problem stringlengths 10 5.15k | answer dict |
|---|---|
Given numbers \( x_{1}, \ldots, x_{n} \in (0,1) \), find the maximum value of the expression
$$
A = \frac{\sqrt[4]{1-x_{1}} + \ldots + \sqrt[4]{1-x_{n}}}{\frac{1}{\sqrt[4]{x_{1}}} + \ldots + \frac{1}{\sqrt[4]{x_{n}}}}
$$ | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The positive integers $A, B, C$, and $D$ form an arithmetic and geometric sequence as follows: $A, B, C$ form an arithmetic sequence, while $B, C, D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{3}$, what is the smallest possible value of $A + B + C + D$? | {
"answer": "76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If two lines $l$ and $m$ have equations $y = -2x + 8$, and $y = -3x + 9$, what is the probability that a point randomly selected in the 1st quadrant and below $l$ will fall between $l$ and $m$? | {
"answer": "0.15625",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(ABCD-A_{1}B_{1}C_{1}D_{1}\) is a cube and \(P-A_{1}B_{1}C_{1}D_{1}\) is a regular tetrahedron, find the cosine of the angle between the skew lines \(A_{1}P\) and \(BC_{1}\), given that the distance from point \(P\) to plane \(ABC\) is \(\frac{3}{2}AB\). | {
"answer": "\\frac{\\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the difference between the sum of the first one hundred positive even integers and the sum of the first one hundred positive multiples of 3. | {
"answer": "-5050",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The second hand on a clock is 8 cm long. How far in centimeters does the tip of the second hand travel during a period of 45 minutes? Express your answer in terms of $\pi$. | {
"answer": "720\\pi",
"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. The function $f(x)=2\cos x\sin (x-A)+\sin A (x\in R)$ reaches its maximum value at $x=\frac{5\pi}{12}$.
(1) Find the range of the function $f(x)$ when $x\in(0,\frac{\pi}{2})$;
(2) If $a=7$ and $\sin B+\sin C=\frac{13\sqrt{3}}{14}$, find the area of $\triangle ABC$. | {
"answer": "10\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The hyperbola $M$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ has left and right foci $F_l$ and $F_2$. The parabola $N$: $y^{2} = 2px (p > 0)$ has a focus at $F_2$. Point $P$ is an intersection point of hyperbola $M$ and parabola $N$. If the midpoint of $PF_1$ lies on the $y$-axis, calculate the eccentricity of this hyperbola. | {
"answer": "\\sqrt{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, and $c$ are the lengths of the sides opposite the angles $A$, $B$, and $C$ in $\triangle ABC$ respectively, with $a=2$, and $$\frac{\sin A - \sin B}{\sin C} = \frac{c - b}{2 + b}.$$ Find the maximum area of $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Maria needs to build a circular fence around a garden. Based on city regulations, the garden's diameter needs to be close to 30 meters, with an allowable error of up to $10\%$. After building, the fence turned out to have a diameter of 33 meters. Calculate the area she thought she was enclosing and the actual area enclosed. What is the percent difference between these two areas? | {
"answer": "21\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An odd function $f(x)$ defined on $R$ satisfies $f(x) = f(2-x)$. When $x \in [0,1]$, $f(x) = ax^{3} + 2x + a + 1$. Find $f(2023)$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Maria buys computer disks at a price of 5 for $6 and sells them at a price of 4 for $7. Find how many computer disks Maria must sell to make a profit of $120. | {
"answer": "219",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given three points in space \\(A(0,2,3)\\), \\(B(-2,1,6)\\), and \\(C(1,-1,5)\\):
\\((1)\\) Find \\(\cos < \overrightarrow{AB}, \overrightarrow{AC} >\\).
\\((2)\\) Find the area of the parallelogram with sides \\(AB\\) and \\(AC\\). | {
"answer": "7\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of $\sin 135^{\circ}\cos 15^{\circ}-\cos 45^{\circ}\sin (-15^{\circ})$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $-{1^{2023}}+|{\sqrt{3}-2}|-3\tan60°$. | {
"answer": "1 - 4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the curve $C$ is given by $\frac{x^2}{4} + \frac{y^2}{3} = 1$. Taking the origin $O$ of the Cartesian coordinate system $xOy$ as the pole and the positive half-axis of $x$ as the polar axis, and using the same unit length, a polar coordinate system is established. It is known that the line $l$ is given by $\rho(\cos\theta - 2\sin\theta) = 6$.
(I) Write the Cartesian coordinate equation of line $l$ and the parametric equation of curve $C$;
(II) Find a point $P$ on curve $C$ such that the distance from point $P$ to line $l$ is maximized, and find this maximum value. | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The value of $a$ is chosen so that the number of roots of the first equation $4^{x}-4^{-x}=2 \cos(a x)$ is 2007. How many roots does the second equation $4^{x}+4^{-x}=2 \cos(a x)+4$ have for the same value of $a$? | {
"answer": "4014",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the volume in cubic centimeters of a truncated cone formed by cutting a smaller cone from a larger cone. The larger cone has a diameter of 8 cm at the base and a height of 10 cm. The smaller cone, which is cut from the top, has a diameter of 4 cm and a height of 4 cm. Express your answer in terms of \(\pi\). | {
"answer": "48\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, the curve $C$ is given by the parametric equations $\begin{cases} & x=2\sqrt{3}\cos \alpha \\ & y=2\sin \alpha \end{cases}$, where $\alpha$ is the parameter, $\alpha \in (0,\pi)$. In the polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis, the point $P$ has polar coordinates $(4\sqrt{2},\frac{\pi}{4})$, and the line $l$ has the polar equation $\rho \sin \left( \theta -\frac{\pi}{4} \right)+5\sqrt{2}=0$.
(I) Find the Cartesian equation of line $l$ and the standard equation of curve $C$;
(II) If $Q$ is a moving point on curve $C$, and $M$ is the midpoint of segment $PQ$, find the maximum distance from point $M$ to line $l$. | {
"answer": "6 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the circle $C:(x-1)^2+(y-2)^2=25$ and the line $l:(2m+1)x+(m+1)y-7m-4=0$, determine the length of the shortest chord intercepted by line $l$ on circle $C$. | {
"answer": "4\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The opposite of the arithmetic square root of $\sqrt{81}$ is ______. | {
"answer": "-9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Complex numbers \( a \), \( b \), and \( c \) form an equilateral triangle with side length 24 in the complex plane. If \( |a + b + c| = 48 \), find \( |ab + ac + bc| \). | {
"answer": "768",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From June to August 1861, a total of 1026 inches of rain fell in Cherrapunji, India, heavily influenced by the monsoon season, and the total duration of these summer months is 92 days and 24 hours per day. Calculate the average rainfall in inches per hour. | {
"answer": "\\frac{1026}{2208}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The increasing sequence of positive integers $a_1, a_2, a_3, \dots$ is defined by the rule
\[a_{n + 2} = a_{n + 1} + a_n\]
for all $n \ge 1.$ If $a_7 = 210$, then find $a_8.$ | {
"answer": "340",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system xOy, the parametric equations of the curve C1 are given by $$\begin{cases} x=t\cos\alpha \\ y=1+t\sin\alpha \end{cases}$$, and the polar coordinate equation of the curve C2 with the origin O as the pole and the positive semi-axis of the x-axis as the polar axis is ρ=2cosθ.
1. If the parameter of curve C1 is α, and C1 intersects C2 at exactly one point, find the Cartesian equation of C1.
2. Given point A(0, 1), if the parameter of curve C1 is t, 0<α<π, and C1 intersects C2 at two distinct points P and Q, find the maximum value of $$\frac {1}{|AP|}+\frac {1}{|AQ|}$$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The edge of a regular tetrahedron is equal to \(\sqrt{2}\). Find the radius of the sphere whose surface touches all the edges of the tetrahedron. | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If point P lies on the graph of the function $y=e^x$ and point Q lies on the graph of the function $y=\ln x$, then the minimum distance between points P and Q is \_\_\_\_\_\_. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are 6 balls of each of the four colors: red, blue, yellow, and green, each numbered from 1 to 6, calculate the number of ways to select 3 balls with distinct numbers, such that no two balls have the same color or consecutive numbers. | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest solution to the equation \[\lfloor x^2 \rfloor - \lfloor x \rfloor^2 = 19.\] | {
"answer": "\\sqrt{109}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $G$ be the set of points $(x, y)$ such that $x$ and $y$ are positive integers less than or equal to 20. Say that a ray in the coordinate plane is *ocular* if it starts at $(0, 0)$ and passes through at least one point in $G$ . Let $A$ be the set of angle measures of acute angles formed by two distinct ocular rays. Determine
\[
\min_{a \in A} \tan a.
\] | {
"answer": "1/722",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a rectangular grid constructed with toothpicks of equal length, with a height of 15 toothpicks and a width of 12 toothpicks, calculate the total number of toothpicks required to build the grid. | {
"answer": "387",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$. It is known that $2a\cos A=c\cos B+b\cos C$.
(Ⅰ) Find the value of $\cos A$;
(Ⅱ) If $a=1$ and $\cos^2 \frac{B}{2}+\cos^2 \frac{C}{2}=1+ \frac{\sqrt{3}}{4}$, find the value of side $c$. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two intersecting circles O: $x^2 + y^2 = 25$ and C: $x^2 + y^2 - 4x - 2y - 20 = 0$, which intersect at points A and B, find the length of the common chord AB. | {
"answer": "\\sqrt{95}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 10.\] | {
"answer": "\\frac{131}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of
\[\frac{2x + 3y + 4}{\sqrt{x^2 + 4y^2 + 2}}\]
over all real numbers \( x \) and \( y \). | {
"answer": "\\sqrt{29}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)=\cos x(\sqrt{3}\sin x-\cos x)+\frac{3}{2}$.
$(1)$ Find the interval on which $f(x)$ is monotonically decreasing on $[0,\pi]$.
$(2)$ If $f(\alpha)=\frac{2}{5}$ and $\alpha\in(\frac{\pi}{3},\frac{5\pi}{6})$, find the value of $\sin 2\alpha$. | {
"answer": "\\frac{-3\\sqrt{3} - 4}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a square with side length $2$ , and let a semicircle with flat side $CD$ be drawn inside the square. Of the remaining area inside the square outside the semi-circle, the largest circle is drawn. What is the radius of this circle? | {
"answer": "4 - 2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We write the equation on the board:
$$
(x-1)(x-2) \ldots (x-2016) = (x-1)(x-2) \ldots (x-2016) .
$$
We want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimal number of factors that need to be erased to achieve this? | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $p$, $q$, $r$, $s$, $t$, and $u$ are integers such that $1728x^3 + 64 = (px^2 + qx + r)(sx^2 + tx + u)$ for all $x$, then what is $p^2+q^2+r^2+s^2+t^2+u^2$? | {
"answer": "23456",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle $ABCD$, $AB=2$, $BC=4$, and points $E$, $F$, and $G$ are located as follows: $E$ is the midpoint of $\overline{BC}$, $F$ is the midpoint of $\overline{CD}$, and $G$ is one fourth of the way down $\overline{AD}$ from $A$. If point $H$ is the midpoint of $\overline{GE}$, what is the area of the shaded region defined by triangle $EHF$?
A) $\dfrac{5}{4}$
B) $\dfrac{3}{2}$
C) $\dfrac{7}{4}$
D) 2
E) $\dfrac{9}{4}$ | {
"answer": "\\dfrac{5}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2x^{2}-3x-\ln x+e^{x-a}+4e^{a-x}$, where $e$ is the base of the natural logarithm, if there exists a real number $x_{0}$ such that $f(x_{0})=3$ holds, then the value of the real number $a$ is \_\_\_\_\_\_. | {
"answer": "1-\\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose
\[\frac{1}{x^3 - 3x^2 - 13x + 15} = \frac{A}{x+3} + \frac{B}{x-1} + \frac{C}{(x-1)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$? | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest square number that, when divided by a cube number, results in a fraction in its simplest form where the numerator is a cube number (other than 1) and the denominator is a square number (other than 1)? | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square has vertices at \((-2a, -2a), (2a, -2a), (-2a, 2a), (2a, 2a)\). The line \( y = -\frac{x}{2} \) cuts this square into two congruent quadrilaterals. Calculate the perimeter of one of these quadrilaterals divided by \( 2a \). Express your answer in simplified radical form. | {
"answer": "4 + \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the constant $d$ such that
$$\left(3x^3 - 2x^2 + x - \frac{5}{4}\right)(ex^3 + dx^2 + cx + f) = 9x^6 - 5x^5 - x^4 + 20x^3 - \frac{25}{4}x^2 + \frac{15}{4}x - \frac{5}{2}$$ | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ have an angle of $45^\circ$ between them, and $|\overrightarrow{a}|=1$, $|2\overrightarrow{a}-\overrightarrow{b}|=\sqrt{10}$, find the magnitude of vector $\overrightarrow{b}$. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Grandma has two balls of yarn: one large and one small. From the large ball, she can either knit a sweater and three socks, or five identical hats. From the small ball, she can either knit half a sweater or two hats. (In both cases, all the yarn will be used up.) What is the maximum number of socks Grandma can knit using both balls of yarn? Justify your answer. | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(Ⅰ) Find the equation of the line that passes through the intersection point of the two lines $2x-3y-3=0$ and $x+y+2=0$, and is perpendicular to the line $3x+y-1=0$.
(Ⅱ) Given the equation of line $l$ in terms of $x$ and $y$ as $mx+y-2(m+1)=0$, find the maximum distance from the origin $O$ to the line $l$. | {
"answer": "2 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jane places six ounces of tea into a ten-ounce cup and six ounces of milk into a second cup of the same size. She then pours two ounces of tea from the first cup to the second and, after stirring thoroughly, pours two ounces of the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now milk?
A) $\frac{1}{8}$
B) $\frac{1}{6}$
C) $\frac{1}{4}$
D) $\frac{1}{3}$
E) $\frac{1}{2}$ | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\tan \alpha=-2$, the focus of the parabola $y^{2}=2px (p > 0)$ is $F(-\sin \alpha\cos \alpha,0)$, and line $l$ passes through point $F$ and intersects the parabola at points $A$ and $B$ with $|AB|=4$, find the distance from the midpoint of segment $AB$ to the line $x=-\frac{1}{2}$. | {
"answer": "\\frac{21}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $b\sin 2A = \sqrt{3}a\sin B$.
$(Ⅰ)$ Find $\angle A$;
$(Ⅱ)$ If the area of $\triangle ABC$ is $3\sqrt{3}$, choose one of the three conditions, condition ①, condition ②, or condition ③, as the given condition to ensure the existence and uniqueness of $\triangle ABC$, and find the value of $a$.
Condition ①: $\sin C = \frac{2\sqrt{7}}{7}$;
Condition ②: $\frac{b}{c} = \frac{3\sqrt{3}}{4}$;
Condition ③: $\cos C = \frac{\sqrt{21}}{7}$
Note: If the chosen condition does not meet the requirements, no points will be awarded for question $(Ⅱ)$. If multiple suitable conditions are chosen and answered separately, the first answer will be scored. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a positive integer $n$ , define $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$ . Find the positive integer $k$ for which $7?9?=5?k?$ .
*Proposed by Tristan Shin* | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $f(x)=\frac{x}{x+a}$ is symmetric about the point $(1,1)$, and the function $g(x)=\log_{10}(10^x+1)+bx$ is even. Find the value of $a+b$. | {
"answer": "-\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of $v_2$ when $x = 2$ for $f(x) = 3x^4 + x^3 + 2x^2 + x + 4$ using Horner's method. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain shopping mall sells two types of products, A and B. The profit margin for each unit of product A is $40\%$, and for each unit of product B is $50\%$. When the quantity of product A sold is $150\%$ of the quantity of product B sold, the total profit margin for selling these two products in the mall is $45\%$. Determine the total profit margin when the quantity of product A sold is $50\%$ of the quantity of product B sold. | {
"answer": "47.5\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right circular cone $(P-ABC)$ with lateral edges $(PA)$, $(PB)$, $(PC)$ being pairwise perpendicular, and base edge $AB = \sqrt{2}$, find the surface area of the circumscribed sphere of the right circular cone $(P-ABC)$. | {
"answer": "3\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be a set of size $11$ . A random $12$ -tuple $(s_1, s_2, . . . , s_{12})$ of elements of $S$ is chosen uniformly at random. Moreover, let $\pi : S \to S$ be a permutation of $S$ chosen uniformly at random. The probability that $s_{i+1}\ne \pi (s_i)$ for all $1 \le i \le 12$ (where $s_{13} = s_1$ ) can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Compute $a$ . | {
"answer": "1000000000004",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos (2x+ \frac {\pi}{4})$, if we shrink the x-coordinates of all points on the graph of $y=f(x)$ to half of their original values while keeping the y-coordinates unchanged; and then shift the resulting graph to the right by $|\varphi|$ units, and the resulting graph is symmetric about the origin, find the value of $\varphi$. | {
"answer": "\\frac {3\\pi}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle \( \triangle ABC \) with \( Q \) as the midpoint of \( BC \), \( P \) on \( AC \) such that \( CP = 3PA \), and \( R \) on \( AB \) such that \( S_{\triangle PQR} = 2 S_{\triangle RBQ} \). If \( S_{\triangle ABC} = 300 \), find \( S_{\triangle PQR} \). | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Given that $0 < x < \dfrac{4}{3}$, find the maximum value of $x(4-3x)$.
(2) Point $(x,y)$ moves along the line $x+2y=3$. Find the minimum value of $2^{x}+4^{y}$. | {
"answer": "4 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polar equation of curve $C$ is $\rho=1$, with the pole as the origin of the Cartesian coordinate system, and the polar axis as the positive half-axis of $x$, establish the Cartesian coordinate system. The parametric equation of line $l$ is $\begin{cases} x=-1+4t \\ y=3t \end{cases}$ (where $t$ is the parameter), find the length of the chord cut by line $l$ on curve $C$. | {
"answer": "\\dfrac {8}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the curves $C\_1$: $\begin{cases} & x=-4+\cos t, \ & y=3+\sin t \ \end{cases}$ (with $t$ as the parameter), and $C\_2$: $\begin{cases} & x=6\cos \theta, \ & y=2\sin \theta \ \end{cases}$ (with $\theta$ as the parameter).
(1) Convert the equations of $C\_1$ and $C\_2$ into general form and explain what type of curves they represent.
(2) If the point $P$ on $C\_1$ corresponds to the parameter $t=\frac{\pi }{2}$, and $Q$ is a moving point on $C\_2$, find the minimum distance from the midpoint $M$ of $PQ$ to the line $C\_3$: $\begin{cases} & x=-3\sqrt{3}+\sqrt{3}\alpha, \ & y=-3-\alpha \ \end{cases}$ (with $\alpha$ as the parameter). | {
"answer": "3\\sqrt{3}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equation of line $l$ as $\begin{cases} x = \frac{\sqrt{2}}{2}t \\ y = \frac{\sqrt{2}}{2}t + 4\sqrt{2} \end{cases}$ (where $t$ is the parameter) and the polar equation of circle $C$ as $\rho = 2\cos (\theta + \frac{\pi}{4})$,
(I) Find the rectangular coordinates of the center of circle $C$.
(II) Find the minimum length of a tangent line drawn from a point on line $l$ to circle $C$. | {
"answer": "2\\sqrt {6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a right-angled triangle \( ACD \) with a point \( B \) on the side \( AC \). The sides of triangle \( ABD \) have lengths 3, 7, and 8. What is the area of triangle \( BCD \)? | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ be a point on the ellipse $\frac{x^2}{16} + \frac{y^2}{9} =1$, and $F_1$, $F_2$ be the left and right foci of the ellipse, respectively. If $\angle F_1 PF_2 = \frac{\pi}{3}$, find the area of $\triangle F_1 PF_2$. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $|\overrightarrow{a}|=|\overrightarrow{b}|=1$, $\overrightarrow{a}\cdot \overrightarrow{b}= \frac{1}{2}$, and $(\overrightarrow{a}- \overrightarrow{c})\cdot(\overrightarrow{b}- \overrightarrow{c})=0$. Then, calculate the maximum value of $|\overrightarrow{c}|$. | {
"answer": "\\frac{\\sqrt{3}+1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle with a radius of 5, its center lies on the x-axis with an integer horizontal coordinate and is tangent to the line 4x + 3y - 29 = 0.
(1) Find the equation of the circle;
(2) If the line ax - y + 5 = 0 (a ≠ 0) intersects the circle at points A and B, does there exist a real number a such that the line l passing through point P(-2, 4) is perpendicularly bisecting chord AB? If such a real number a exists, find its value; otherwise, explain the reason. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify the expression first, then evaluate it: \\(5(3a^{2}b-ab^{2})-(ab^{2}+3a^{2}b)\\), where \\(a= \frac {1}{2}\\), \\(b= \frac {1}{3}\\). | {
"answer": "\\frac {2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A teacher received letters on Monday to Friday with counts of $10$, $6$, $8$, $5$, $6$ respectively. Calculate the standard deviation of this data set. | {
"answer": "\\dfrac {4 \\sqrt {5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A lighthouse emits a yellow signal every 15 seconds and a red signal every 28 seconds. The yellow signal is first seen 2 seconds after midnight, and the red signal is first seen 8 seconds after midnight. At what time will both signals be seen together for the first time? | {
"answer": "92",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular solid has three adjacent faces with areas of $1$, $2$, and $2$, respectively. All the vertices of the rectangular solid are located on the same sphere. Find the volume of this sphere. | {
"answer": "\\sqrt{6}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
1. Simplify and evaluate the expression: $\log_{\frac{1}{3}} \sqrt{27} + \lg 25 + \lg 4 + 7^{-\log_{7} 2} + (-0.98)^0$
2. Given a point $P(\sqrt{2}, -\sqrt{6})$ on the terminal side of angle $\alpha$, evaluate: $\frac{\cos \left( \frac{\pi}{2} + \alpha \right) \cos \left( 2\pi - \alpha \right) + \sin \left( -\alpha - \frac{\pi}{2} \right) \cos \left( \pi - \alpha \right)}{\sin \left( \pi + \alpha \right) \cos \left( \frac{\pi}{2} - \alpha \right)}$ | {
"answer": "\\frac{-\\sqrt{3} - 1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terminal side of angle $θ$ lies on the ray $y=2x(x≤0)$, find the value of $\sin θ + \cos θ$. | {
"answer": "-\\frac{3\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x$ is a multiple of $3456$, what is the greatest common divisor of $f(x)=(5x+3)(11x+2)(14x+7)(3x+8)$ and $x$? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), perpendiculars are drawn from the right focus $F(2\sqrt{2}, 0)$ to the two asymptotes, with the feet of the perpendiculars being $A$ and $B$, respectively. Let point $O$ be the origin. If the area of quadrilateral $OAFB$ is $4$, determine the eccentricity of the hyperbola. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system, a pole coordinate system is established with the origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. Given the curve $C$: ${p}^{2}=\frac{12}{2+{\mathrm{cos}}^{}θ}$ and the line $l$: $2p\mathrm{cos}\left(θ-\frac{π}{6}\right)=\sqrt{3}$.
1. Write the rectangular coordinate equations for the line $l$ and the curve $C$.
2. Let points $A$ and $B$ be the two intersection points of line $l$ and curve $C$. Find the value of $|AB|$. | {
"answer": "\\frac{4\\sqrt{10}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the mathematical expectation of the area of the projection of a cube with edge of length $1$ onto a plane with an isotropically distributed random direction of projection. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the 21 top-rated water slides in the world. In the stem-and-leaf plot, $1 \ 45$ represents 1 minute, 45 seconds, which is equivalent to 105 seconds. What is the median of this data set? Express your answer in seconds.
\begin{tabular}{c|cccccc}
0&15&30&45&55&&\\
1&00&20&35&45&55&\\
2&10&15&30&45&50&55\\
3&05&10&15&&&\\
\end{tabular} | {
"answer": "135",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of real solutions of the equation
\[\frac{x}{50} = \cos x.\] | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \sqrt {3}\sin (\omega x+\varphi)-\cos (\omega x+\varphi)$ $(\omega > 0,0 < \varphi < \pi)$ is an even function, and the distance between two adjacent axes of symmetry of its graph is $\dfrac {\pi}{2}$, then the value of $f(- \dfrac {\pi}{8})$ is \_\_\_\_\_\_. | {
"answer": "\\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $\angle C= \frac{\pi}{2}$, $\angle B= \frac{\pi}{6}$, and $AC=2$. $M$ is the midpoint of $AB$. $\triangle ACM$ is folded along $CM$ such that the distance between $A$ and $B$ is $2\sqrt{2}$. The surface area of the circumscribed sphere of the tetrahedron $M-ABC$ is \_\_\_\_\_\_. | {
"answer": "16\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If 2006 integers $a_1, a_2, \ldots a_{2006}$ satisfy the following conditions: $a_1=0$, $|a_2|=|a_1+2|$, $|a_3|=|a_2+2|$, $\ldots$, $|a_{2006}|=|a_{2005}+2|$, then the minimum value of $a_1+a_2+\ldots+a_{2005}$ is. | {
"answer": "-2004",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$ , $k \geq 7$ , and for which the following equalities hold: $$ d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1 $$ *Proposed by Mykyta Kharin* | {
"answer": "2024",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin (\frac{\pi }{3}-\theta )=\frac{3}{4}$, find $\cos (\frac{\pi }{3}+2\theta )$. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form. | {
"answer": "1 + \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the following infinite geometric series: $$\frac{7}{8}-\frac{14}{27}+\frac{28}{81}-\dots$$ Find the common ratio of this series. | {
"answer": "-\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jo, Blair, and Parker take turns counting from 1, increasing by one more than the last number said by the previous person. What is the $100^{\text{th}}$ number said? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a new operation $\star$ such that for positive integers $a, b, c$, $a \star b \star c = \frac{a \times b + c}{a + b + c}$. Calculate the value of $4 \star 8 \star 2$.
**A)** $\frac{34}{14}$ **B)** $\frac{16}{7}$ **C)** $\frac{17}{7}$ **D)** $\frac{32}{14}$ **E)** $2$ | {
"answer": "\\frac{17}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( m = 2^{40}5^{24} \). How many positive integer divisors of \( m^2 \) are less than \( m \) but do not divide \( m \)? | {
"answer": "959",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two different numbers are randomly selected from the set $\{-3, -2, 0, 0, 5, 6, 7\}$. What is the probability that the product of these two numbers is $0$?
**A)** $\frac{1}{4}$
**B)** $\frac{1}{5}$
**C)** $\frac{5}{21}$
**D)** $\frac{1}{3}$
**E)** $\frac{1}{2}$ | {
"answer": "\\frac{5}{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the function $f(x)=a- \frac {2}{2^{x}+1}(a\in\mathbb{R})$
$(1)$ Determine the monotonicity of the function $f(x)$ and provide a proof;
$(2)$ If there exists a real number $a$ such that the function $f(x)$ is an odd function, find $a$;
$(3)$ For the $a$ found in $(2)$, if $f(x)\geqslant \frac {m}{2^{x}}$ holds true for all $x\in[2,3]$, find the maximum value of $m$. | {
"answer": "\\frac {12}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate:<br/>$(1)-1^{2023}+8×(-\frac{1}{2})^{3}+|-3|$;<br/>$(2)(-25)×\frac{3}{2}-(-25)×\frac{5}{8}+(-25)÷8($simplified calculation). | {
"answer": "-25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\frac{\cos A}{\cos C}=\frac{a}{2b-c}$, find:<br/>
$(1)$ The measure of angle $A$;<br/>
$(2)$ If $a=\sqrt{7}$, $c=3$, and $D$ is the midpoint of $BC$, find the length of $AD$. | {
"answer": "\\frac{\\sqrt{19}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the twelve letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "TEST"? Express your answer as a common fraction. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line $2x+3y-6=0$ intersects the $x$-axis and $y$-axis at points A and B, respectively. Point P is on the line $y=-x-1$. The minimum value of $|PA|+|PB|$ is ________. | {
"answer": "\\sqrt{37}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that \[\operatorname{lcm}(1024,2016)=\operatorname{lcm}(1024,2016,x_1,x_2,\ldots,x_n),\] with $x_1$ , $x_2$ , $\cdots$ , $x_n$ are distinct postive integers. Find the maximum value of $n$ .
*Proposed by Le Duc Minh* | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $({\frac{{x-1}}{x}-\frac{{x-2}}{{x+1}}})÷\frac{{2{x^2}-x}}{{{x^2}+2x+1}}$, where $x$ satisfies $x^{2}-2x-2=0$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A class has a group of 7 people, and now 3 of them are chosen to swap seats with each other, while the remaining 4 people's seats remain unchanged. Calculate the number of different rearrangement plans. | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( a > 0 \), if \( f(g(h(a))) = 17 \), where \( f(x) = x^2 + 5 \), \( g(x) = x^2 - 3 \), and \( h(x) = 2x + 1 \), what is the value of \( a \)? | {
"answer": "\\frac{-1 + \\sqrt{3 + 2\\sqrt{3}}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bicycle factory plans to produce a batch of bicycles of the same model, planning to produce $220$ bicycles per day. However, due to various reasons, the actual daily production will differ from the planned quantity. The table below shows the production situation of the workers in a certain week: (Exceeding $220$ bicycles is recorded as positive, falling short of $220$ bicycles is recorded as negative)
| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
|-----------------|--------|---------|-----------|----------|--------|----------|--------|
| Production Change (bicycles) | $+5$ | $-2$ | $-4$ | $+13$ | $-10$ | $+16$ | $-9$ |
$(1)$ According to the records, the total production in the first four days was ______ bicycles;<br/>
$(2)$ How many more bicycles were produced on the day with the highest production compared to the day with the lowest production?<br/>
$(3)$ The factory implements a piece-rate wage system, where each bicycle produced earns $100. For each additional bicycle produced beyond the daily planned production, an extra $20 is awarded, and for each bicycle less produced, $20 is deducted. What is the total wage of the workers for this week? | {
"answer": "155080",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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