problem stringlengths 10 5.15k | answer dict |
|---|---|
A hospital's internal medicine ward has 15 nurses, who work in pairs, rotating shifts every 8 hours. After two specific nurses work the same shift together, calculate the maximum number of days required for them to work the same shift again. | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f\left(x\right)=e^{x}-ax+\frac{1}{2}{x}^{2}$, where $a \gt -1$.<br/>$(Ⅰ)$ When $a=0$, find the equation of the tangent line to the curve $y=f\left(x\right)$ at the point $\left(0,f\left(0\right)\right)$;<br/>$(Ⅱ)$ When $a=1$, find the extreme values of the function $f\left(x\right)$;<br/>$(Ⅲ)$ If $f(x)≥\frac{1}{2}x^2+x+b$ holds for all $x\in R$, find the maximum value of $b-a$. | {
"answer": "1 + \\frac{1}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two types of anti-inflammatory drugs must be selected from $X_{1}$, $X_{2}$, $X_{3}$, $X_{4}$, $X_{5}$, with the restriction that $X_{1}$ and $X_{2}$ must be used together, and one type of antipyretic drug must be selected from $T_{1}$, $T_{2}$, $T_{3}$, $T_{4}$, with the further restriction that $X_{3}$ and $T_{4}$ cannot be used at the same time. Calculate the number of different test schemes. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with a focal length of $2$, and point $Q( \frac{a^{2}}{ \sqrt{a^{2}-b^{2}}},0)$ on the line $l$: $x=2$.
(1) Find the standard equation of the ellipse $C$;
(2) Let $O$ be the coordinate origin, $P$ a moving point on line $l$, and $l'$ a line passing through point $P$ that is tangent to the ellipse at point $A$. Find the minimum value of the area $S$ of $\triangle POA$. | {
"answer": "\\frac{ \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For point M on the ellipse $\frac {x^{2}}{9}+ \frac {y^{2}}{4}=1$, find the minimum distance from M to the line $x+2y-10=0$. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A trapezoid field is uniformly planted with wheat. The trapezoid has one pair of parallel sides measuring 80 m and 160 m, respectively, with the longer side on the bottom. The other two non-parallel sides each measure 120 m. The angle between a slanted side and the longer base is $45^\circ$. At harvest, the wheat at any point in the field is brought to the nearest point on the field's perimeter. What fraction of the crop is brought to the longest side? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distance \( AB \) is 100 km. Cyclists depart simultaneously from \( A \) and \( B \) towards each other with speeds of 20 km/h and 30 km/h respectively. Along with the first cyclist from \( A \), a fly departs with a speed of 50 km/h. The fly travels until it meets the cyclist from \( B \), then turns around and flies back to meet the cyclist from \( A \), and continues this pattern. How many kilometers will the fly travel in the direction from \( A \) to \( B \) before the cyclists meet? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given point $A(1,2)$ and circle $C: x^{2}+y^{2}+2mx+2y+2=0$.
$(1)$ If there are two tangents passing through point $A$, find the range of $m$.
$(2)$ When $m=-2$, a point $P$ on the line $2x-y+3=0$ is chosen to form two tangents $PM$ and $PN$ to the circle. Find the minimum area of quadrilateral $PMCN$. | {
"answer": "\\frac{7\\sqrt{15}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$. | {
"answer": "110.8333",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All vertices of a regular 2016-gon are initially white. What is the least number of them that can be painted black so that:
(a) There is no right triangle
(b) There is no acute triangle
having all vertices in the vertices of the 2016-gon that are still white? | {
"answer": "1008",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A moving point $A$ is on the circle $C$: $(x-1)^{2}+y^{2}=1$, and a moving point $B$ is on the line $l:x+y-4=0$. The coordinates of the fixed point $P$ are $P(-2,2)$. The minimum value of $|PB|+|AB|$ is ______. | {
"answer": "\\sqrt{37}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate plane $(xOy)$, a point $A(2,0)$, a moving point $B$ on the curve $y= \sqrt {1-x^{2}}$, and a point $C$ in the first quadrant form an isosceles right triangle $ABC$ with $\angle A=90^{\circ}$. The maximum length of the line segment $OC$ is _______. | {
"answer": "1+2 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area bounded by the graph of $y = \arccos(\cos x)$ and the $x$-axis on the interval $0 \leq x \leq 2\pi$. | {
"answer": "\\pi^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $\cos A= \frac {4}{5}$, $\cos B= -\frac { \sqrt {2}}{10}$.
$(1)$ Find $C$;
$(2)$ If $c=5$, find the area of $\triangle ABC$. | {
"answer": "\\frac {21}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangular pyramid P-ABC, where PA, PB, and PC are mutually perpendicular and PA=1, the center of the circumscribed sphere is O. Find the distance from O to plane ABC. | {
"answer": "\\frac{\\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $(\frac{{2x}^{2}+2x}{{x}^{2}-1}-\frac{{x}^{2}-x}{{x}^{2}-2x+1})÷\frac{x}{x+1}$, where $x=|\sqrt{3}-2|+(\frac{1}{2})^{-1}-(π-3.14)^0-\sqrt[3]{27}+1$. | {
"answer": "-\\frac{2\\sqrt{3}}{3} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $\lim_{n \to \infty} \frac{C_n^2}{n^2+1}$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In acute triangle $\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to angles $A$, $B$, $C$ respectively, and $4a\sin B = \sqrt{7}b$.
$(1)$ If $a = 6$ and $b+c = 8$, find the area of $\triangle ABC$.
$(2)$ Find the value of $\sin (2A+\frac{2\pi}{3})$. | {
"answer": "\\frac{\\sqrt{3}-3\\sqrt{7}}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ and $y$ are positive integers such that $xy - 2x + 5y = 111$, what is the minimal possible value of $|x - y|$? | {
"answer": "93",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Julia's garden has a 3:7 ratio of tulips to daisies. She currently has 35 daisies. She plans to add 30 more daisies and wants to plant additional tulips to maintain the original ratio. How many tulips will she have after this addition? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is randomly thrown onto the interval $[6, 10]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-3k-10\right)x^{2}+(3k-8)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$. | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin \theta$ and $\cos \theta$ are the two roots of the equation $4x^{2}-4mx+2m-1=0$, and $\frac {3\pi}{2} < \theta < 2\pi$, find the angle $\theta$. | {
"answer": "\\frac {5\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, triangle $\triangle ABC$ has vertices $A(-6, 0)$ and $C(6, 0)$. Vertex $B$ lies on the left branch of the hyperbola $\frac{x^2}{25} - \frac{y^2}{11} = 1$. Find the value of $\frac{\sin A - \sin C}{\sin B}$. | {
"answer": "\\frac{5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A barrel with a height of 1.5 meters is completely filled with water and sealed with a lid. The mass of the water in the barrel is 1000 kg. A long, thin vertical tube with a cross-section of $1 \mathrm{~cm}^{2}$ is inserted into the lid of the barrel and completely filled with water. Find the length of the tube if it is known that after filling, the pressure at the bottom of the barrel increased by 2 times. The density of water is 1000 kg/m³. | {
"answer": "1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the quadratic equation \( ax^2 + bx + c \) and the table of values \( 6300, 6481, 6664, 6851, 7040, 7231, 7424, 7619, 7816 \) for a sequence of equally spaced increasing values of \( x \), determine the function value that does not belong to the table. | {
"answer": "6851",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line $l$ does not pass through the origin $O$ and intersects an ellipse $\frac{x^{2}}{2}+y^{2}=1$ at points $A$ and $B$. $M$ is the midpoint of segment $AB$. Determine the product of the slopes of line $AB$ and line $OM$. | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the geometric sequence {a_n}, a_6 and a_{10} are the two roots of the equation x^2+6x+2=0. Determine the value of a_8. | {
"answer": "-\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two dice are rolled consecutively, and the numbers obtained are denoted as $a$ and $b$.
(Ⅰ) Find the probability that the point $(a, b)$ lies on the graph of the function $y=2^x$.
(Ⅱ) Using the values of $a$, $b$, and $4$ as the lengths of three line segments, find the probability that these three segments can form an isosceles triangle. | {
"answer": "\\frac{7}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\frac {π}{2}＜α＜ \frac {3π}{2}$, points A, B, and C are in the same plane rectangular coordinate system with coordinates A(3, 0), B(0, 3), and C(cosα, sinα) respectively.
(1) If $| \overrightarrow {AC}|=| \overrightarrow {BC}|$, find the value of angle α;
(2) When $\overrightarrow {AC}\cdot \overrightarrow {BC}=-1$, find the value of $\frac {2sin^{2}α+sin(2α)}{1+tan\alpha }$. | {
"answer": "- \\frac {5}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Take one point $M$ on the curve $y=\ln x$ and another point $N$ on the line $y=2x+6$, respectively. The minimum value of $|MN|$ is ______. | {
"answer": "\\dfrac {(7+\\ln 2) \\sqrt {5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the random variable $\xi$ follows a normal distribution $N(0,\sigma^{2})$, and $P(\xi > 2)=0.023$, determine $P(-2\leqslant \xi\leqslant 2)$. | {
"answer": "0.954",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seven thousand twenty-two can be expressed as the sum of a two-digit number and a four-digit number. | {
"answer": "7022",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $\cos S = 0.5$ in the diagram below. What is $ST$?
[asy]
pair P,S,T;
P = (0,0);
S = (6,0);
T = (0,6*tan(acos(0.5)));
draw(P--S--T--P);
draw(rightanglemark(S,P,T,18));
label("$P$",P,SW);
label("$S$",S,SE);
label("$T$",T,N);
label("$10$",S/2,S);
[/asy] | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \\(n\\) be a positive integer, and \\(f(n) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}\\). It is calculated that \\(f(2) = \frac{3}{2}\\), \\(f(4) > 2\\), \\(f(8) > \frac{5}{2}\\), and \\(f(16) > 3\\). Observing the results above, according to the pattern, it can be inferred that \\(f(128) > \_\_\_\_\_\_\_\_． | {
"answer": "\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $(xOy)$, a curve $C_{1}$ is defined by the parametric equations $x=\cos{\theta}$ and $y=\sin{\theta}$, and a line $l$ is defined by the polar equation $\rho(2\cos{\theta} - \sin{\theta}) = 6$.
1. Find the Cartesian equations for the curve $C_{1}$ and the line $l$.
2. Find a point $P$ on the curve $C_{1}$ such that the distance from $P$ to the line $l$ is minimized, and compute this minimum distance. | {
"answer": "\\frac{6\\sqrt{5}}{5} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the total number of stations is 6 and 3 of them are selected for getting off, calculate the probability that person A and person B get off at different stations. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many digits are there in the number \(N\) if \(N=2^{12} \times 5^{8}\) ?
If \(\left(2^{48}-1\right)\) is divisible by two whole numbers between 60 and 70, find them.
Given \(2^{\frac{1}{2}} \times 9^{\frac{1}{9}}\) and \(3^{\frac{1}{3}} \times 8^{\frac{1}{8}}\), what is the greatest number? | {
"answer": "3^{\\frac{1}{3}} \\times 8^{\\frac{1}{8}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can the number 210 be factored into a product of four natural numbers? The order of the factors does not matter. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40, m] = 120$ and $\mathop{\text{lcm}}[m, 45] = 180$, what is $m$? | {
"answer": "m = 36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system $xOy$, the equation of the hyperbola $C$ is $x^{2}-y^{2}=1$. Find all real numbers $a$ greater than 1 that satisfy the following requirement: Through the point $(a, 0)$, draw any two mutually perpendicular lines $l_{1}$ and $l_{2}$. If $l_{1}$ intersects the hyperbola $C$ at points $P$ and $Q$, and $l_{2}$ intersects $C$ at points $R$ and $S$, then $|PQ| = |RS|$ always holds. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
While hiking in a valley, a hiker first walks 15 miles north, then 8 miles east, then 9 miles south, and finally 2 miles east. How far is the hiker from the starting point after completing these movements? | {
"answer": "2\\sqrt{34}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the probability of having $2$ adjacent white balls or $2$ adjacent blue balls in a random arrangement of $3$ red, $2$ white and $2$ blue balls? | {
"answer": "$\\dfrac{10}{21}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F\_1$ is the left focus of the hyperbola $C$: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, point $B$ has coordinates $(0, b)$, and the line $F\_1B$ intersects with the two asymptotes of hyperbola $C$ at points $P$ and $Q$. If $\overrightarrow{QP} = 4\overrightarrow{PF\_1}$, find the eccentricity of the hyperbola $C$. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya has 9 different books by Arkady and Boris Strugatsky, each containing a single work by the authors. Vasya wants to arrange these books on a shelf in such a way that:
(a) The novels "Beetle in the Anthill" and "Waves Extinguish the Wind" are next to each other (in any order).
(b) The stories "Restlessness" and "A Story About Friendship and Non-friendship" are next to each other (in any order).
In how many ways can Vasya do this?
Choose the correct answer:
a) \(4 \cdot 7!\);
b) \(9!\);
c) \(\frac{9!}{4!}\);
d) \(4! \cdot 7!\);
e) another answer. | {
"answer": "4 \\cdot 7!",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A geometric progression with 10 terms starts with the first term as 2 and has a common ratio of 3. Calculate the sum of the new geometric progression formed by taking the reciprocal of each term in the original progression.
A) $\frac{29523}{59049}$
B) $\frac{29524}{59049}$
C) $\frac{29525}{59049}$
D) $\frac{29526}{59049}$ | {
"answer": "\\frac{29524}{59049}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xinjiang region has a dry climate and is one of the three major cotton-producing areas in China, producing high-quality long-staple cotton. In an experiment on the germination rate of a certain variety of long-staple cotton seeds, research institute staff selected experimental fields with basically the same conditions, sowed seeds simultaneously, and determined the germination rate, obtaining the following data:
| Number of<br/>cotton seeds| $100$ | $200$ | $500$ | $1000$ | $2000$ | $5000$ | $10000$ |
|---|---|---|---|---|---|---|---|
| Number of<br/>germinated seeds| $98$ | $192$ | $478$ | $953$ | $1902$ | $4758$ | $9507$ |
Then the germination rate of this variety of long-staple cotton seeds is approximately ______ (rounded to $0.01$). | {
"answer": "0.95",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a modified Ackermann function \( A(m, n) \) with the same recursive relationships as the original problem:
\[ A(m,n) = \left\{
\begin{aligned}
&n+1& \text{ if } m = 0 \\
&A(m-1, 1) & \text{ if } m > 0 \text{ and } n = 0 \\
&A(m-1, A(m, n-1))&\text{ if } m > 0 \text{ and } n > 0.
\end{aligned}
\right.\]
Compute \( A(3, 2) \). | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the minimum possible value of the sum
\[
\frac{a}{3b} + \frac{b}{6c} + \frac{c}{9a},
\]
where \( a, b, \) and \( c \) are positive real numbers. | {
"answer": "\\frac{1}{3\\sqrt[3]{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two lines $l_{1}$: $(m+1)x+2y+2m-2=0$ and $l_{2}$: $2x+(m-2)y+2=0$, if $l_{1} \parallel l_{2}$, then $m=$ ______. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles with centers \( M \) and \( N \), lying on the side \( AB \) of triangle \( ABC \), are tangent to each other and intersect the sides \( AC \) and \( BC \) at points \( A, P \) and \( B, Q \) respectively. Additionally, \( AM = PM = 2 \) and \( BN = QN = 5 \). Find the radius of the circumcircle of triangle \( ABC \), given that the ratio of the area of triangle \( AQn \) to the area of triangle \( MPB \) is \(\frac{15 \sqrt{2 + \sqrt{3}}}{5 \sqrt{3}}\). | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
find all $k$ distinct integers $a_1,a_2,...,a_k$ such that there exists an injective function $f$ from reals to themselves such that for each positive integer $n$ we have $$ \{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\} $$ . | {
"answer": "{0}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( ABCDEF \) be a hexagon such that the diagonals \( AD, BE, \) and \( CF \) intersect at the point \( O \), and the area of the triangle formed by any three adjacent points is 2 (for example, the area of \(\triangle BCD\) is 2). Find the area of the hexagon. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify and find the value of: $a^{2}-\left(a-\dfrac{2a}{a+1}\right)\div \dfrac{a^{2}-2a+1}{a^{2}-1}$, where $a$ is a solution of the equation $x^{2}-x-\dfrac{7}{2}=0$. | {
"answer": "\\dfrac{7}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The complex numbers corresponding to the vertices $O$, $A$, and $C$ of the parallelogram $OABC$ are $0$, $3+2i$, and $-2+4i$, respectively.<br/>$(1)$ Find the complex number corresponding to point $B$;<br/>$(2)$ In triangle $OAB$, find the height $h$ on side $OB$. | {
"answer": "\\frac{16\\sqrt{37}}{37}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the graph of the function $f(x)=3\sin(2x+\varphi)$ is symmetric about the point $\left(\frac{\pi}{3},0\right)$ $(|\varphi| < \frac{\pi}{2})$, determine the equation of one of the axes of symmetry of the graph of $f(x)$. | {
"answer": "\\frac{\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the least positive period $q$ of the functions $g$ such that $g(x+2) + g(x-2) = g(x)$ for all real $x$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sqrt{3}\sin x\cos x-\cos (\pi +2x)$.
(1) Find the interval(s) where $f(x)$ is monotonically increasing.
(2) In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $f(C)=1,c=\sqrt{3},a+b=2\sqrt{3}$, find the area of $\Delta ABC$. | {
"answer": "\\frac{3\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a=3 \sqrt {3}$, $c=2$, $B=150^{\circ}$, find the length of side $b$ and the area of $\triangle ABC$. | {
"answer": "\\frac{3 \\sqrt {3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among all triangles $ABC,$ find the maximum value of $\cos A + \cos B \cos C.$ | {
"answer": "\\frac{1}{\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers from 1 to 9 are placed at the vertices of a cube such that the sum of the four numbers on each face is the same. Find the common sum. | {
"answer": "22.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $C$ be a point on the parabola $y = x^2 - 4x + 7,$ and let $D$ be a point on the line $y = 3x - 5.$ Find the shortest distance $CD$ and also ensure that the projection of point $C$ over line $y = 3x - 5$ lands on point $D$. | {
"answer": "\\frac{0.25}{\\sqrt{10}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, if $a=3$, $b= \sqrt {3}$, $\angle A= \dfrac {\pi}{3}$, then the size of $\angle C$ is \_\_\_\_\_\_. | {
"answer": "\\dfrac {\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The probability that students A and B stand together among three students A, B, and C lined up in a row is what? | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polar equation of curve $C_1$ is $\rho^2=\frac {2}{3+\cos2\theta}$, establish a rectangular coordinate system with the pole O as the origin and the polar axis as the positive direction of the x-axis. After stretching all the x-coordinates of points on curve $C_1$ to twice their original values and shortening all the y-coordinates to half of their original values, we obtain curve $C_2$.
1. Write down the rectangular coordinate equation of curve $C_1$.
2. Take any point R on curve $C_2$ and find the maximum distance from point R to the line $l: x + y - 5 = 0$. | {
"answer": "\\frac{13\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are ten digits: $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$.
$(1)$ How many unique three-digit numbers can be formed without repetition?
$(2)$ How many unique four-digit even numbers can be formed without repetition? | {
"answer": "2296",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A coordinate system and parametric equations problem (4-4):
In the rectangular coordinate system $xoy$, a curve $C_1$ is defined by the parametric equations $\begin{cases} x = 1 + \cos \alpha \\ y = \sin^2 \alpha - \frac{9}{4} \end{cases}$, where $\alpha$ is the parameter and $\alpha \in \mathbb{R}$. In the polar coordinate system with the origin $O$ as the pole and the nonnegative $x$-axis as the polar axis (using the same length units), there are two other curves: $C_2: \rho \sin \left( \theta + \frac{\pi}{4} \right) = -\frac{\sqrt{2}}{2}$ and $C_3: \rho = 2 \cos \theta$.
(I) Find the rectangular coordinates of the intersection point $M$ of the curves $C_1$ and $C_2$.
(II) Let $A$ and $B$ be moving points on the curves $C_2$ and $C_3$, respectively. Find the minimum value of the distance $|AB|$. | {
"answer": "\\sqrt{2} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression $1 - \frac{1}{1 + \sqrt{5}} + \frac{1}{1 - \sqrt{5}}$. | {
"answer": "1 - \\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a function $f(x)$ that satisfies $f(x+3)=-f(x)$, when $x\in \left[-3,0\right)$, $f(x)=2^{x}+\sin \frac{πx}{3}$, determine the value of $f(2023)$. | {
"answer": "-\\frac{1}{4} + \\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the integer $n,$ $-180 < n < 180,$ such that $\tan n^\circ = \tan 345^\circ.$ | {
"answer": "-15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
2 diagonals of a regular decagon (a 10-sided polygon) are chosen. What is the probability that their intersection lies inside the decagon? | {
"answer": "\\dfrac{42}{119}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $n$ such that $2^6 \cdot 3^3 \cdot n = 10!$. | {
"answer": "350",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=- \sqrt {3}\sin ^{2}x+\sin x\cos x$.
(1) Find the value of $f( \frac {25π}{6})$.
(2) Find the smallest positive period of the function $f(x)$ and its maximum and minimum values in the interval $[0, \frac {π}{2}]$. | {
"answer": "-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area, in square units, of a triangle that has sides of $5, 3,$ and $3$ units? Express your answer in simplest radical form. | {
"answer": "\\frac{5\\sqrt{11}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of $$ \int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx $$ over all continuously differentiable functions $f:[0,1]\to\mathbb R$ with $f(0)=0$ and $$ \int^1_0|f'(x)|^2dx\le1. $$ | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Thirty clever students from 6th, 7th, 8th, 9th, and 10th grades were tasked with creating forty problems for an olympiad. Any two students from the same grade came up with the same number of problems, while any two students from different grades came up with a different number of problems. How many students came up with one problem each? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line $y = 2$ intersects the graph of $y = 3x^2 + 2x - 5$ at the points $C$ and $D$. Determine the distance between $C$ and $D$ and express this distance in the form $\frac{\sqrt{p}}{q}$, where $p$ and $q$ are coprime positive integers. | {
"answer": "\\frac{2\\sqrt{22}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola C: y^2 = 4x, point B(3,0), and the focus F, point A lies on C. If |AF| = |BF|, calculate the length of |AB|. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that the variance of each of the given independent random variables does not exceed 4. Determine the number of such variables for which the probability that the deviation of the arithmetic mean of the random variable from the arithmetic mean of their mathematical expectations by no more than 0.25 exceeds 0.99. | {
"answer": "6400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A checker can move in one direction on a divided strip into cells, moving either to the adjacent cell or skipping one cell in one move. In how many ways can it move 10 cells? 11 cells? | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An engineer invested $\$15,\!000$ in a nine-month savings certificate that paid a simple annual interest rate of $9\%$. After nine months, she invested the total value of her investment in another nine-month certificate. After another nine months, the investment was worth $\$17,\!218.50$. If the annual interest rate of the second certificate is $s\%,$ what is $s$? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_n\}$ that satisfies $a_1=33$ and $a_{n+1}-a_n=2n$, find the minimum value of $\frac {a_{n}}{n}$. | {
"answer": "\\frac {21}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is known that $\overrightarrow{a}=(\cos A,\cos B)$, $\overrightarrow{b}=(a,2c-b)$, and $\overrightarrow{a} \parallel \overrightarrow{b}$.
(Ⅰ) Find the magnitude of angle $A$;
(Ⅱ) If $b=3$ and the area of $\triangle ABC$, $S_{\triangle ABC}=3 \sqrt {3}$, find the value of $a$. | {
"answer": "\\sqrt {13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | {
"answer": "12\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When the municipal government investigated the relationship between changes in citizens' income and tourism demand, a sample of 5000 people was randomly selected using the independence test method. The calculation showed that $K^{2}=6.109$. Based on this data, the municipal government asserts that the credibility of the relationship between changes in citizens' income and tourism demand is ______ $\%.$
Attached: Common small probability values and critical values table:
| $P(K^{2}\geqslant k_{0})$ | $0.15$ | $0.10$ | $0.05$ | $0.025$ | $0.010$ | $0.001$ |
|---------------------------|--------|--------|--------|---------|---------|---------|
| $k_{0}$ | $2.072$| $2.706$| $3.841$| $5.024$ | $6.635$ | $10.828$| | {
"answer": "97.5\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, establish a polar coordinate system with the origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis, using the same unit of length in both coordinate systems. Given that circle $C$ has a center at point ($2$, $\frac{7π}{6}$) in the polar coordinate system and a radius of $\sqrt{3}$, and line $l$ has parametric equations $\begin{cases} x=- \frac{1}{2}t \\ y=-2+ \frac{\sqrt{3}}{2}t\end{cases}$.
(1) Find the Cartesian coordinate equations for circle $C$ and line $l$.
(2) If line $l$ intersects circle $C$ at points $M$ and $N$, find the area of triangle $MON$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | {
"answer": "12\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. It is given that $\frac{b}{c} = \frac{2\sqrt{3}}{3}$ and $A + 3C = \pi$.
$(1)$ Find the value of $\cos C$;
$(2)$ Find the value of $\sin B$;
$(3)$ If $b = 3\sqrt{3}$, find the area of $\triangle ABC$. | {
"answer": "\\frac{9\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of six-letter words where the first and last two letters are the same (e.g., "aabbaa"). | {
"answer": "456976",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, the circle is inscribed in the square. This means that the circle and the square share points \(S, T, U,\) and \(V\), and the width of the square is exactly equal to the diameter of the circle. Rounded to the nearest tenth, what percentage of line segment \(XY\) is outside the circle? | {
"answer": "29.3",
"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{119}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $f\left(x\right)=\ln |a+\frac{1}{{1-x}}|+b$ is an odd function, then $a=$____, $b=$____. | {
"answer": "\\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Nanjing Youth Olympic Games are about to open, and a clothing store owner, Mr. Chen, spent 3600 yuan to purchase two types of sportswear, A and B, and sold them out quickly. When Mr. Chen went to purchase the same types and quantities of clothing again, he found that the purchase prices of types A and B had increased by 20 yuan/piece and 5 yuan/piece, respectively, resulting in an additional expenditure of 400 yuan. Let the number of type A clothing purchased by Mr. Chen each time be $x$ pieces, and the number of type B clothing be $y$ pieces.
(1) Please write down the function relationship between $y$ and $x$ directly: .
(2) After calculating, Mr. Chen found that the average unit price of types A and B clothing had increased by 8 yuan during the second purchase compared to the first.
① Find the values of $x$ and $y$.
② After selling all the clothing purchased for the second time at a 35% profit, Mr. Chen took all the sales proceeds to purchase more goods. At this time, the prices of both types of clothing had returned to their original prices, so Mr. Chen spent 3000 yuan to purchase type B clothing, and the rest of the money was used to purchase type A clothing. As a result, the quantities of types A and B clothing purchased were exactly equal. How many pieces of clothing did Mr. Chen purchase in total this time? | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the three internal angles are $A$, $B$, and $C$. If $\dfrac{\sqrt{3}\cos A + \sin A}{\sqrt{3}\sin A - \cos A} = \tan(-\dfrac{7}{12}\pi)$, find the maximum value of $2\cos B + \sin 2C$. | {
"answer": "\\dfrac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( n \geq 2 \) be a fixed integer. Find the least constant \( C \) such that the inequality
\[ \sum_{i<j} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq C\left(\sum_{i} x_{i}\right)^{4} \]
holds for every \( x_{1}, \ldots, x_{n} \geq 0 \) (the sum on the left consists of \(\binom{n}{2}\) summands). For this constant \( \bar{C} \), characterize the instances of equality. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum area of the part bounded by the parabola $ y\equal{}a^3x^2\minus{}a^4x\ (a>0)$ and the line $ y\equal{}x$ . | {
"answer": "\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line passing through the points (3,9) and (-1,1) has an x-intercept of ( ). | {
"answer": "-\\frac{3}{2}",
"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 $a\sin 2B= \sqrt{3}b\sin A$.
1. Find $B$;
2. If $\cos A= \dfrac{1}{3}$, find the value of $\sin C$. | {
"answer": "\\dfrac{2\\sqrt{6}+1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an equilateral triangle $ABC$ with side length $6$, point $D$ is the midpoint of $BC$. Calculate $\tan{\angle BAD}$. | {
"answer": "\\frac{1}{\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Amy, Beth, and Claire each have some sweets. Amy gives one third of her sweets to Beth. Beth gives one third of all the sweets she now has to Claire. Then Claire gives one third of all the sweets she now has to Amy. All the girls end up having the same number of sweets.
Claire begins with 40 sweets. How many sweets does Beth have originally? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Given $\tan(\alpha+\beta)= \frac{2}{5}$ and $\tan\left(\beta- \frac{\pi}{4}\right)= \frac{1}{4}$, find the value of $\frac{\cos\alpha+\sin\alpha}{\cos\alpha-\sin\alpha}$;
(2) Given $\alpha$ and $\beta$ are acute angles, and $\cos(\alpha+\beta)= \frac{\sqrt{5}}{5}$, $\sin(\alpha-\beta)= \frac{\sqrt{10}}{10}$, find $2\beta$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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