problem stringlengths 10 5.15k | answer dict |
|---|---|
Floyd looked at a standard $12$ hour analogue clock at $2\!:\!36$ . When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved added to $247$ degrees. How many minutes after $3\!:\!00$ was that? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a shape made from ten squares of side-length \(1 \mathrm{~cm}\), joined edge to edge. What is the length of its perimeter, in centimetres?
A) 14
B) 18
C) 30
D) 32
E) 40 | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of the following expressions:
(1) $(2 \frac {7}{9})^{0.5}+0.1^{-2}+(2 \frac {10}{27})^{- \frac {2}{3}}-3\pi^{0}+ \frac {37}{48}$;
(2) $(-3 \frac {3}{8})^{- \frac {2}{3}}+(0.002)^{- \frac {1}{2}}-10(\sqrt {5}-2)^{-1}+(\sqrt {2}- \sqrt {3})^{0}$. | {
"answer": "- \\frac {167}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that both $α$ and $β$ are acute angles, $\cos α= \frac {1}{7}$, and $\cos (α+β)=- \frac {11}{14}$, find the value of $\cos β$. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many three-digit numbers can you subtract 297 and obtain a second three-digit number which is the original three-digit number reversed? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin x+a\cos x(x∈R)$ whose one symmetric axis is $x=- \frac {π}{4}$.
(I) Find the value of $a$ and the monotonically increasing interval of the function $f(x)$;
(II) If $α$, $β∈(0, \frac {π}{2})$, and $f(α+ \frac {π}{4})= \frac { \sqrt {10}}{5}$, $f(β+ \frac {3π}{4})= \frac {3 \sqrt {5}}{5}$, find $\sin (α+β)$ | {
"answer": "\\frac { \\sqrt {2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polar equation of curve $C$ is $\rho \sin^2\theta = 4\cos\theta$, and the lines $l_1: \theta= \frac{\pi}{3}$, $l_2: \rho\sin\theta=4\sqrt{3}$ intersect curve $C$ at points $A$ and $B$ (with $A$ not being the pole),
(Ⅰ) Find the polar coordinates of points $A$ and $B$;
(Ⅱ) If $O$ is the pole, find the area of $\Delta AOB$. | {
"answer": "\\frac{16}{3}\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regional frisbee league, teams have 7 members and each of the 5 teams takes turns hosting matches. At each match, each team selects three members of that team to be on the match committee, except the host team, which selects four members. How many possible 13-member match committees are there? | {
"answer": "262,609,375",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A school has between 150 and 250 students enrolled. Each day, all the students split into eight different sections for a special workshop. If two students are absent, each section can contain an equal number of students. Find the sum of all possible values of student enrollment at the school. | {
"answer": "2626",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Quadrilateral $ABCD$ has right angles at $A$ and $C$, with diagonal $AC = 5$. If $AB = BC$ and sides $AD$ and $DC$ are of distinct integer lengths, what is the area of quadrilateral $ABCD$? Express your answer in simplest radical form. | {
"answer": "12.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function f(x) = sinωx + cosωx, if there exists a real number x₁ such that for any real number x, f(x₁) ≤ f(x) ≤ f(x₁ + 2018) holds true, find the minimum positive value of ω. | {
"answer": "\\frac{\\pi}{2018}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many arithmetic sequences, where the common difference is a natural number greater than 2, satisfy the conditions that the first term is 1783, the last term is 1993, and the number of terms is at least 3? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $P(\sqrt{3}, 1)$, $Q(\cos x, \sin x)$, and $O$ as the origin of coordinates, the function $f(x) = \overrightarrow{OP} \cdot \overrightarrow{QP}$
(Ⅰ) Find the smallest positive period of the function $f(x)$;
(Ⅱ) If $A$ is an internal angle of $\triangle ABC$, $f(A) = 4$, $BC = 3$, and the area of $\triangle ABC$ is $\frac{3\sqrt{3}}{4}$, find the perimeter of $\triangle ABC$. | {
"answer": "3 + 2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a function $f(x)$ such that for any $x$, $f(x+2)=f(x+1)-f(x)$, and $f(1)=\log_3-\log_2$, $f(2)=\log_3+\log_5$, calculate the value of $f(2010)$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $(xOy)$, the polar coordinate system is established with $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The polar coordinate equation of circle $C$ is $ρ=2 \sqrt{2}\cos \left(θ+\frac{π}{4} \right)$, and the parametric equation of line $l$ is $\begin{cases} x=t \\ y=-1+2 \sqrt{2}t \end{cases}(t\text{ is the parameter})$. Line $l$ intersects circle $C$ at points $A$ and $B$, and $P$ is any point on circle $C$ different from $A$ and $B$.
(1) Find the rectangular coordinates of the circle center.
(2) Find the maximum area of $\triangle PAB$. | {
"answer": "\\frac{10 \\sqrt{5}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the square roots of a number are $2a+3$ and $a-18$, then this number is ____. | {
"answer": "169",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate:<br/>$(1)(\sqrt{50}-\sqrt{8})÷\sqrt{2}$;<br/>$(2)\sqrt{\frac{3}{4}}×\sqrt{2\frac{2}{3}}$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right triangle $PQR$ with $\angle PQR = 90^\circ$, suppose $\cos Q = 0.6$ and $PQ = 15$. What is the length of $QR$? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two arithmetic sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$, with respective sums of the first $n$ terms denoted by $S_{n}$ and $T_{n}$. Given that $\dfrac{S_{n}}{T_{n}} = \dfrac{3n}{2n+1}$, find the value of $\dfrac{a_{1}+a_{2}+a_{14}+a_{19}}{b_{1}+b_{3}+b_{17}+b_{19}}$.
A) $\dfrac{27}{19}$
B) $\dfrac{18}{13}$
C) $\dfrac{10}{7}$
D) $\dfrac{17}{13}$ | {
"answer": "\\dfrac{17}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the functions $f(x)=2(x+1)$ and $g(x)=x+ \ln x$, points $A$ and $B$ are located on the graphs of $f(x)$ and $g(x)$ respectively, and their y-coordinates are always equal. Calculate the minimum distance between points $A$ and $B$. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Observation: Given $\sqrt{5}≈2.236$, $\sqrt{50}≈7.071$, $\sqrt[3]{6.137}≈1.8308$, $\sqrt[3]{6137}≈18.308$; fill in the blanks:<br/>① If $\sqrt{0.5}\approx \_\_\_\_\_\_.$<br/>② If $\sqrt[3]{x}≈-0.18308$, then $x\approx \_\_\_\_\_\_$. | {
"answer": "-0.006137",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x \gt 0$, $y \gt 0$, when $x=$______, the maximum value of $\sqrt{xy}(1-x-2y)$ is _______. | {
"answer": "\\frac{\\sqrt{2}}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For positive real numbers \(a\), \(b\), and \(c\), compute the maximum value of
\[
\frac{abc(a + b + c + ab)}{(a + b)^3 (b + c)^3}.
\] | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the arithmetic sequence $\{a_{n}\}$, if $\frac{{a}_{9}}{{a}_{8}}＜-1$, and its sum of the first $n$ terms $S_{n}$ has a minimum value, determine the minimum value of $n$ for which $S_{n} \gt 0$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $7 \cdot 9\frac{2}{5}$. | {
"answer": "65\\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=2\sin \left(3x+ \dfrac{\pi}{4}\right)$, determine the shift required to obtain its graph from the graph of the function $y=2\sin 3x$. | {
"answer": "\\dfrac{\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that M is a point on the parabola $y^2 = 2px$ ($p > 0$), F is the focus of the parabola $C$, and $|MF| = p$. K is the intersection point of the directrix of the parabola $C$ and the x-axis. Calculate the measure of angle $\angle MKF$. | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \log_{m}(m - x)$, if the maximum value in the interval $[3, 5]$ is 1 greater than the minimum value, determine the real number $m$. | {
"answer": "3 + \\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest integer \( k \) such that when \( 5 \) is multiplied by a number consisting of \( k \) digits of \( 7 \) (i.e., \( 777\ldots7 \) with \( k \) sevens), the resulting product has digits summing to \( 800 \).
A) 86
B) 87
C) 88
D) 89
E) 90 | {
"answer": "88",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Any seven points are taken inside or on a square with side length $2$. Determine $b$, the smallest possible number with the property that it is always possible to select one pair of points from these seven such that the distance between them is equal to or less than $b$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two rows of seats, with 4 seats in the front row and 5 seats in the back row. Now, we need to arrange seating for 2 people, and these 2 people cannot sit next to each other (sitting one in front and one behind is also considered as not adjacent). How many different seating arrangements are there? | {
"answer": "58",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|+1?$
A) $\frac{\pi}{2} + 2$
B) $\frac{3\pi}{2}$
C) $\frac{3\pi}{2} + 2$
D) $2\pi + 2$ | {
"answer": "\\frac{3\\pi}{2} + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{a_n\}$ that satisfies $a_1=1$, $a_2=2$, and $2na_n=(n-1)a_{n-1}+(n+1)a_{n+1}$ for $n \geq 2$ and $n \in \mathbb{N}^*$, find the value of $a_{18}$. | {
"answer": "\\frac{26}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are three environmental knowledge quiz questions, $A$, $B$, and $C$. The table below shows the statistics of the quiz results. The number of people who answered exactly two questions correctly is $\qquad$, and the number of people who answered only one question correctly is $\qquad$.
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline & Total number of people & Correctly answered $A$ & Correctly answered $B$ & Correctly answered $C$ & All incorrect & All correct \\
\hline Number of people & 40 & 10 & 13 & 15 & 15 & 1 \\
\hline
\end{tabular} | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given any two positive integers, a certain operation (denoted by the operator $\oplus$) is defined as follows: when $m$ and $n$ are both positive even numbers or both positive odd numbers, $m \oplus n = m + n$; when one of $m$ and $n$ is a positive even number and the other is a positive odd number, $m \oplus n = m \cdot n$. The number of elements in the set $M = {(a, b) \mid a \oplus b = 12, a, b \in \mathbb{N}^*}$ is $\_\_\_\_\_\_$. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Translate the graph of the function $f(x)=\sin(2x+\varphi)$ ($|\varphi| < \frac{\pi}{2}$) to the left by $\frac{\pi}{6}$ units. If the resulting graph is symmetric about the origin, determine the minimum value of the function $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$. | {
"answer": "-\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An acronym XYZ is drawn within a 2x4 rectangular grid with grid lines spaced 1 unit apart. The letter X is formed by two diagonals crossing in a $1 \times 1$ square. Y consists of a vertical line segment and two slanted segments each forming 45° with the vertical line, making up a symmetric letter. Z is formed by a horizontal segment at the top and bottom of a $1 \times 2$ rectangle, with a diagonal connecting these segments. In units, what is the total length of the line segments forming the acronym XYZ?
A) $5 + 4\sqrt{2} + \sqrt{5}$
B) $5 + 2\sqrt{2} + 3\sqrt{5}$
C) $6 + 3\sqrt{2} + 2\sqrt{5}$
D) $7 + 4\sqrt{2} + \sqrt{3}$
E) $4 + 5\sqrt{2} + \sqrt{5}$ | {
"answer": "5 + 4\\sqrt{2} + \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $2a^{2}\sin B\sin C=\sqrt{3}(a^{2}+b^{2}-c^{2})\sin A$. Find:
$(1)$ Angle $C$;
$(2)$ If $a=1$, $b=2$, and the midpoint of side $AB$ is $D$, find the length of $CD$. | {
"answer": "\\frac{\\sqrt{7}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $2020\times 2020$ squares, and at most one piece is placed in each square. Find the minimum possible number of pieces to be used when placing a piece in a way that satisfies the following conditions.
・For any square, there are at least two pieces that are on the diagonals containing that square.
Note : We say the square $(a,b)$ is on the diagonals containing the square $(c,d)$ when $|a-c|=|b-d|$ . | {
"answer": "2020",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A linear function \( f(x) \) is given. It is known that the distance between the points of intersection of the graphs \( y = x^{2} \) and \( y = f(x) \) is \( 2 \sqrt{3} \), and the distance between the points of intersection of the graphs \( y = x^{2}-2 \) and \( y = f(x)+1 \) is \( \sqrt{60} \). Find the distance between the points of intersection of the graphs \( y = x^{2}-1 \) and \( y = f(x)+1 \). | {
"answer": "2 \\sqrt{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\log_9 \Big(\log_4 (\log_3 x) \Big) = 1$, calculate the value of $x^{-2/3}$. | {
"answer": "3^{-174762.6667}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a quadratic function $f(x)$ with a second-degree coefficient $a$, and the inequality $f(x) > -2x$ has the solution set $(1,3)$:
(1) If the function $y = f(x) + 6a$ has exactly one zero, find the explicit form of $f(x)$.
(2) Let $h(a)$ be the maximum value of $f(x)$, find the minimum value of $h(a)$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the last two digits of the decimal representation of $9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}$ ? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For some constants \( c \) and \( d \), let
\[ g(x) = \left\{
\begin{array}{cl}
cx + d & \text{if } x < 3, \\
10 - 2x & \text{if } x \ge 3.
\end{array}
\right.\]
The function \( g \) has the property that \( g(g(x)) = x \) for all \( x \). What is \( c + d \)? | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a triangle $XYZ$, $\angle XYZ = \angle YXZ$. If $XZ=8$ and $YZ=11$, what is the perimeter of $\triangle XYZ$? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x = \frac{3}{4}$ is a solution to the equation $108x^2 - 35x - 77 = 0$, what is the other value of $x$ that will solve the equation? Express your answer as a common fraction. | {
"answer": "-\\frac{23}{54}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin a= \frac{ \sqrt{5}}{5}$, $a\in\left( \frac{\pi}{2},\pi\right)$, find:
$(1)$ The value of $\sin 2a$;
$(2)$ The value of $\tan \left( \frac{\pi}{3}+a\right)$. | {
"answer": "5 \\sqrt{3}-8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, the equation of curve C is $\rho^2\cos2\theta=9$. Point P is $(2\sqrt{3}, \frac{\pi}{6})$. Establish a Cartesian coordinate system with the pole O as the origin and the positive half-axis of the x-axis as the polar axis.
(1) Find the parametric equation of line OP and the Cartesian equation of curve C;
(2) If line OP intersects curve C at points A and B, find the value of $\frac{1}{|PA|} + \frac{1}{|PB|}$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid $ABCD$, the sides $AB$ and $CD$ are equal. The length of the base $BC$ is 10 units, and the height from $AB$ to $CD$ is 5 units. The base $AD$ is 22 units. Calculate the perimeter of $ABCD$. | {
"answer": "2\\sqrt{61} + 32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | {
"answer": "\\sqrt{55}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $E: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, and point $M$ is on $E$ with $MF_1$ perpendicular to the x-axis and $\sin \angle MF_2F_1 = \frac{1}{3}$, find the eccentricity of $E$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Metropolitan County, there are $25$ cities. From a given bar chart, the average population per city is indicated midway between $5,200$ and $5,800$. If two of these cities, due to a recent demographic survey, were found to exceed the average by double, calculate the closest total population of all these cities. | {
"answer": "148,500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On an $8 \times 8$ chessboard, some squares are marked with asterisks such that:
(1) No two squares with asterisks share a common edge or vertex;
(2) Every unmarked square shares a common edge or vertex with at least one marked square.
What is the minimum number of squares that need to be marked with asterisks? Explain the reasoning. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \\(\theta\\) is an angle in the fourth quadrant, and \\(\sin (\theta+ \frac {\pi}{4})= \frac {3}{5}\\), then \\(\tan (\theta- \frac {\pi}{4})=\\) \_\_\_\_\_\_ . | {
"answer": "- \\frac {4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A high school has three math teachers. To facilitate students, math teachers are scheduled for duty from Monday to Friday, with two teachers on duty on Monday. If each teacher is on duty for two days a week, then there are ________ possible duty schedules for a week. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A quadrilateral is inscribed in a circle with a radius of 13. The diagonals of the quadrilateral are perpendicular to each other. One of the diagonals is 18, and the distance from the center of the circle to the point where the diagonals intersect is \( 4 \sqrt{6} \).
Find the area of the quadrilateral. | {
"answer": "18 \\sqrt{161}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sets $A, B$ , and $C$ satisfy $|A| = 92$ , $|B| = 35$ , $|C| = 63$ , $|A\cap B| = 16$ , $|A\cap C| = 51$ , $|B\cap C| = 19$ . Compute the number of possible values of $ |A \cap B \cap C|$ . | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $\triangle DEF$ has a right angle at $F$, $\angle D = 60^\circ$, and $DF=12$. Find the radius of the incircle of $\triangle DEF$. | {
"answer": "6(\\sqrt{3}-1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Máté is always in a hurry. He observed that it takes 1.5 minutes to get to the subway when he stands on the moving escalator, while it takes 1 minute to run down the stationary stairs. How long does it take Máté to get down if he can run down the moving escalator? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\alpha$ is an angle in the second quadrant, simplify the function $f(\alpha) = \frac{\tan(\alpha - \pi) \cos(2\pi - \alpha) \sin(-\alpha + \frac{3\pi}{2})}{\cos(-\alpha - \pi) \tan(\pi + \alpha)}$. Then, if $\cos(\alpha + \frac{\pi}{2}) = -\frac{1}{5}$, find $f(\alpha)$. | {
"answer": "-\\frac{2\\sqrt{6}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the line $x=\dfrac{\pi }{6}$ is the axis of symmetry of the graph of the function $f\left(x\right)=\sin \left(2x+\varphi \right)\left(|\varphi | \lt \dfrac{\pi }{2}\right)$, determine the horizontal shift required to transform the graph of the function $y=\sin 2x$ into the graph of $y=f\left(x\right)$. | {
"answer": "\\dfrac{\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive real number $d,$ such that for all nonnegative real numbers $x, y,$ and $z,$
\[
\sqrt{xyz} + d |x^2 - y^2 + z^2| \ge \frac{x + y + z}{3}.
\] | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the polar coordinate equation of circle C is ρ² + 2$\sqrt {2}$ρsin(θ + $\frac {π}{4}$) + 1 = 0, and the origin O of the rectangular coordinate system xOy coincides with the pole, and the positive semi-axis of the x-axis coincides with the polar axis. (1) Find the standard equation and a parametric equation of circle C; (2) Let P(x, y) be any point on circle C, find the maximum value of xy. | {
"answer": "\\frac {3}{2} + \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A wire of length $80$cm is randomly cut into three segments. The probability that each segment is no less than $20$cm is $\_\_\_\_\_\_\_.$ | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac{x^{2}}{4}-y^{2}=1$ with its right focus $F$, and points $P_{1}$, $P_{2}$, …, $P_{n}$ on its right upper part where $2\leqslant x\leqslant 2 \sqrt {5}, y\geqslant 0$. The length of the line segment $|P_{k}F|$ is $a_{k}$, $(k=1,2,3,…,n)$. If the sequence $\{a_{n}\}$ is an arithmetic sequence with the common difference $d\in( \frac{1}{5}, \frac{ {\sqrt {5}}}{5})$, find the maximum value of $n$. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right triangular prism $ABC-A_{1}B_{1}C_{1}$ has all its vertices on the surface of a sphere. Given that $AB=3$, $AC=5$, $BC=7$, and $AA_{1}=2$, find the surface area of the sphere. | {
"answer": "\\frac{208\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right triangle has sides of lengths 5 cm and 11 cm. Calculate the length of the remaining side if the side of length 5 cm is a leg of the triangle. Provide your answer as an exact value and as a decimal rounded to two decimal places. | {
"answer": "9.80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $E$: $x^2 + \frac{y^2}{b^2} = 1 (0 < b < 1)$, and the line $l$ passing through $F_1$ intersects $E$ at points $A$ and $B$. If the sequence $|AF_2|, |AB|, |BF_2|$ forms an arithmetic progression, then:
(1) Find $|AB|$;
(2) If the slope of line $l$ is $1$, find the value of $b$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The fractional equation $\dfrac{x-5}{x+2}=\dfrac{m}{x+2}$ has a root, determine the value of $m$. | {
"answer": "-7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the reciprocal of $\frac{3}{4} + \frac{4}{5}$?
A) $\frac{31}{20}$
B) $\frac{20}{31}$
C) $\frac{19}{20}$
D) $\frac{20}{19}$ | {
"answer": "\\frac{20}{31}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A digital clock displays time in a 24-hour format (from 00:00 to 23:59). Find the largest possible sum of the digits in this time display. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Someone collected data relating the average temperature x (℃) during the Spring Festival to the sales y (ten thousand yuan) of a certain heating product. The data pairs (x, y) are as follows: (-2, 20), (-3, 23), (-5, 27), (-6, 30). Based on the data, using linear regression, the linear regression equation between sales y and average temperature x is found to be $y=bx+a$ with a coefficient $b=-2.4$. Predict the sales amount when the average temperature is -8℃. | {
"answer": "34.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the complex plane, $z,$ $z^2,$ $z^3$ represent, in some order, three vertices of a non-degenerate equilateral triangle. Determine all possible perimeters of the triangle. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, a polar coordinate system is established with the origin $O$ as the pole and the positive half-axis of the x-axis as the polar axis. It is known that the point $P(\sqrt {2}, \frac {7\pi}{4})$ lies on the line $l: \rho\cos\theta +2\rho\sin\theta +a=0$ ($a\in\mathbb{R}$).
(Ⅰ) Find the Cartesian equation of line $l$.
(Ⅱ) If point $A$ lies on the line $l$, and point $B$ lies on the curve $C: \begin{cases} x=t \\ y=\frac{1}{4}t^2 \end{cases}$ (where $t$ is a parameter), find the minimum value of $|AB|$. | {
"answer": "\\frac{\\sqrt{5}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let S<sub>n</sub> be the sum of the first n terms of the arithmetic sequence {a<sub>n</sub>}, given that a<sub>7</sub> = 5 and S<sub>5</sub> = -55.
1. Find S<sub>n</sub>.
2. Let b<sub>n</sub> = $$\frac {S_{n}}{n}$$, find the sum of the first 19 terms, T<sub>19</sub>, of the sequence { $$\frac {1}{b_{n}b_{n+1}}$$}. | {
"answer": "-\\frac {1}{19}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$ , $\rho_2$ , $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$ . Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$ , $C(\rho_2, 0)$ , $D(\rho_3, 0)$ . If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$ , find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained.
*Brazitikos Silouanos, Greece* | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distances from three points lying in a horizontal plane to the base of a television tower are 800 m, 700 m, and 500 m, respectively. From each of these three points, the tower is visible (from the base to the top) at certain angles, with the sum of these three angles being $90^{\circ}$.
A) Find the height of the television tower (in meters).
B) Round the answer to the nearest whole number of meters. | {
"answer": "374",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the system of equations
\[
8x - 6y = c,
\]
\[
12y - 18x = d.
\]
If this system has a solution \((x, y)\) where both \(x\) and \(y\) are nonzero, find the value of \(\frac{c}{d}\), assuming \(d\) is nonzero. | {
"answer": "-\\frac{4}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $P(-2,-3)$ and $Q(5, 3)$ in the $xy$-plane; point $R(2,m)$ is taken so that $PR+RQ$ is minimized. Determine the value of $m$.
A) $\frac{3}{5}$
B) $\frac{2}{5}$
C) $\frac{3}{7}$
D) $\frac{1}{5}$ | {
"answer": "\\frac{3}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \\(f(x) = x^2 + 2ax + 4\\) and the interval \\([-3,5]\\), calculate the probability that the function has no real roots. | {
"answer": "\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
By using equations, recurring decimals can be converted into fractions. For example, when converting $0.\overline{3}$ into a fraction, we can let $0.\overline{3} = x$. From $0.\overline{3} = 0.333\ldots$, we know that $10x = 3.333\ldots$. Therefore, $10x = 3 + 0.\overline{3}$. So, $10x = 3 + x$. Solving this equation, we get $x = \frac{1}{3}$, which means $0.\overline{3} = \frac{1}{3}$.
$(1)$ Convert $0.\overline{4}\overline{5}$ into a fraction, fill in the blanks below:
Let $0.\overline{4}\overline{5} = x$. From $0.\overline{4}\overline{5} = 0.4545\ldots$, we have $100x = 45.4545\ldots$.
So, $100x = 45 + 0.\overline{4}\overline{5}$. Therefore, ______. Solving this equation, we get $x = \_\_\_\_\_\_$.
$(2)$ Convert $0.2\overline{4}\overline{5}$ into a fraction. | {
"answer": "\\frac{27}{110}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate plane $(xOy)$, two acute angles $\alpha$ and $\beta$ are formed with the non-negative semi-axis of $Ox$ as the initial side. Their terminal sides intersect the unit circle at points $A$ and $B$ respectively. The vertical coordinates of $A$ and $B$ are $\frac{\sqrt{5}}{5}$ and $\frac{3\sqrt{10}}{10}$ respectively.
1. Find $\alpha - \beta$.
2. Find the value of $\cos(2\alpha - \beta)$. | {
"answer": "\\frac{3\\sqrt{10}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The stem-and-leaf plot shows the duration of songs (in minutes and seconds) played during a concert by a band. There are 15 songs listed in the plot. In the stem-and-leaf plot, $3 \ 15$ represents $3$ minutes, $15$ seconds, which is the same as $195$ seconds. Find the median duration of the songs. Express your answer in seconds.
\begin{tabular}{c|ccccc}
1&30&45&50&&\\
2&10&20&30&35&50\\
3&00&15&15&30&45\\
\end{tabular} | {
"answer": "170",
"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 2(tanA + tanB) = $\frac{\text{tanA}}{\text{cosB}} + \frac{\text{tanB}}{\text{cosA}}$.
(1) Find the value of $\frac{a+b}{c}$;
(2) If c = 2 and C = $\frac{\pi}{3}$, find the area of triangle ABC. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(0 \leq x_0 < 1\), let
\[
x_n = \left\{ \begin{array}{ll}
2x_{n-1} & \text{if } 2x_{n-1} < 1 \\
2x_{n-1} - 1 & \text{if } 2x_{n-1} \geq 1
\end{array} \right.
\]
for all integers \(n > 0\). Determine the number of initial values of \(x_0\) that satisfy \(x_0 = x_6\). | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a country there are $15$ cities, some pairs of which are connected by a single two-way airline of a company. There are $3$ companies and if any of them cancels all its flights, then it would still be possible to reach every city from every other city using the other two companies. At least how many two-way airlines are there? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The bases of an isosceles trapezoid are in the ratio 3:2. A circle is constructed on the larger base as its diameter, and this circle intersects the smaller base such that the segment cut off on the smaller base is equal to half of the smaller base. In what ratio does the circle divide the non-parallel sides of the trapezoid? | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle ABC, the angles A, B, and C are represented by vectors AB and BC with an angle θ between them. Given that the dot product of AB and BC is 6, and that $6(2-\sqrt{3})\leq|\overrightarrow{AB}||\overrightarrow{BC}|\sin(\pi-\theta)\leq6\sqrt{3}$.
(I) Find the value of $\tan 15^\circ$ and the range of values for θ.
(II) Find the maximum value of the function $f(\theta)=\frac{1-\sqrt{2}\cos(2\theta-\frac{\pi}{4})}{\sin\theta}$. | {
"answer": "\\sqrt{3}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a function \\(f(x)\\) defined on \\(\mathbb{R}\\) that satisfies: the graph of \\(y=f(x-1)\\) is symmetric about the point \\((1,0)\\), and when \\(x \geqslant 0\\), it always holds that \\(f(x+2)=f(x)\\). When \\(x \in [0,2)\\), \\(f(x)=e^{x}-1\\), where \\(e\\) is the base of the natural logarithm, evaluate \\(f(2016)+f(-2017)\\). | {
"answer": "1-e",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=\sqrt{2}$, and $\overrightarrow{a} \bot (\overrightarrow{a}-\overrightarrow{b})$, find the projection of vector $\overrightarrow{a}$ in the direction of vector $\overrightarrow{b}$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression $3 - (-3)^{-\frac{2}{3}}$. | {
"answer": "3 - \\frac{1}{\\sqrt[3]{9}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the polar coordinate equation of curve C is ρ - 4cosθ = 0, establish a rectangular coordinate system with the pole as the origin and the polar axis as the positive semi-axis. Line l passes through point M(3, 0) with a slope angle of $\frac{\pi}{6}$.
(I) Find the rectangular coordinate equation of curve C and the parametric equation of line l;
(II) If 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"
} |
The sequence $\{a_n\}$ satisfies $a_{n+1}+(-1)^n a_n = 2n-1$. Find the sum of the first $80$ terms of $\{a_n\}$. | {
"answer": "3240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An entire floor is tiled with blue and white tiles. The floor has a repeated tiling pattern that forms every $8 \times 8$ square. Each of the four corners of this square features an asymmetrical arrangement of tiles where the bottom left $4 \times 4$ segment within each $8 \times 8$ square consists of blue tiles except for a $2 \times 2$ section of white tiles at its center. What fraction of the floor is made up of blue tiles?
A) $\frac{1}{2}$
B) $\frac{5}{8}$
C) $\frac{3}{4}$
D) $\frac{7}{8}$
E) $\frac{2}{3}$ | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a random variable $X\sim N(2, \sigma ^{2})$, $P(X\leqslant 0)=0.15$, calculate $P(2\leqslant X\leqslant 4)$. | {
"answer": "0.35",
"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 $a^{2}+b^{2}+4 \sqrt {2}=c^{2}$ and $ab=4$, find the minimum value of $\frac {\sin C}{\tan ^{2}A\cdot \sin 2B}$. | {
"answer": "\\frac {3 \\sqrt {2}}{2}+2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Use the Euclidean algorithm to find the greatest common divisor (GCD) of 117 and 182, and verify it using the method of successive subtraction.
(2) Use the Horner's method to calculate the value of the polynomial \\(f(x)=1-9x+8x^{2}-4x^{4}+5x^{5}+3x^{6}\\) at \\(x=-1\\). | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the eccentricity of the conic section \(C\): \(x^{2}+my^{2}=1\) is \(2\), determine the value of \(m\). | {
"answer": "-\\dfrac {1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $y=kx+b$ is a tangent to the curve $f\left(x\right)=\ln x+2$ and also a tangent to the curve $g\left(x\right)=\ln \left(x+1\right)$, determine the value of $k-b$. | {
"answer": "1 + \\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a point $P(x,y)$ moving on the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$, let $d = \sqrt{x^{2} + y^{2} + 4y + 4} - \frac{x}{2}$. Find the minimum value of $d$.
A) $\sqrt{5} - 2$
B) $2\sqrt{2} - 1$
C) $\sqrt{5} - 1$
D) $\sqrt{6} - 1$ | {
"answer": "2\\sqrt{2} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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