\pi}{4}, \frac{3\pi}{4} \right]\). | {
"answer": "\\frac{11}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the ellipse $C: \frac{x^2}{2} + \frac{y^2}{3} = 1$ has a focus on the positive y-axis denoted as $F$. A line $l$ passing through $F$ with a slope angle of $\frac{3\pi}{4}$ intersects $C$ at points $M$ and $N$. The quadrilateral $OMPN$ is a parallelogram.
(1) Determine the positional relationship between point $P$ and the ellipse;
(2) Calculate the area of the parallelogram $OMPN$. | {
"answer": "\\frac{4}{5}\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given point $A(0,2)$, and $P$ is any point on the ellipse $\frac{x^2}{4}+y^2=1$, then the maximum value of $|PA|$ is ______. | {
"answer": "\\frac{2\\sqrt{21}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin \alpha + \cos \alpha = -\frac{\sqrt{5}}{2}$ and $\frac{5\pi}{4} < \alpha < \frac{3\pi}{2}$, find the value of $\cos \alpha - \sin \alpha$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, the curve $C_1$: $\rho=2\cos\theta$, and the curve $$C_{2}：\rho\sin^{2}\theta=4\cos\theta$$．Establish a Cartesian coordinate system xOy with the pole as the origin and the polar axis as the positive half-axis of x, the parametric equation of curve C is $$\begin{cases} x=2+ \frac {1}{2}t \\ y= \frac { \sqrt {3}}{2}t\end{cases}$$（t is the parameter）．
（I）Find the Cartesian equations of $C_1$ and $C_2$；
（II）C intersects with $C_1$ and $C_2$ at four different points, and the sequence of these four points on C is P, Q, R, S. Find the value of $||PQ|-|RS||$． | {
"answer": "\\frac {11}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_n\}$ where the sum of the first $n$ terms is $S_n = (a+1)n^2 + a$, if the sides of a certain triangle are $a_2$, $a_3$, and $a_4$, then the area of this triangle is ________. | {
"answer": "\\frac{15}{4} \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
You want to paint some edges of a regular dodecahedron red so that each face has an even number of painted edges (which can be zero). Determine from How many ways this coloration can be done.
Note: A regular dodecahedron has twelve pentagonal faces and in each vertex concur three edges. The edges of the dodecahedron are all different for the purpose of the coloring . In this way, two colorings are the same only if the painted edges they are the same. | {
"answer": "2048",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the longest side of the polygon formed by the system:
$$
\begin{cases}
x + y \leq 4 \\
x + 2y \geq 4 \\
x \geq 0 \\
y \geq 0
\end{cases}
$$ | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $sn(α+ \frac {π}{6})= \frac {1}{3}$, and $\frac {π}{3} < α < \pi$, find $\sin ( \frac {π}{12}-α)$. | {
"answer": "- \\frac {4+ \\sqrt {2}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(a\), \(b\), and \(c\) are positive numbers with \(ab = 24\sqrt[3]{3}\), \(ac = 40\sqrt[3]{3}\), and \(bc = 15\sqrt[3]{3}\), find the value of \(abc\). | {
"answer": "120\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\vartriangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, given that $c=2$, $C=\dfrac{\pi }{3}$.
(1) If the area of $\vartriangle ABC$ is equal to $\sqrt{3}$, find $a$ and $b$;
(2) If $\sin B=2\sin A$, find the area of $\vartriangle ABC$. | {
"answer": "\\dfrac{2 \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest positive integer $n$ such that $4n$ is a perfect square and $5n$ is a perfect cube. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with radius $\frac{\sqrt{2}}{2}$ and a regular hexagon with side length 1 share the same center. Calculate the area inside the circle, but outside the hexagon. | {
"answer": "\\frac{\\pi}{2} - \\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point $(x,y)$ is randomly and uniformly chosen inside the rectangle with vertices (0,0), (0,3), (4,3), and (4,0). What is the probability that $x + 2y < 6$? | {
"answer": "\\dfrac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line parallel to the side $AC$ of a triangle $ABC$ with $\angle C = 90$ intersects side $AB$ at $M$ and side $BC$ at $N$ , so that $CN/BN = AC/BC = 2/1$ . The segments $CM$ and $AN$ meet at $O$ . Let $K$ be a point on the segment $ON$ such that $MO+OK = KN$ . The bisector of $\angle ABC$ meets the line through $K$ perpendicular to $AN$ at point $T$ .
Determine $\angle MTB$ .
| {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of $n (n > 0)$ such that the function \\(f(x)= \begin{vmatrix} \sqrt {3} & \sin x \\\\ 1 & \cos x\\end{vmatrix} \\) when shifted $n$ units to the left becomes an even function. | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ is isosceles with $AB=AC$ . The bisectors of angles $ABC$ and $ACB$ meet at $I$ . If the measure of angle $CIA$ is $130^\circ$ , compute the measure of angle $CAB$ .
*Proposed by Connor Gordon* | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x) = \cos(x + \theta) + \sqrt{2}\sin(x + \phi)$ be an even function, where $\theta$ and $\phi$ are acute angles, and $\cos\theta = \frac{\sqrt{6}}{3}\sin\phi$. Then, evaluate the value of $\theta + \phi$. | {
"answer": "\\frac{7\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Stock investor Li Jin bought shares of a certain company last Saturday for $27 per share. The table below shows the price changes of the stock within the week.
| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
|-----------------|--------|---------|-----------|----------|--------|----------|
| Price Change per Share (Compared to Previous Day) | $-1.5$ | $-1$ | $+1.5$ | $+0.5$ | $+1$ | $-0.5$ |
At the close of trading on Wednesday, the price per share was ____ dollars; the highest price during the week was ____ dollars per share; and the lowest price was ____ dollars per share. | {
"answer": "24.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a similar game setup, there are 30 boxes, each containing one of the following values: \begin{tabular}{|c|c|}\hline\$.01&\$1,000\\\hline\$1&\$5,000\\\hline\$5&\$10,000\\\hline\$10&\$25,000\\\hline\$25&\$50,000\\\hline\$50&\$75,000\\\hline\$75&\$100,000\\\hline\$100&\$200,000\\\hline\$200&\$300,000\\\hline\$300&\$400,000\\\hline\$400&\$500,000\\\hline\$500&\$750,000\\\hline\$750&\$1,000,000\\\hline\end{tabular} What is the minimum number of boxes a participant needs to eliminate to have at least a 50% chance of holding a box containing no less than $200,000? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
City A has 2 attractions, $A$ and $B$, while City B has 3 attractions, $C$, $D$, and $E$. When randomly selecting attractions to visit, find the probability of the following events:
1. Selecting exactly 1 attraction in City A.
2. Selecting exactly 2 attractions in the same city. | {
"answer": "\\frac{2}{5}",
"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 $a\sin 2B=\sqrt{3}b\sin A$.
$(1)$ Find the magnitude of angle $B$;
$(2)$ If $\cos A=\frac{1}{3}$, find the value of $\sin C$. | {
"answer": "\\frac{2\\sqrt{6}+1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that an ellipse and a hyperbola $(x^{2}-y^{2}=1)$ share the same foci and the eccentricity is $\frac{\sqrt{2}}{2}$.
(I) Find the standard equation of the ellipse;
(II) A line passing through point $P(0,1)$ intersects the ellipse at points $A$ and $B$. $O$ is the origin. If $\overrightarrow{AP}=2\overrightarrow{PB}$, find the area of $\triangle AOB$. | {
"answer": "\\frac{\\sqrt{126}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of the area of a square inscribed in a semicircle to the area of a square inscribed in a full circle is: | {
"answer": "2: 5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an inverted cone with a base radius of $15 \mathrm{cm}$ and a height of $15 \mathrm{cm}$, and a cylinder with a horizontal base radius of $18 \mathrm{cm}$, determine the height in centimeters of the water in the cylinder after $10\%$ of the water is lost from the cone. | {
"answer": "3.125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a function $g$ from the positive integers to the positive integers with the following properties:
(i) $g$ is increasing.
(ii) $g(mn) = g(m)g(n)$ for all positive integers $m$ and $n$.
(iii) If $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.
Compute all possible values of $g(88).$ | {
"answer": "7744",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If it costs two cents for each plastic digit used to number each locker and it costs $294.94 to label all lockers up to a certain number, calculate the highest locker number labeled. | {
"answer": "3963",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of all real numbers $x$ that are not in the domain of the function $$g(x) = \frac{1}{2 + \frac{1}{2 + \frac{1}{x}}}.$$ | {
"answer": "-\\frac{9}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the equation of the line that passes through the intersection of the lines $2x+3y+5=0$ and $2x+5y+7=0$, and is parallel to the line $x+3y=0$. Also, calculate the distance between these two parallel lines. | {
"answer": "\\frac{2\\sqrt{10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A woman invests in a property for $12,000 with the aim of receiving a $6\%$ return on her investment after covering all expenses including taxes and insurance. She pays $360 annually in taxes and $240 annually for insurance. She also keeps aside $10\%$ of each month's rent for maintenance. Calculate the monthly rent. | {
"answer": "122.22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$(1)$ Find the value of $x$: $4\left(x+1\right)^{2}=49$;<br/>$(2)$ Calculate: $\sqrt{9}-{({-1})^{2018}}-\sqrt[3]{{27}}+|{2-\sqrt{5}}|$. | {
"answer": "\\sqrt{5} - 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hexadecimal system, determine the product of $A$ and $B$. | {
"answer": "6E",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle passes through the three vertices of an isosceles triangle that has two sides of length 5 and a base of length 4. What is the area of this circle? Express your answer in terms of $\pi$. | {
"answer": "\\frac{13125}{1764}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cyclic quadrilateral $ABCD$ satisfies $\angle ABD = 70^\circ$ , $\angle ADB=50^\circ$ , and $BC=CD$ . Suppose $AB$ intersects $CD$ at point $P$ , while $AD$ intersects $BC$ at point $Q$ . Compute $\angle APQ-\angle AQP$ . | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given P(A) = 0.65, P(B) = 0.2, and P(C) = 0.1, calculate the probability of the event "the drawn product is not a first-class product". | {
"answer": "0.35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A train took $X$ minutes ($0 < X < 60$) to travel from platform A to platform B. Find $X$ if it's known that at both the moment of departure from A and the moment of arrival at B, the angle between the hour and minute hands of the clock was $X$ degrees. | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $(a_1, a_2, a_3,\ldots,a_{13})$ be a permutation of $(1,2,3,\ldots,13)$ for which
$$a_1 > a_2 > a_3 > a_4 > a_5 > a_6 > a_7 \mathrm{\ and \ } a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12} < a_{13}.$$
Find the number of such permutations. | {
"answer": "924",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A larger grid is considered for the next challenge, where each segment must still only be traversed in a rightward or downward direction. Starting from point $A$, located at the top-left of a 3x3 grid, to point $B$ at the bottom-right. How many different routes can be taken? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose
\[\frac{1}{x^3 - 2x^2 - 13x + 10} = \frac{A}{x+2} + \frac{B}{x-1} + \frac{C}{(x-1)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$? | {
"answer": "\\frac{1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 24$ and $X$ is an integer, what is the smallest possible value of $X$? | {
"answer": "4625",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pair of standard $6$-sided dice is rolled to determine the side length of a square. What is the probability that the numerical value of the area of the square is less than the numerical value of the perimeter? | {
"answer": "\\frac{1}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a stack of 200 cards, numbered from 1 to 200 from top to bottom. Starting with the top card, the following operations are performed in sequence: remove the top card, then place the next card at the bottom of the stack; remove the new top card, then place the next card at the bottom of the stack… This process is repeated. Which card remains at the end, out of the original 200 cards? | {
"answer": "145",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α∈[\dfrac{π}{2}, \dfrac{3π}{2}]$, $β∈[-\dfrac{π}{2}, 0]$, and the equations $(α-\dfrac{π}{2})^{3}-\sin α-2=0$ and $8β^{3}+2\cos^{2}β+1=0$ hold, find the value of $\sin(\dfrac{α}{2}+β)$. | {
"answer": "\\dfrac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction. | {
"answer": "\\frac{335}{2011}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(a\) and \(b\) are real numbers, and the polynomial \(x^{4} + a x^{3} + b x^{2} + a x + 1 = 0\) has at least one real root, determine the minimum value of \(a^{2} + b^{2}\). | {
"answer": "4/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equations:<br/>$(1)2\left(x-1\right)^{2}=1-x$;<br/>$(2)4{x}^{2}-2\sqrt{3}x-1=0$. | {
"answer": "\\frac{\\sqrt{3} - \\sqrt{7}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the 3 vertices of a triangle and 7 points inside it (a total of 10 points), how many smaller triangles can the original triangle be divided into?
(1985 Shanghai Junior High School Math Competition, China;
1988 Jiangsu Province Junior High School Math Competition, China) | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fill in the blanks with appropriate numbers.
6.8 + 4.1 + __ = 12 __ + 6.2 + 7.6 = 20 19.9 - __ - 5.6 = 10 | {
"answer": "4.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x, y \) be nonnegative integers such that \( x + 2y \) is a multiple of 5, \( x + y \) is a multiple of 3, and \( 2x + y \geq 99 \). Find the minimum possible value of \( 7x + 5y \). | {
"answer": "366",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive multiples of 6 that are less than 150 have a units digit of 6? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin\left(\theta - \frac{\pi}{6}\right) = \frac{1}{4}$ with $\theta \in \left( \frac{\pi}{6}, \frac{2\pi}{3}\right)$, calculate the value of $\cos\left(\frac{3\pi}{2} + \theta\right)$. | {
"answer": "\\frac{\\sqrt{15} + \\sqrt{3}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the left focus of the ellipse $$\frac {x^{2}}{a^{2}}$$+ $$\frac {y^{2}}{b^{2}}$$\=1 (a＞b＞0) is F, the left vertex is A, and the upper vertex is B. If the distance from point F to line AB is $$\frac {2b}{ \sqrt {17}}$$, then the eccentricity of the ellipse is \_\_\_\_\_\_. | {
"answer": "\\frac {1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, side lengths are $XY = 30$, $YZ = 45$, and $XZ = 51$. Points P and Q are on $XY$ and $XZ$ respectively, such that $XP = 18$ and $XQ = 15$. Determine the ratio of the area of triangle $XPQ$ to the area of quadrilateral $PQZY$.
A) $\frac{459}{625}$
B) $\frac{1}{2}$
C) $\frac{3}{5}$
D) $\frac{459}{675}$ | {
"answer": "\\frac{459}{625}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One end of a bus route is at Station $A$ and the other end is at Station $B$. The bus company has the following rules:
(1) Each bus must complete a one-way trip within 50 minutes (including the stopping time at intermediate stations), and it stops for 10 minutes when reaching either end.
(2) A bus departs from both Station $A$ and Station $B$ every 6 minutes. Determine the minimum number of buses required for this bus route. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
**Circle $T$ has a circumference of $12\pi$ inches, and segment $XY$ is a diameter. If the measure of angle $TXZ$ is $45^{\circ}$, what is the length, in inches, of segment $XZ$?** | {
"answer": "6\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x$ is a multiple of $2520$, what is the greatest common divisor of $g(x) = (4x+5)(5x+2)(11x+8)(3x+7)$ and $x$? | {
"answer": "280",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given point $O$ inside $\triangle ABC$, and $\overrightarrow{OA}+\overrightarrow{OC}+2 \overrightarrow{OB}=0$, calculate the ratio of the area of $\triangle AOC$ to the area of $\triangle ABC$. | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among all triangles $ABC$, find the maximum value of $\cos A + \cos B \cos C$. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \begin{cases} 2^{x}, & x < 2 \\ f(x-1), & x\geqslant 2 \end{cases}$, then $f(\log_{2}7)=$ ______. | {
"answer": "\\frac {7}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{m}=(\sin A, \frac {1}{2})$ and $\overrightarrow{n}=(3,\sin A+ \sqrt {3}\cos A)$ are collinear, where $A$ is an internal angle of $\triangle ABC$.
$(1)$ Find the size of angle $A$;
$(2)$ If $BC=2$, find the maximum value of the area $S$ of $\triangle ABC$, and determine the shape of $\triangle ABC$ when $S$ reaches its maximum value. | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the function $G$ has a maximum value of $M$ and a minimum value of $N$ on $m\leqslant x\leqslant n\left(m \lt n\right)$, and satisfies $M-N=2$, then the function is called the "range function" on $m\leqslant x\leqslant n$. <br/>$(1)$ Functions ① $y=2x-1$; ② $y=x^{2}$, of which function ______ is the "range function" on $1\leqslant x\leqslant 2$; (Fill in the number) <br/>$(2)$ Given the function $G:y=ax^{2}-4ax+3a\left(a \gt 0\right)$. <br/>① When $a=1$, the function $G$ is the "range function" on $t\leqslant x\leqslant t+1$, find the value of $t$; <br/>② If the function $G$ is the "range function" on $m+2\leqslant x\leqslant 2m+1(m$ is an integer), and $\frac{M}{N}$ is an integer, find the value of $a$. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $ABC$ with sides $AB = 7$, $AC = 8$, and $BC = 5$, find the value of
\[\frac{\cos \frac{A - B}{2}}{\sin \frac{C}{2}} - \frac{\sin \frac{A - B}{2}}{\cos \frac{C}{2}}.\] | {
"answer": "\\frac{16}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to the two points $\left(0, -\sqrt{3}\right)$ and $\left(0, \sqrt{3}\right)$ is $4$. Let the trajectory of point $P$ be $C$.
(Ⅰ) Find the equation of curve $C$;
(Ⅱ) Find the coordinates of the vertices, the lengths of the major and minor axes, and the eccentricity of the ellipse. | {
"answer": "\\dfrac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the game show $\text{\emph{Wheel of Fortune II}}$, you observe a spinner with the labels ["Bankrupt", "$\$700$", "$\$900$", "$\$200$", "$\$3000$", "$\$800$"]. Given that each region has equal area, determine the probability of earning exactly $\$2400$ in your first three spins. | {
"answer": "\\frac{1}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A product originally priced at \$120 receives a discount of 8%. Calculate the percentage increase needed to return the reduced price to its original amount. | {
"answer": "8.7\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $(X-2)^8 = a + a_1(x-1) + \ldots + a_8(x-1)^8$, then the value of $\left(a_2 + a_4 + \ldots + a_8\right)^2 - \left(a_1 + a_3 + \ldots + a_7\right)^2$ is (Answer in digits). | {
"answer": "-255",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle $EFGH$ has area $4024$. An ellipse with area $4024\pi$ passes through $E$ and $G$ and has foci at $F$ and $H$. What is the perimeter of the rectangle? | {
"answer": "8\\sqrt{2012}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a \lt 0$, the graph of the function $f\left(x\right)=a^{2}\sin 2x+\left(a-2\right)\cos 2x$ is symmetric with respect to the line $x=-\frac{π}{8}$. Find the maximum value of $f\left(x\right)$. | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \(12^2\) and \(13^2\)? | {
"answer": "165",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the inequality (e-a)e^x + x + b + 1 ≤ 0, where e is the natural constant, find the maximum value of $\frac{b+1}{a}$. | {
"answer": "\\frac{1}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $A(\sin\theta, 1)$, $B(\cos\theta, 0)$, $C(-\sin\theta, 2)$, and $\overset{→}{AB}=\overset{→}{BP}$.
(I) Consider the function $f\left(\theta\right)=\overset{→}{BP}\cdot\overset{→}{CA}$, $\theta\in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, discuss the monotonicity of the function and find its range.
(II) If points $O$, $P$, $C$ are collinear, find the value of $\left|\overset{→}{OA}+\overset{→}{OB}\right|$. | {
"answer": "\\frac{\\sqrt{74}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A set \( \mathcal{T} \) of distinct positive integers has the property that for every integer \( y \) in \( \mathcal{T}, \) the arithmetic mean of the set of values obtained by deleting \( y \) from \( \mathcal{T} \) is an integer. Given that 2 belongs to \( \mathcal{T} \) and that 3003 is the largest element of \( \mathcal{T}, \) what is the greatest number of elements that \( \mathcal{T} \) can have? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Lori makes a list of all the numbers between $1$ and $999$ inclusive. She first colors all the multiples of $5$ red. Then she colors blue every number which is adjacent to a red number. How many numbers in her list are left uncolored? | {
"answer": "402",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a version of SHORT BINGO, a $5\times5$ card has specific ranges for numbers placed in each column. In the first column, 5 distinct numbers must be chosen from the set $1-15$, but they must all be prime numbers. As before, the middle square is labeled as WILD and the other columns have specific number ranges as in the original game. How many distinct possibilities are there for the values in the first column of this SHORT BINGO card? | {
"answer": "720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In right triangle \\(ABC\\), where \\(\angle C = 90^{\circ}\\) and \\(\angle A = 30^{\circ}\\), the eccentricity of the ellipse, which has \\(A\\) and \\(B\\) as its foci and passes through point \\(C\\), is \_\_\_\_\_. | {
"answer": "\\sqrt{3} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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