problem stringlengths 10 5.15k | answer dict |
|---|---|
The moisture content of freshly cut grass is $70\%$, while the moisture content of hay is $16\%. How much grass needs to be cut to obtain 1 ton of hay? | {
"answer": "2800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mateo receives $20 every hour for one week, and Sydney receives $400 every day for one week. Calculate the difference between the total amounts of money that Mateo and Sydney receive over the one week period. | {
"answer": "560",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows two 10 by 14 rectangles which are edge-to-edge and share a common vertex. It also shows the center \( O \) of one rectangle and the midpoint \( M \) of one edge of the other. What is the distance \( OM \)?
A) 12
B) 15
C) 18
D) 21
E) 24 | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two dice are thrown one after the other, and the numbers obtained are denoted as $a$ and $b$.
(Ⅰ) Find the probability that $a^2 + b^2 = 25$;
(Ⅱ) Given that the lengths of three line segments are $a$, $b$, and $5$, find the probability that these three line segments can form an isosceles triangle. | {
"answer": "\\dfrac{7}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that acute angles $\alpha$ and $\beta$ satisfy $\alpha+2\beta=\frac{2\pi}{3}$ and $\tan\frac{\alpha}{2}\tan\beta=2-\sqrt{3}$, find the value of $\alpha +\beta$. | {
"answer": "\\frac{5\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F$ is the right focus of the ellipse $C:\frac{x^2}{4}+\frac{y^2}{3}=1$, $P$ is a point on the ellipse $C$, and $A(1,2\sqrt{2})$, find the maximum value of $|PA|+|PF|$. | {
"answer": "4 + 2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Math City plans to add more streets and now has 10 streets, but two pairs of these streets are parallel to each other. No other streets are parallel, and no street is parallel to more than one other street. What is the greatest number of police officers needed at intersections, assuming that each intersection has exactly one police officer stationed? | {
"answer": "43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles have centers at $(3,5)$ and $(20,15)$. Both circles are tangent to the x-axis. Determine the distance between the closest points of the two circles. | {
"answer": "\\sqrt{389} - 20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the arithmetic sequence $\{a_n\}$, it is known that $a_1=10$, and the sum of the first $n$ terms is $S_n$. If $S_9=S_{12}$, find the maximum value of $S_n$ and the corresponding value of $n$. | {
"answer": "55",
"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 it is given that \\(A < B < C\\) and \\(C = 2A\\).
\\((1)\\) If \\(c = \sqrt{3}a\\), find the measure of angle \\(A\\).
\\((2)\\) If \\(a\\), \\(b\\), and \\(c\\) are three consecutive positive integers, find the area of \\(\triangle ABC\\). | {
"answer": "\\dfrac{15\\sqrt{7}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line $l: x - 2y + 2 = 0$ passes through the left focus F<sub>1</sub> and a vertex B of an ellipse. Find the eccentricity of the ellipse. | {
"answer": "\\frac{2\\sqrt{5}}{5}",
"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. If $$a=b\cos C+\frac{\sqrt{3}}{3}c\sin B$$.
(1) Find the value of angle B.
(2) If the area of triangle ABC is S=$$5\sqrt{3}$$, and a=5, find the value of b. | {
"answer": "\\sqrt{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Chelsea leads by 60 points halfway through a 120-shot archery tournament, scores at least 5 points per shot, and scores at least 10 points for each of her next n shots, determine the minimum number of shots, n, she must get as bullseyes to guarantee her victory. | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jennifer plans a profit of 20% on the selling price of an item, and her expenses are 10% of the selling price. There is also a sales tax of 5% on the selling price of the item. The item sells for $10.00. Calculate the rate of markup on cost of this item. | {
"answer": "53.85\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the radius \( AO \) of a circle centered at \( O \), a point \( M \) is chosen. On one side of \( AO \), points \( B \) and \( C \) are chosen on the circle such that \( \angle AMB = \angle OMC = \alpha \). Find the length of \( BC \) if the radius of the circle is 10 and \( \cos \alpha = \frac{4}{5} \). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain school, 3 teachers are chosen from a group of 6 to give support teaching in 3 remote areas, with each area receiving one teacher. There are restrictions such that teacher A and teacher B cannot go together, and teacher A can only go with teacher C or not go at all. How many different dispatch plans are there? | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a sequence defined as $500, x, 500 - x, \ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ that leads to the longest sequence. | {
"answer": "309",
"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 $c\sin A= \sqrt {3}a\cos C$ and $(a-c)(a+c)=b(b-c)$, consider the function $f(x)=2\sin x\cos ( \frac {π}{2}-x)- \sqrt {3}\sin (π+x)\cos x+\sin ( \frac {π}{2}+x)\cos x$.
(1) Find the period and the equation of the axis of symmetry of the function $y=f(x)$.
(2) Find the value of $f(B)$. | {
"answer": "\\frac {5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a point $P$ on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with foci $F_{1}$, $F_{2}$, find the eccentricity of the ellipse given that $\overrightarrow{PF_{1}} \cdot \overrightarrow{PF_{2}}=0$ and $\tan \angle PF_{1}F_{2}= \frac{1}{2}$. | {
"answer": "\\frac{\\sqrt{5}}{3}",
"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": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a bag containing 12 green marbles and 8 purple marbles, Phil draws a marble at random, records its color, replaces it, and repeats this process until he has drawn 10 marbles. What is the probability that exactly five of the marbles he draws are green? Express your answer as a decimal rounded to the nearest thousandth. | {
"answer": "0.201",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular hexagon is inscribed in another regular hexagon such that each vertex of the inscribed hexagon divides a side of the original hexagon into two parts in the ratio 2:1. Find the ratio of the area of the inscribed hexagon to the area of the larger hexagon. | {
"answer": "7/9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given (1+i)x=1+yi, find |x+yi|. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $O$ is any point in space, and $A$, $B$, $C$, $D$ are four points such that no three of them are collinear, but they are coplanar, and $\overrightarrow{OA}=2x\cdot \overrightarrow{BO}+3y\cdot \overrightarrow{CO}+4z\cdot \overrightarrow{DO}$, find the value of $2x+3y+4z$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there? | {
"answer": "115",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)= \begin{cases} \sin \frac{\pi}{3}x, & x\leqslant 2011, \\ f(x-4), & x > 2011, \end{cases}$ find $f(2012)$. | {
"answer": "-\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=4\cos (3x+φ)(|φ| < \dfrac{π}{2})$, its graph is symmetric about the line $x=\dfrac{11π}{12}$. When $x\_1$, $x\_2∈(−\dfrac{7π}{12},−\dfrac{π}{12})$, $x\_1≠x\_2$, and $f(x\_1)=f(x\_2)$, determine the value of $f(x\_1+x\_2)$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the Shanghai World Expo, 20 volunteers were recruited, with each volunteer assigned a unique number from 1 to 20. If four individuals are to be selected randomly from this group and divided into two teams according to their numbers, with the smaller numbers in one team and the larger numbers in another, what is the total number of ways to ensure that both volunteers number 5 and number 14 are selected and placed on the same team? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \\(a > 0\\), the function \\(f(x)= \frac {1}{3}x^{3}+ \frac {1-a}{2}x^{2}-ax-a\\).
\\((1)\\) Discuss the monotonicity of \\(f(x)\\);
\\((2)\\) When \\(a=1\\), let the function \\(g(t)\\) represent the difference between the maximum and minimum values of \\(f(x)\\) on the interval \\([t,t+3]\\). Find the minimum value of \\(g(t)\\) on the interval \\([-3,-1]\\). | {
"answer": "\\frac {4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a triangle $ABC$ , let $I$ denote the incenter. Let the lines $AI,BI$ and $CI$ intersect the incircle at $P,Q$ and $R$ , respectively. If $\angle BAC = 40^o$ , what is the value of $\angle QPR$ in degrees ? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jeff wants to calculate the product $0.52 \times 7.35$ using a calculator. However, he mistakenly inputs the numbers as $52 \times 735$ without the decimal points. The calculator then shows a product of $38220$. What would be the correct product if Jeff had correctly entered the decimal points?
A) $0.3822$
B) $38.22$
C) $3.822$
D) $0.03822$
E) $382.2$ | {
"answer": "3.822",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\{a_n\}$ be an arithmetic sequence, and $S_n$ be the sum of its first $n$ terms, given that $S_5 < S_6$, $S_6=S_7 > S_8$, then the correct conclusion(s) is/are \_\_\_\_\_\_
$(1) d < 0$
$(2) a_7=0$
$(3) S_9 > S_5$
$(4) S_6$ and $S_7$ are both the maximum value of $S_n$. | {
"answer": "(1)(2)(4)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the greatest positive integer $A$ with the following property: For every permutation of $\{1001,1002,...,2000\}$ , the sum of some ten consecutive terms is great than or equal to $A$ . | {
"answer": "10055",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $3$ by $3$ determinant has three entries equal to $2$ , three entries equal to $5$ , and three entries equal to $8$ . Find the maximum possible value of the determinant. | {
"answer": "405",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression $\frac{\sqrt{3}\tan 12^\circ - 3}{\sin 12^\circ (4\cos^2 12^\circ - 2)}=\_\_\_\_\_\_\_\_.$ | {
"answer": "-4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the random variable $X$ follows a normal distribution $N(2, \sigma^2)$, and $P(X \leq 4) = 0.84$, determine the value of $P(X < 0)$. | {
"answer": "0.16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $(1-2x)^{2017} = a_0 + a_1(x-1) + a_2(x-1)^2 + \ldots + a_{2016}(x-1)^{2016} + a_{2017}(x-1)^{2017}$ ($x \in \mathbb{R}$), find the value of $a_1 - 2a_2 + 3a_3 - 4a_4 + \ldots - 2016a_{2016} + 2017a_{2017}$. | {
"answer": "-4034",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate:
1. $(2\frac{3}{5})^{0} + 2^{-2}\cdot(2\frac{1}{4})^{-\frac{1}{2}} + (\frac{25}{36})^{0.5} + \sqrt{(-2)^{2}}$;
2. $\frac{1}{2}\log\frac{32}{49} - \frac{4}{3}\log\sqrt{8} + \log\sqrt{245}$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The minimum distance from any integer-coordinate point on the plane to the line \( y = \frac{5}{3} x + \frac{4}{5} \) is to be determined. | {
"answer": "\\frac{\\sqrt{34}}{85}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_{n}\}$ that satisfies the equation: ${a_{n+1}}+{({-1})^n}{a_n}=3n-1$ ($n∈{N^*}$), calculate the sum of the first $60$ terms of the sequence $\{a_{n}\}$. | {
"answer": "2760",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sample contains 5 individuals with values a, 0, 1, 2, 3, and the average value of the sample is 1, calculate the standard deviation of the sample. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, it is known that the line
$$
\begin{cases}
x=-\frac{3}{2}+\frac{\sqrt{2}}{2}l\\
y=\frac{\sqrt{2}}{2}l
\end{cases}
$$
(with $l$ as the parameter) intersects with the curve
$$
\begin{cases}
x=\frac{1}{8}t^{2}\\
y=t
\end{cases}
$$
(with $t$ as the parameter) at points $A$ and $B$. Find the length of the segment $AB$. | {
"answer": "4\\sqrt{2}",
"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 + \\sqrt{105}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the expression \((\sqrt{2(1+\cos 2x)} - \sqrt{3-\sqrt{2}} \sin x + 1) \cdot (3 + 2\sqrt{7-\sqrt{2}} \cos y - \cos 2y)\). If the answer is not an integer, round it to the nearest whole number. | {
"answer": "-9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $(xOy)$, point $P(1, 2)$ is on a line $l$ with a slant angle of $\alpha$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. The equation of curve $C$ is $\rho = 6 \sin \theta$.
(1) Write the parametric equation of $l$ and the rectangular coordinate equation of $C$;
(2) Suppose $l$ intersects $C$ at points $A$ and $B$. Find the minimum value of $\frac{1}{|PA|} + \frac{1}{|PB|}$. | {
"answer": "\\frac{2 \\sqrt{7}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle PQRS has a perimeter of 24 meters and side PQ is fixed at 7 meters. Find the minimum diagonal PR of the rectangle. | {
"answer": "\\sqrt{74}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, where $C=60 ^{\circ}$, $AB= \sqrt {3}$, and the height from $AB$ is $\frac {4}{3}$, find the value of $AC+BC$. | {
"answer": "\\sqrt {11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $1=d_1<d_2<d_3<\dots<d_k=n$ be the divisors of $n$ . Find all values of $n$ such that $n=d_2^2+d_3^3$ . | {
"answer": "68",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos α=-\dfrac{4}{5}\left(\dfrac{π}{2}<α<π\right)$, find $\cos\left(\dfrac{π}{6}-α\right)$ and $\cos\left(\dfrac{π}{6}+α\right)$. | {
"answer": "-\\dfrac{3+4\\sqrt{3}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A = (-4, 0),$ $B=(-3,2),$ $C=(3,2),$ and $D=(4,0).$ Suppose that point $P$ satisfies \[PA + PD = PB + PC = 10.\]Find the $y-$coordinate of $P,$ and express it in its simplest form. | {
"answer": "\\frac{6}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $0 < α < \dfrac{\pi }{2}$, $-\dfrac{\pi }{2} < β < 0$, $\cos \left( \dfrac{\pi }{4}+α \right)=\dfrac{1}{3}$, and $\cos \left( \dfrac{\pi }{4}-\dfrac{\beta }{2} \right)=\dfrac{\sqrt{3}}{3}$, calculate the value of $\cos \left( α +\dfrac{\beta }{2} \right)$. | {
"answer": "\\dfrac{5\\sqrt{3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to the centers of two circles $C_1$ and $C_2$ is equal to 4, where $C_1: x^2+y^2-2\sqrt{3}y+2=0$, $C_2: x^2+y^2+2\sqrt{3}y-3=0$. Let the trajectory of point $P$ be $C$.
(1) Find the equation of $C$;
(2) Suppose the line $y=kx+1$ intersects $C$ at points $A$ and $B$. What is the value of $k$ when $\overrightarrow{OA} \perp \overrightarrow{OB}$? What is the value of $|\overrightarrow{AB}|$ at this time? | {
"answer": "\\frac{4\\sqrt{65}}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Isabella took 9 math tests and received 9 different scores, each an integer between 88 and 100, inclusive. After each test, she noticed that the average of her test scores was always an integer. Her score on the ninth test was 93. What was her score on the eighth test? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\{a_{n}\}$ be an arithmetic sequence with a common difference of $d$, and $d \gt 1$. Define $b_{n}=\frac{{n}^{2}+n}{{a}_{n}}$, and let $S_{n}$ and $T_{n}$ be the sums of the first $n$ terms of the sequences $\{a_{n}\}$ and $\{b_{n}\}$, respectively.
$(1)$ If $3a_{2}=3a_{1}+a_{3}$ and $S_{3}+T_{3}=21$, find the general formula for $\{a_{n}\}$.
$(2)$ If $\{b_{n}\}$ is an arithmetic sequence and $S_{99}-T_{99}=99$, find $d$. | {
"answer": "\\frac{51}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For rational numbers $a$ and $b$, define the operation "$\otimes$" as $a \otimes b = ab - a - b - 2$.
(1) Calculate the value of $(-2) \otimes 3$;
(2) Compare the size of $4 \otimes (-2)$ and $(-2) \otimes 4$. | {
"answer": "-12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is
$$
\begin{cases}
x = -1 + \frac {\sqrt {2}}{2}t \\
y = 1 + \frac {\sqrt {2}}{2}t
\end{cases}
(t \text{ is the parameter}),
$$
and the equation of circle $C$ is $(x-2)^{2} + (y-1)^{2} = 5$. Establish a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis.
(Ⅰ) Find the polar equations of line $l$ and circle $C$.
(Ⅱ) If line $l$ intersects circle $C$ at points $A$ and $B$, find the value of $\cos ∠AOB$. | {
"answer": "\\frac{3\\sqrt{10}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The polynomial \( f(x) \) satisfies the equation \( f(x) - f(x-2) = (2x-1)^{2} \) for all \( x \). Find the sum of the coefficients of \( x^{2} \) and \( x \) in \( f(x) \). | {
"answer": "\\frac{5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, each of four identical circles touch three others. The circumference of each circle is 48. Calculate the perimeter of the shaded region formed within the central area where all four circles touch. Assume the circles are arranged symmetrically like petals of a flower. | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an ${8}$ × ${8}$ squares chart , we dig out $n$ squares , then we cannot cut a "T"shaped-5-squares out of the surplus chart .
Then find the mininum value of $n$ . | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\sum_{n = 0}^{\infty}\sin^{2n}\theta = 4$, what is the value of $\sin{2\theta}$? | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two types of steel plates need to be cut into three sizes $A$, $B$, and $C$. The number of each size that can be obtained from each type of steel plate is shown in the table:
\begin{tabular}{|l|c|c|c|}
\hline & Size $A$ & Size $B$ & Size $C$ \\
\hline First type of steel plate & 2 & 1 & 1 \\
\hline Second type of steel plate & 1 & 2 & 3 \\
\hline
\end{tabular}
If we need 15 pieces of size $A$, 18 pieces of size $B$, and 27 pieces of size $C$, find the minimum number of plates $m$ and $n$ of the two types required, so that $m + n$ is minimized. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$n$ coins are simultaneously flipped. The probability that two or fewer of them show tails is $\frac{1}{4}$. Find $n$. | {
"answer": "n = 5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a square and $P$ be a point on the shorter arc $AB$ of the circumcircle of the square. Which values can the expression $\frac{AP+BP}{CP+DP}$ take? | {
"answer": "\\sqrt{2} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $m\in R$, the moving straight line passing through the fixed point $A$ with equation $x+my-2=0$ intersects the moving straight line passing through the fixed point $B$ with equation $mx-y+4=0$ at point $P\left(x,y\right)$. Find the maximum value of $|PA|\cdot |PB|$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
795. Calculate the double integral \(\iint_{D} x y \, dx \, dy\), where region \(D\) is:
1) A rectangle bounded by the lines \(x=0, x=a\), \(y=0, y=b\);
2) An ellipse \(4x^2 + y^2 \leq 4\);
3) Bounded by the line \(y=x-4\) and the parabola \(y^2=2x\). | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Line $l_{1}$: $mx+2y-3=0$ is parallel to line $l_{2}$: $3x+\left(m-1\right)y+m-6=0$. Find the value of $m$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ is an integer, find the largest integer that always divides the expression \[(12x + 2)(8x + 14)(10x + 10)\] when $x$ is odd. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangle $ABC$ , $| BC | = 1$ and there is exactly one point $D$ on the side $BC$ such that $|DA|^2 = |DB| \cdot |DC|$ . Determine all possible values of the perimeter of the triangle $ABC$ . | {
"answer": "\\sqrt{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A psychic is faced with a deck of 36 cards placed face down (four suits, with nine cards of each suit). He names the suit of the top card, after which the card is revealed and shown to him. Then the psychic names the suit of the next card, and so on. The goal is for the psychic to guess as many suits correctly as possible.
The backs of the cards are asymmetric, and the psychic can see the orientation in which the top card is placed. The psychic's assistant knows the order of the cards in the deck but cannot change it. However, the assistant can place the backs of each card in one of two orientations.
Could the psychic have agreed with the assistant before the assistant knew the order of the cards to ensure guessing the suit of at least a) 19 cards; b) 23 cards?
If you have devised a method to guess a number of cards greater than 19, please describe it. | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to points $F_1(0, -\sqrt{3})$ and $F_2(0, \sqrt{3})$ is equal to 4. Let the trajectory of point $P$ be $C$.
(1) Find the equation of trajectory $C$;
(2) Let line $l: y=kx+1$ intersect curve $C$ at points $A$ and $B$. For what value of $k$ is $|\vec{OA} + \vec{OB}| = |\vec{AB}|$ (where $O$ is the origin)? What is the value of $|\vec{AB}|$ at this time? | {
"answer": "\\frac{4\\sqrt{65}}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an isosceles triangle $ABC$ satisfying $AB=AC$, $\sqrt{3}BC=2AB$, and point $D$ is on side $BC$ with $AD=BD$, then the value of $\sin \angle ADB$ is ______. | {
"answer": "\\frac{2 \\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $AB$ in the triangle below.
[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0.5,sqrt(3)/2);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$18$", (A+C)/2,W);
label("$30^\circ$", (0.3,0),N);
[/asy] | {
"answer": "18\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fisherman Vasya caught several fish. He placed the three largest fish, which constitute 35% of the total weight of the catch, in the refrigerator. He gave the three smallest fish, which constitute 5/13 of the weight of the remaining fish, to the cat. Vasya ate all the rest of the caught fish himself. How many fish did Vasya catch? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square with a perimeter of 36 is inscribed in a square with a perimeter of 40. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?
A) $\sqrt{101}$
B) $9\sqrt{2}$
C) $8\sqrt{2}$
D) $\sqrt{90}$
E) $10$ | {
"answer": "9\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two teachers and 4 students need to be divided into 2 groups, each consisting of 1 teacher and 2 students. Calculate the number of different arrangements. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cookie Monster now finds a bigger cookie with the boundary described by the equation $x^2 + y^2 - 8 = 2x + 4y$. He wants to know both the radius and the area of this cookie to determine if it's enough for his dessert. | {
"answer": "13\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five identical right-angled triangles can be arranged so that their larger acute angles touch to form a star. It is also possible to form a different star by arranging more of these triangles so that their smaller acute angles touch. How many triangles are needed to form the second star? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate $\frac{5}{a+b}$ where $a=7$ and $b=3$.
A) $\frac{1}{2}$
B) $1$
C) $10$
D) $-8$
E) Meaningless | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin(α + \sqrt{2}\cos(α) = \sqrt{3})$, find the value of $\tan(α)$.
A) $\frac{\sqrt{2}}{2}$
B) $\sqrt{2}$
C) $-\frac{\sqrt{2}}{2}$
D) $-\sqrt{2}$ | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many distinct arrangements of the letters in the word "basics" are there, specifically those beginning with a vowel? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle XYZ$, $\angle XYZ = 30^\circ$, $XY = 12$, and $XZ = 8$. Points $P$ and $Q$ lie on $\overline{XY}$ and $\overline{XZ}$ respectively. What is the minimum possible value of $YP + PQ + QZ$?
A) $\sqrt{154}$
B) $\sqrt{208 + 96\sqrt{3}}$
C) $16$
D) $\sqrt{208}$ | {
"answer": "\\sqrt{208 + 96\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lengths of a pair of corresponding medians of two similar triangles are 10cm and 4cm, respectively, and the sum of their perimeters is 140cm. The perimeters of these two triangles are , and the ratio of their areas is . | {
"answer": "25:4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The perimeter of a semicircle with an area of ______ square meters is 15.42 meters. | {
"answer": "14.13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polar equation of circle $E$ is $\rho=4\sin \theta$, with the pole as the origin and the polar axis as the positive half of the $x$-axis, establish a Cartesian coordinate system with the same unit length (where $(\rho,\theta)$, $\rho \geqslant 0$, $\theta \in [0,2\pi)$).
$(1)$ Line $l$ passes through the origin, and its inclination angle $\alpha= \dfrac {3\pi}{4}$. Find the polar coordinates of the intersection point $A$ of $l$ and circle $E$ (point $A$ is not the origin);
$(2)$ Line $m$ passes through the midpoint $M$ of segment $OA$, and line $m$ intersects circle $E$ at points $B$ and $C$. Find the maximum value of $||MB|-|MC||$. | {
"answer": "2 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and $a^{2}$, $b^{2}$, $c^{2}$ form an arithmetic sequence. Calculate the maximum value of $\sin B$. | {
"answer": "\\dfrac{ \\sqrt {3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $128^3 = 16^x$, what is the value of $2^{-x}$? Express your answer as a common fraction. | {
"answer": "\\frac{1}{2^{5.25}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A traffic light cycles as follows: green for 45 seconds, then yellow for 5 seconds, and then red for 50 seconds. Mark chooses a random five-second interval to observe the light. What is the probability that the color changes during his observation? | {
"answer": "\\frac{3}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum value of $\frac{(3^t - 2t)t}{9^t}$ for integer values of $t$?
**A)** $\frac{1}{8}$
**B)** $\frac{1}{10}$
**C)** $\frac{1}{12}$
**D)** $\frac{1}{16}$
**E)** $\frac{1}{18}$ | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\tan(\omega x+\phi)$ $(\omega>0, 0<|\phi|<\frac{\pi}{2})$, where two adjacent branches of the graph intersect the coordinate axes at points $A(\frac{\pi}{6},0)$ and $B(\frac{2\pi}{3},0)$. Find the sum of all solutions of the equation $f(x)=\sin(2x-\frac{\pi}{3})$ for $x\in [0,\pi]$. | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\angle BAC = 60^{\circ}$, $D$ is a point on side $BC$ such that $AD = \sqrt{7}$, and $BD:DC = 2c:b$, then the minimum value of the area of $\triangle ABC$ is ____. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an experiment, a certain constant \( c \) is measured to be 2.43865 with an error range of \(\pm 0.00312\). The experimenter wants to publish the value of \( c \), with each digit being significant. This means that regardless of how large \( c \) is, the announced value of \( c \) (with \( n \) digits) must match the first \( n \) digits of the true value of \( c \). What is the most precise value of \( c \) that the experimenter can publish? | {
"answer": "2.44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that P and Q are points on the graphs of the functions $2x-y+6=0$ and $y=2\ln x+2$ respectively, find the minimum value of the line segment |PQ|. | {
"answer": "\\frac{6\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=a^{x+4}+2$ with $a \gt 0$ and $a \gt 1$, find the value of $\sin \alpha$ if the terminal side of angle $\alpha$ passes through a point on the graph of the function. | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volume of the solid of revolution generated by rotating the region bounded by the curve $y= \sqrt{2x}$, the line $y=x-4$, and the x-axis around the x-axis is \_\_\_\_\_\_. | {
"answer": "\\frac{128\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a function $f(x)$ on $\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \in [-3,-1)$, $f(x)=-(x+2)^{2}$, and when $x \in [-1,3)$, $f(x)=x$. Find the value of $f(1)+f(2)+f(3)+\ldots+f(2016)$. | {
"answer": "336",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the two-digit numbers less than 20, the largest prime number is ____, and the largest composite number is ____. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a circular track, Alphonse is at point \( A \) and Beryl is diametrically opposite at point \( B \). Alphonse runs counterclockwise and Beryl runs clockwise. They run at constant, but different, speeds. After running for a while they notice that when they pass each other it is always at the same three places on the track. What is the ratio of their speeds? | {
"answer": "2:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In January 1859, an eight-year-old boy dropped a newly-hatched eel into a well in Sweden. The eel, named Ale, finally died in August 2014. How many years old was Åle when it died? | {
"answer": "155",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the fractional equation involving $x$, $\frac{x+m}{x-2}+\frac{1}{2-x}=3$, if it has a root with an increase, then $m=\_\_\_\_\_\_$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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