problem stringlengths 10 5.15k | answer dict |
|---|---|
In a large box of ribbons, $\frac{1}{4}$ are yellow, $\frac{1}{3}$ are purple, $\frac{1}{6}$ are orange, and the remaining 40 ribbons are black. How many of the ribbons are purple? | {
"answer": "53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $f(x)=3 \sqrt {2}\cos (x+\varphi)+\sin x$, where $x\in \mathbb{R}$ and $\varphi\in\left(- \frac {\pi}{2}, \frac {\pi}{2}\right)$, passes through the point $\left( \frac {\pi}{2},4\right)$. Find the minimum value of $f(x)$. | {
"answer": "-5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polar equation of a line is $ρ\sin(θ+ \frac{π}{4})= \frac{\sqrt{2}}{2}$, and the parametric equation of the circle $M$ is $\begin{cases} x = 2\cosθ \\ y = -2 + 2\sinθ \end{cases}$, where $θ$ is the parameter.
(I) Convert the line's polar equation into a Cartesian coordinate equation;
(II) Determine the minimum distance from a point on the circle $M$ to the line. | {
"answer": "\\frac{3\\sqrt{2}}{2} - 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$P(x)=ax^2+bx+c$ has exactly $1$ different real root where $a,b,c$ are real numbers. If $P(P(P(x)))$ has exactly $3$ different real roots, what is the minimum possible value of $abc$ ? | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $m$ be a real number where $m > 0$. If for any $x \in (1, +\infty)$, the inequality $2e^{2mx} - \frac{ln x}{m} ≥ 0$ always holds, then find the minimum value of the real number $m$. | {
"answer": "\\frac{1}{2e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the greatest integer less than 150 for which the greatest common divisor of that integer and 18 is 6? | {
"answer": "138",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $(xOy)$, the parametric equation of line $l$ is given by $ \begin{cases} x=-\frac{1}{2}t \\ y=2+\frac{\sqrt{3}}{2}t \end{cases} (t\text{ is the parameter})$, and a circle $C$ with polar coordinate equation $\rho=4\cos\theta$ is established with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. Let $M$ be any point on circle $C$, and connect $OM$ and extend it to $Q$ such that $|OM|=|MQ|$.
(I) Find the rectangular coordinate equation of the trajectory of point $Q$;
(II) If line $l$ intersects the trajectory of point $Q$ at points $A$ and $B$, and the rectangular coordinates of point $P$ are $(0,2)$, find the value of $|PA|+|PB|$. | {
"answer": "4+2\\sqrt{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. Given vectors $\overrightarrow{m} = (b+c, a^2 + bc)$ and $\overrightarrow{n} = (b+c, -1)$ with $\overrightarrow{m} \cdot \overrightarrow{n} = 0$.
(1) Find the size of angle $A$;
(2) If $a = \sqrt{3}$, find the maximum area of $\triangle ABC$. | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of six consecutive positive integers, all of which are nonprime? | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular hexagon has an area of $150\sqrt{3}$ cm². If each side of the hexagon is decreased by 3 cm, by how many square centimeters is the area decreased? | {
"answer": "76.5\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α \in (0, \frac{π}{3})$ satisfies the equation $\sqrt{6} \sin α + \sqrt{2} \cos α = \sqrt{3}$, find the values of:
1. $\cos (α + \frac{π}{6})$
2. $\cos (2α + \frac{π}{12})$ | {
"answer": "\\frac{\\sqrt{30} + \\sqrt{2}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Choose $4-4$: Parameter Equation Lecture. In the plane rectangular coordinate system $xOy$, with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, establish a polar coordinate system. The polar coordinates of point $P$ are $(2\sqrt{3}, \dfrac{\pi}{6})$. The polar coordinate equation of curve $C$ is $\rho ^{2}+2\sqrt{3}\rho \sin \theta = 1$.
(Ⅰ) Write down the rectangular coordinates of point $P$ and the general equation of curve $C.
(Ⅱ) If $Q$ is a moving point on $C$, find the minimum value of the distance from the midpoint $M$ of $PQ$ to the line $l: \left\{\begin{array}{l}x=3+2t\\y=-2+t\end{array}\right.$ (where $t$ is a parameter). | {
"answer": "\\dfrac{11\\sqrt{5}}{10} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$ and $b$ are positive numbers, and $a+b=1$, find the value of $a$ when $a=$____, such that the minimum value of the algebraic expression $\frac{{2{a^2}+1}}{{ab}}-2$ is ____. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let complex numbers $\omega_{1}=-\frac{1}{2}+\frac{\sqrt{3}}{2}i$ and $\omega_{2}=\cos\frac{\pi}{12}+\sin\frac{\pi}{12}i$. If $z=\omega_{1}\cdot\omega_{2}$, find the imaginary part of the complex number $z$. | {
"answer": "\\frac { \\sqrt {2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $x - 3y + m = 0$ ($m \neq 0$) and the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), let points $A$ and $B$ be the intersections of the line with the two asymptotes of the hyperbola. If point $P(m, 0)$ satisfies $|PA| = |PB|$, find the eccentricity of the hyperbola. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the coordinates of the three vertices of $\triangle ABC$ are $A(0,1)$, $B(1,0)$, $C(0,-2)$, and $O$ is the origin, if a moving point $M$ satisfies $|\overrightarrow{CM}|=1$, calculate the maximum value of $|\overrightarrow{OA}+ \overrightarrow{OB}+ \overrightarrow{OM}|$. | {
"answer": "\\sqrt{2}+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=(ax^{2}+bx+c)e^{x}$ $(a > 0)$, the derivative $y=f′(x)$ has two zeros at $-3$ and $0$.
(Ⅰ) Determine the intervals of monotonicity for $f(x)$.
(Ⅱ) If the minimum value of $f(x)$ is $-1$, find the maximum value of $f(x)$. | {
"answer": "\\dfrac {5}{e^{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 4 male and 2 female volunteers, a total of 6 volunteers, and 2 elderly people standing in a row for a group photo. The photographer requests that the two elderly people stand next to each other in the very center, with the two female volunteers standing immediately to the left and right of the elderly people. The number of different ways they can stand is: | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the following equations using appropriate methods:<br/>$(1)2x^{2}-3x+1=0$;<br/>$(2)\left(y-2\right)^{2}=\left(2y+3\right)^{2}$. | {
"answer": "-\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α$ is an angle in the first quadrant, it satisfies $\sin α - \cos α = \frac{\sqrt{10}}{5}$. Find $\cos 2α$. | {
"answer": "- \\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x$ is a multiple of $12600$, what is the greatest common divisor of $g(x) = (5x + 7)(11x + 3)(17x + 8)(4x + 5)$ and $x$? | {
"answer": "840",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a class of 50 students, it is decided to use systematic sampling to select 10 students. The 50 students are randomly assigned numbers from 1 to 50 and divided into groups, with the first group being 1-5, the second group 6-10, ..., and the tenth group 45-50. If a student with the number 12 is selected from the third group, then the student selected from the eighth group will have the number \_\_\_\_\_\_. | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer $n$ such that $\frac{n}{n+50}$ is equal to a terminating decimal? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the function $f(x)$ satisfies $f(3x) = f\left(3x - \frac{3}{2}\right)$ for all $x \in \mathbb{R}$, then the smallest positive period of $f(x)$ is \_\_\_\_\_\_. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a four-digit natural number $A$, if the digit in the thousands place is $5$ more than the digit in the tens place, and the digit in the hundreds place is $3$ more than the digit in the units place, then $A$ is called a "five-three number." For example, for the four-digit number $6714$, since $6-1=5$ and $7-4=3$, therefore $6714$ is a "five-three number"; for the four-digit number $8821$, since $8-2\neq 5$, therefore $8421$ is not a "five-three number". The difference between the largest and smallest "five-three numbers" is ______; for a "five-three number" $A$ with the digit in the thousands place being $a$, the digit in the hundreds place being $b$, the digit in the tens place being $c$, and the digit in the units place being $d$, let $M(A)=a+c+2(b+d)$ and $N(A)=b-3$. If $\frac{M(A)}{N(A)}$ is divisible by $5$, then the value of $A$ that satisfies the condition is ______. | {
"answer": "5401",
"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. Given vectors $\overrightarrow{m}=(\cos A,\cos B)$ and $\overrightarrow{n}=(a,2c-b)$, and $\overrightarrow{m} \parallel \overrightarrow{n}$.
(Ⅰ) Find the magnitude of angle $A$;
(Ⅱ) Find the maximum value of $\sin B+\sin C$ and determine the shape of $\triangle ABC$ at this value. | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square has sides of length 8, and a circle centered at one of its vertices has a radius of 12. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$. | {
"answer": "64 + 108\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and $P$ is a point on the ellipse such that $PF\_2$ is perpendicular to the $x$-axis. If $|F\_1F\_2| = 2|PF\_2|$, calculate the eccentricity of the ellipse. | {
"answer": "\\frac{\\sqrt{5} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If two sides of a triangle are 8 and 15 units, and the angle between them is 30 degrees, what is the length of the third side? | {
"answer": "\\sqrt{289 - 120\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $y=f(x)$ is a quadratic function, and $f(0)=-5$, $f(-1)=-4$, $f(2)=-5$,
(1) Find the analytical expression of this quadratic function.
(2) Find the maximum and minimum values of the function $f(x)$ when $x \in [0,5]$. | {
"answer": "- \\frac {16}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an isosceles trapezoid with \(AB = 24\) units, \(CD = 10\) units, and legs \(AD\) and \(BC\) each measuring \(13\) units. Find the length of diagonal \(AC\). | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a math competition, a team of $8$ students has $2$ hours to solve $30$ problems. If each problem needs to be solved by $2$ students, on average how many minutes can a student spend on a problem? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three boys played a "Word" game in which they each wrote down ten words. For each word a boy wrote, he scored three points if neither of the other boys had the same word; he scored one point if only one of the other boys had the same word. No points were awarded for words which all three boys had. When they added up their scores, they found that they each had different scores. Sam had the smallest score (19 points), and James scored the most. How many points did James score? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin(\omega x+\varphi)$, which is monotonically increasing on the interval ($\frac{\pi}{6}$,$\frac{{2\pi}}{3}$), and the lines $x=\frac{\pi}{6}$ and $x=\frac{{2\pi}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, find $f(-\frac{{5\pi}}{{12}})$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volume of the solid formed by rotating an isosceles right triangle with legs of length 1 around its hypotenuse is __________. | {
"answer": "\\frac{\\sqrt{2}}{6}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = x^2 - 2\cos{\theta}x + 1$, where $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{2}\right]$.
(1) When $\theta = \frac{\pi}{3}$, find the maximum and minimum values of $f(x)$.
(2) If $f(x)$ is a monotonous function on $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{2}\right]$ and $\theta \in [0, 2\pi)$, find the range of $\theta$.
(3) If $\sin{\alpha}$ and $\cos{\alpha}$ are the two real roots of the equation $f(x) = \frac{1}{4} + \cos{\theta}$, find the value of $\frac{\tan^2{\alpha} + 1}{\tan{\alpha}}$. | {
"answer": "\\frac{16 + 4\\sqrt{11}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given 5 people stand in a row, and there is exactly 1 person between person A and person B, determine the total number of possible arrangements. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCDEFGH$ is a cube where each side has length $a$. Find $\sin \angle GAC$. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\cos \left(40^{\circ}-\theta \right)+\cos \left(40^{\circ}+\theta \right)+\cos \left(80^{\circ}-\theta \right)=0$, calculate the value of $\tan \theta$. | {
"answer": "-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle has a radius of 3 units. There are many line segments of length 4 units that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.
A) $3\pi$
B) $5\pi$
C) $4\pi$
D) $7\pi$
E) $6\pi$ | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When a student used a calculator to find the average of 30 data points, they mistakenly entered one of the data points, 105, as 15. Find the difference between the calculated average and the actual average. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\sin \theta= \frac {3}{5}$ and $\frac {5\pi}{2} < \theta < 3\pi$, then $\sin \frac {\theta}{2}=$ ______. | {
"answer": "-\\frac {3 \\sqrt {10}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a_n$ denote the angle opposite to the side of length $4n^2$ units in an integer right angled triangle with lengths of sides of the triangle being $4n^2, 4n^4+1$ and $4n^4-1$ where $n \in N$ . Then find the value of $\lim_{p \to \infty} \sum_{n=1}^p a_n$ | {
"answer": "$\\pi/2$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number $18!=6,402,373,705,728,000$ has many positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
A) $\frac{1}{16}$
B) $\frac{1}{18}$
C) $\frac{1}{15}$
D) $\frac{1}{20}$
E) $\frac{1}{21}$ | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Carolyn and Paul are playing a game starting with a list of the integers $1$ to $n.$ The rules of the game are:
$\bullet$ Carolyn always has the first turn.
$\bullet$ Carolyn and Paul alternate turns.
$\bullet$ On each of her turns, Carolyn must remove one number from the list such that this number has at least one positive divisor other than itself remaining in the list.
$\bullet$ On each of his turns, Paul must remove from the list all of the positive divisors of the number that Carolyn has just removed, and he can optionally choose one additional number that is a multiple of any divisor he is removing.
$\bullet$ If Carolyn cannot remove any more numbers, then Paul removes the rest of the numbers.
For example, if $n=8,$ a possible sequence of moves could be considered.
Suppose that $n=8$ and Carolyn removes the integer $4$ on her first turn. Determine the sum of the numbers that Carolyn removes. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that fits inside the cylinder? | {
"answer": "2\\sqrt{61}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a, b \in \mathbb{R}^+$, and $a+b=1$. Find the minimum value of $\sqrt{a^2+1} + \sqrt{b^2+4}$. | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of solutions to the equation
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,150 \pi)$. | {
"answer": "75",
"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 $\sqrt{3}a=2b\sin A$.
$(1)$ Find angle $B$;
$(2)$ If $b=\sqrt{7}$, $c=3$, and $D$ is the midpoint of side $AC$, find $BD$. | {
"answer": "\\frac{\\sqrt{19}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 52 students in a class. Now, using the systematic sampling method, a sample of size 4 is drawn. It is known that the seat numbers in the sample are 6, X, 30, and 42. What should be the seat number X? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle has its length increased by $30\%$ and its width increased by $15\%$. Determine the percentage increase in the area of the rectangle. | {
"answer": "49.5\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parabola, given by the equation $y^{2}=2px (p > 0)$, has a focus that lies on the line $l$. This line intersects the parabola at two points, $A$ and $B$. A circle with the chord $AB$ as its diameter has the equation $(x-3)^{2}+(y-2)^{2}=16$. Find the value of $p$. | {
"answer": "p = 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the first day, Barry Sotter used his magic wand to make an object's length increase by $\frac{1}{3}$. Meaning if the length of the object was originally $x$, then after the first day, it is $x + \frac{1}{3} x.$ On the second day, he increased the object's new length from the previous day by $\frac{1}{4}$; on the third day by $\frac{1}{5}$, and so on, with each day increasing the object's length by the next increment in the series $\frac{1}{n+3}$ for the $n^{\text{th}}$ day. If by the $n^{\text{th}}$ day Barry wants the object's length to be exactly 50 times its original length, what is the value of $n$? | {
"answer": "147",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$ to be a decreasing function on the interval $(0, +\infty)$, then $m = \boxed{\text{answer}}$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$ , with $i = 1,2, ..., 2015$ . | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle \( \triangle ABC \) with \(\angle B = 90^\circ\). The incircle touches sides \(BC\), \(CA\), and \(AB\) at points \(D\), \(E\), and \(F\) respectively. Line \(AD\) intersects the incircle at another point \(P\), and \(PF \perp PC\). Find the ratio of the side lengths of \(\triangle ABC\). | {
"answer": "3:4:5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five volunteers participate in community service for two days, Saturday and Sunday. Each day, two people are selected to serve. Find the number of ways to select exactly one person to serve for both days. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then find the value of the algebraic expression $\frac{a}{{{a^2}-2a+1}}÷({1+\frac{1}{{a-1}}})$, where $a=\sqrt{2}$. | {
"answer": "\\sqrt{2}+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the rectangular coordinate system xOy, establish a polar coordinate system with O as the pole and the non-negative semi-axis of the x-axis as the polar axis. The line l passes through point P(-1, 2) with an inclination angle of $\frac{2π}{3}$, and the polar coordinate equation of circle C is $ρ = 2\cos(θ + \frac{π}{3})$.
(I) Find the general equation of circle C and the parametric equation of line l;
(II) Suppose line l intersects circle C at points M and N. Find the value of |PM|•|PN|. | {
"answer": "6 + 2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a hexagon \( A B C D E F \) with an area of 60 that is inscribed in a circle \( \odot O \), where \( AB = BC, CD = DE, \) and \( EF = AF \). What is the area of \( \triangle B D F \)? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$? | {
"answer": "16.67\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a pot, there are 6 sesame-filled dumplings, 5 peanut-filled dumplings, and 4 red bean paste-filled dumplings. These three types of dumplings look exactly the same from the outside. If 4 dumplings are randomly selected from the pot, the probability that at least one dumpling of each type is selected is. | {
"answer": "\\dfrac{48}{91}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a multiplication error involving two positive integers $a$ and $b$, Ron mistakenly reversed the digits of the three-digit number $a$. The erroneous product obtained was $396$. Determine the correct value of the product $ab$. | {
"answer": "693",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Parallelogram $ABCD$ has vertices $A(3,4)$, $B(-2,1)$, $C(-5,-2)$, and $D(0,1)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is left of the $y$-axis? Express your answer as a common fraction. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Assume a deck of 27 cards where each card features one of three symbols (star, circle, or square), each symbol painted in one of three colors (red, yellow, or blue), and each color applied in one of three intensities (light, medium, or dark). Each symbol-color-intensity combination is unique across the cards. A set of three cards is defined as complementary if:
i. Each card has a different symbol or all have the same symbol.
ii. Each card has a different color or all have the same color.
iii. Each card has a different intensity or all have the same intensity.
Determine the number of different complementary three-card sets available. | {
"answer": "117",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two plane vectors, the angle between them is $120^\circ$, and $a=1$, $|b|=2$. If the plane vector $m$ satisfies $m\cdot a=m\cdot b=1$, then $|m|=$ ______. | {
"answer": "\\frac{ \\sqrt{21}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify the product \[\frac{9}{3}\cdot\frac{15}{9}\cdot\frac{21}{15} \dotsm \frac{3n+6}{3n} \dotsm \frac{3003}{2997}.\] | {
"answer": "1001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f\left(x\right)=ax^{2}-1$ and $g\left(x\right)=\ln \left(ax\right)$ have an "$S$ point", then find the value of $a$. | {
"answer": "\\frac{2}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two lines with slopes 3 and -1 intersect at the point $(3, 1)$. What is the area of the triangle enclosed by these two lines and the horizontal line $y = 8$?
- **A)** $\frac{25}{4}$
- **B)** $\frac{98}{3}$
- **C)** $\frac{50}{3}$
- **D)** 36
- **E)** $\frac{200}{9}$ | {
"answer": "\\frac{98}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{a_n\}$ where $a_1 = \frac{1}{2}$ and $a_{n+1} = \frac{1+a_n}{1-a_n}$ for $n \in N^*$, find the smallest value of $n$ such that $a_1+a_2+a_3+…+a_n \geqslant 72$. | {
"answer": "238",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mia is researching a yeast population. There are 50 yeast cells present at 10:00 a.m. and the population triples every 5 minutes. Assuming none of the yeast cells die, how many yeast cells are present at 10:18 a.m. the same day? | {
"answer": "1350",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Select 5 people from 4 boys and 5 girls to participate in a math extracurricular group. How many different ways are there to select under the following conditions?
(1) Select 2 boys and 3 girls, and girl A must be selected;
(2) Select at most 4 girls, and boy A and girl B cannot be selected at the same time. | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The arithmetic mean of a set of $60$ numbers is $42$. If three numbers of the set, namely $40$, $50$, and $60$, are discarded, the arithmetic mean of the remaining set of numbers is:
**A)** 41.3
**B)** 41.4
**C)** 41.5
**D)** 41.6
**E)** 41.7 | {
"answer": "41.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using five distinct digits, $1$, $4$, $5$, $8$, and $9$, determine the $51\text{st}$ number in the sequence when arranged in ascending order.
A) $51489$
B) $51498$
C) $51849$
D) $51948$ | {
"answer": "51849",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a moving circle $C$ that passes through points $A(4,0)$ and $B(0,-2)$, and intersects with the line passing through point $M(1,-2)$ at points $E$ and $F$. Find the minimum value of $|EF|$ when the area of circle $C$ is at its minimum. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose we need to divide 12 dogs into three groups, where one group contains 4 dogs, another contains 6 dogs, and the last contains 2 dogs. How many ways can we form the groups so that Rover is in the 4-dog group and Spot is in the 6-dog group? | {
"answer": "2520",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_n\}$ with a common difference $d = -2$, and $a_1 + a_4 + a_7 + \ldots + a_{97} = 50$, find the value of $a_3 + a_6 + a_9 + \ldots + a_{99}$. | {
"answer": "-66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $θ \in (0,π)$, and $\sin ( \frac {π}{4}-θ)= \frac { \sqrt {2}}{10}$, find $\tan 2θ$. | {
"answer": "\\frac {24}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Abby finished the softball season with a total of 45 hits, among which were 2 home runs, 3 triples, and 7 doubles, calculate the percentage of her hits that were singles. | {
"answer": "73.33\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The inclination angle of the line $\sqrt{3}x+y+2024=0$ is $\tan^{-1}\left(-\frac{\sqrt{3}}{1}\right)$. Calculate the angle in radians. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the graph of the function $f(x) = (x^2 - ax - 5)(x^2 - ax + 3)$ is symmetric about the line $x=2$, then the minimum value of $f(x)$ is \_\_\_\_\_\_. | {
"answer": "-16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse, if its two foci and the two vertices on its minor axis form a square, calculate its eccentricity. | {
"answer": "\\dfrac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers are in the list $ -48, -41, -34, \ldots, 65, 72?$ | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equation of line $l$ as $ \begin{cases} x=m+ \frac { \sqrt {2}}{2}t \\ y= \frac { \sqrt {2}}{2}t \end{cases} (t\text{ is the parameter})$, establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of the ellipse $(C)$ is $ρ^{2}\cos ^{2}θ+3ρ^{2}\sin ^{2}θ=12$. The left focus $(F)$ of the ellipse is located on line $(l)$.
$(1)$ If line $(l)$ intersects ellipse $(C)$ at points $A$ and $B$, find the value of $|FA|⋅|FB|$;
$(2)$ Find the maximum value of the perimeter of the inscribed rectangle in ellipse $(C)$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the digits $a, b, c, d, e$ such that the two five-digit numbers formed with them satisfy the equation $\overline{a b c d e} \cdot 9 = \overline{e d c b a}$. | {
"answer": "10989",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of primes $p$ between $100$ and $200$ for which $x^{11}+y^{16}\equiv 2013\pmod p$ has a solution in integers $x$ and $y$ . | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation of a line is $Ax+By=0$, choose two different numbers from the set $\{1, 2, 3, 4, 5\}$ to be the values of $A$ and $B$ each time, and find the number of different lines obtained. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}| = |\overrightarrow{b}| = |\overrightarrow{a} + \overrightarrow{b}|$, the cosine of the angle between $\overrightarrow{a}$ and $2\overrightarrow{a} - \overrightarrow{b}$ is ______. | {
"answer": "\\frac{5\\sqrt{7}}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two ferries cross a river with constant speeds, turning at the shores without losing time. They start simultaneously from opposite shores and meet for the first time 700 feet from one shore. They continue to the shores, return, and meet for the second time 400 feet from the opposite shore. Determine the width of the river. | {
"answer": "1400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer $n$ such that for any $n$ mutually coprime integers greater than 1 and not exceeding 2009, there is at least one prime number among them. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=x^{2}-2x$ where $x \in [-2,a]$. Find the minimum value of $f(x)$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card?
- A) 35
- B) 42
- C) 48
- D) 53
- E) 82 | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parallelogram with area $1$ and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the $8$ lines formed we have a octagon inside of the parallelogram. Determine the area of this octagon | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the expansion of $(1+x){(x-\frac{2}{x})}^{3}$, calculate the coefficient of $x$. | {
"answer": "-6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each edge of a regular tetrahedron is given a stripe. The choice of which edge to stripe is made at random. What is the probability that there is at least one triangle face with all its edges striped? | {
"answer": "\\frac{1695}{4096}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the greatest integer less than or equal to \[\frac{5^{105} + 4^{105}}{5^{99} + 4^{99}}?\] | {
"answer": "15624",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $D$ be the circle with equation $x^2 + 4y - 16 = -y^2 + 12x + 16$. Find the values of $(c,d)$, the center of $D$, and $s$, the radius of $D$, and calculate $c+d+s$. | {
"answer": "4 + 6\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(-2,4)$. If a point is chosen at random from the region defined by the parallelogram, what is the probability that the point lies below or on the line $y = -1$? Express your answer as a common fraction. | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A total of 1000 senior high school students from a certain school participated in a mathematics exam. The scores in this exam follow a normal distribution N(90, σ²). If the probability of a score being within the interval (70, 110] is 0.7, estimate the number of students with scores not exceeding 70. | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a one-way single-lane highway, cars travel at the same speed and maintain a safety distance such that for every 20 kilometers per hour or part thereof in speed, there is a distance of one car length between the back of one car and the front of the next. Each car is 5 meters long. A sensor on the side of the road counts the number of cars that pass in one hour. Let $N$ be the maximum whole number of cars that can pass the sensor in one hour. Determine the quotient when $N$ is divided by 10. | {
"answer": "400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.