problem stringlengths 10 5.15k | answer dict |
|---|---|
Given an ellipse $C$ with its center at the origin and its foci on the $x$-axis, and its eccentricity equal to $\frac{1}{2}$. One of its vertices is exactly the focus of the parabola $x^{2}=8\sqrt{3}y$.
(Ⅰ) Find the standard equation of the ellipse $C$;
(Ⅱ) If the line $x=-2$ intersects the ellipse at points $P$ and $Q$, and $A$, $B$ are points on the ellipse located on either side of the line $x=-2$.
(i) If the slope of line $AB$ is $\frac{1}{2}$, find the maximum area of the quadrilateral $APBQ$;
(ii) When the points $A$, $B$ satisfy $\angle APQ = \angle BPQ$, does the slope of line $AB$ have a fixed value? Please explain your reasoning. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $y=kx+b\left(k \gt 0\right)$ is tangent to the circle $x^{2}+y^{2}=1$ and the circle $\left(x-4\right)^{2}+y^{2}=1$, find $k=$____ and $b=$____. | {
"answer": "-\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is chosen at random on the number line between 0 and 1, and the point is colored red. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored blue. What is the probability that the number of the blue point is greater than the number of the red point, but less than three times the number of the red point? | {
"answer": "\\frac{1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Juan rolls a fair regular octahedral die marked with the numbers 1 through 8. Amal now rolls a fair twelve-sided die marked with the numbers 1 through 12. What is the probability that the product of the two rolls is a multiple of 4? | {
"answer": "\\frac{7}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Sichuan, the new college entrance examination was launched in 2022, and the first new college entrance examination will be implemented in 2025. The new college entrance examination adopts the "$3+1+2$" mode. "$3$" refers to the three national unified examination subjects of Chinese, Mathematics, and Foreign Languages, regardless of whether they are arts or sciences; "$1$" refers to choosing one subject from Physics and History; "$2$" refers to choosing two subjects from Political Science, Geography, Chemistry, and Biology. The subject selection situation of the first-year high school students in a certain school in 2022 is shown in the table below:
| Subject Combination | Physics, Chemistry, Biology | Physics, Chemistry, Political Science | Physics, Chemistry, Geography | History, Political Science, Geography | History, Political Science, Biology | History, Chemistry, Political Science | Total |
|---------------------|-----------------------------|-------------------------------------|-----------------------------|--------------------------------------|----------------------------------|--------------------------------------|-------|
| Male | 180 | 80 | 40 | 90 | 30 | 20 | 440 |
| Female | 150 | 70 | 60 | 120 | 40 | 20 | 460 |
| Total | 330 | 150 | 100 | 210 | 70 | 40 | 900 |
$(1)$ Complete the $2\times 2$ contingency table below and determine if there is a $99\%$ certainty that "choosing Physics is related to the gender of the student":
| | Choose Physics | Do not choose Physics | Total |
|-------------------|----------------|-----------------------|-------|
| Male | | | |
| Female | | | |
| Total | | | |
$(2)$ From the female students who chose the combinations of History, Political Science, Biology and History, Chemistry, Political Science, select 6 students using stratified sampling to participate in a history knowledge competition. Find the probability that the 2 selected female students are from the same combination.
Given table and formula: ${K^2}=\frac{{n{{(ad-bc)}^2}}}{{(a+b)(c+d)(a+c)(b+d)}}$
| $P(K^{2}\geqslant k_{0})$ | 0.15 | 0.1 | 0.05 | 0.01 |
|---------------------------|------|------|------|------|
| $k_{0}$ | 2.072| 2.706| 3.841| 6.635| | {
"answer": "\\frac{7}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose the estimated €25 billion (Euros) cost to send a person to the planet Mars is shared equally by the 300 million people in a consortium of countries. Given the exchange rate of 1 Euro = 1.2 dollars, calculate each person's share in dollars. | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Mike walks to his college, averaging 70 steps per minute with each step being 80 cm long, and it takes him 20 minutes to get there, determine how long it takes Tom to reach the college, given that he averages 120 steps per minute, but his steps are only 50 cm long. | {
"answer": "18.67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a \) and \( b \) be positive integers such that \( 15a + 16b \) and \( 16a - 15b \) are both perfect squares. Find the smallest possible value among these squares. | {
"answer": "481^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $$f(x)=2\sin(wx+\varphi+ \frac {\pi}{3})+1$$ where $|\varphi|< \frac {\pi}{2}$ and $w>0$, is an even function, and the distance between two adjacent axes of symmetry of the function $f(x)$ is $$\frac {\pi}{2}$$.
(1) Find the value of $$f( \frac {\pi}{8})$$.
(2) When $x\in(-\frac {\pi}{2}, \frac {3\pi}{2})$, find the sum of the real roots of the equation $f(x)= \frac {5}{4}$. | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The positive integer divisors of 252, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 14? | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram \(PQRS\) is a rhombus. Point \(T\) is the midpoint of \(PS\) and point \(W\) is the midpoint of \(SR\). What is the ratio of the unshaded area to the shaded area? | {
"answer": "1:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the inequality $\ln (x+1)-(a+2)x\leqslant b-2$ that always holds, find the minimum value of $\frac {b-3}{a+2}$. | {
"answer": "1-e",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Select 5 elements from the set $\{x|1\leq x \leq 11, \text{ and } x \in \mathbb{N}^*\}$ to form a subset of this set, and any two elements in this subset do not sum up to 12. How many different subsets like this are there? (Answer with a number). | {
"answer": "112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Jennifer plans to build a fence around her garden in the shape of a rectangle, with $24$ fence posts, and evenly distributing the remaining along the edges, with $6$ yards between each post, and with the longer side of the garden, including corners, having three times as many posts as the shorter side, calculate the area, in square yards, of Jennifer’s garden. | {
"answer": "855",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area, in square units, of a trapezoid bounded by the lines $y = x$, $y = 15$, $y = 5$ and the line $x = 5$? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The least common multiple of $x$ and $y$ is $18$, and the least common multiple of $y$ and $z$ is $20$. Determine the least possible value of the least common multiple of $x$ and $z$. | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $C$: $y^2=2px (p > 0)$ with focus $F$ and directrix $l$. A line perpendicular to $l$ at point $A$ on the parabola $C$ at $A(4,y_0)$ intersects $l$ at $A_1$. If $\angle A_1AF=\frac{2\pi}{3}$, determine the value of $p$. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seventy percent of a train's passengers are men, and fifteen percent of those men are in the business class. What is the number of men in the business class if the train is carrying 300 passengers? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the spring college entrance examination of Shanghai in 2011, there were 8 universities enrolling students. If exactly 3 students were admitted by 2 of these universities, the number of ways this could happen is ____. | {
"answer": "168",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $ABC$, $\overrightarrow{CA}•\overrightarrow{CB}=1$, the area of the triangle is $S=\frac{1}{2}$,<br/>$(1)$ Find the value of angle $C$;<br/>$(2)$ If $\sin A\cos A=\frac{{\sqrt{3}}}{4}$, $a=2$, find $c$. | {
"answer": "\\frac{2\\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A set of sample data $11$, $13$, $15$, $a$, $19$ has an average of $15$. Calculate the standard deviation of this data set. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\pi$. | {
"answer": "\\frac{9\\pi}{2} - 9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, and it satisfies $S_{2014} > 0$, $S_{2015} < 0$. For any positive integer $n$, it holds that $|a_n| \geqslant |a_k|$, determine the value of $k$. | {
"answer": "1008",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( PQ = 4 \), \( QR = 8 \), \( RS = 8 \), and \( ST = 3 \), if \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \), calculate the distance from \( P \) to \( T \). | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a, b, c, d \) be integers such that \( a > b > c > d \geq -2021 \) and
\[ \frac{a+b}{b+c} = \frac{c+d}{d+a} \]
(and \( b+c \neq 0 \neq d+a \)). What is the maximum possible value of \( a \cdot c \)? | {
"answer": "510050",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the squares of four consecutive positive integers is 9340. What is the sum of the cubes of these four integers? | {
"answer": "457064",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of the base of a hemisphere is $144\pi$. The hemisphere is mounted on top of a cylinder that has the same radius as the hemisphere and a height of 10. What is the total surface area of the combined solid? Express your answer in terms of $\pi$. | {
"answer": "672\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $y^2 = 2px (0 < p < 4)$, with a focus at point $F$, and a point $P$ moving along $C$. Let $A(4,0)$ and $B(p, \sqrt{2}p)$ with the minimum value of $|PA|$ being $\sqrt{15}$, find the value of $|BF|$. | {
"answer": "\\dfrac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function f(x) = 2cos^2(x) - 2$\sqrt{3}$sin(x)cos(x).
(I) Find the monotonically decreasing interval of the function f(x);
(II) Find the sum of all the real roots of the equation f(x) = $- \frac{1}{3}$ in the interval [0, $\frac{\pi}{2}$]. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What are the first three digits to the right of the decimal point in the decimal representation of $(10^{100} + 1)^{5/3}$? | {
"answer": "666",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $40\%$ of the birds were pigeons, $20\%$ were sparrows, $15\%$ were crows, and the remaining were parakeets, calculate the percent of the birds that were not sparrows and were crows. | {
"answer": "18.75\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and $P$ is a point on the ellipse, with $\overrightarrow{PF_{1}} \cdot (\overrightarrow{OF_{1}} + \overrightarrow{OP}) = 0$, if $|\overrightarrow{PF_{1}}| = \sqrt{2}|\overrightarrow{PF_{2}}|$, determine the eccentricity of the ellipse. | {
"answer": "\\sqrt{6} - \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a square \(PQRS\) with edges of length 1, and four arcs, each of which is a quarter of a circle. Arc \(TRU\) has center \(P\); arc \(VPW\) has center \(R\); arc \(UV\) has center \(S\); and arc \(WT\) has center \(Q\). What is the length of the perimeter of the shaded region?
A) 6
B) \((2 \sqrt{2} - 1) \pi\)
C) \(\left(\sqrt{2} - \frac{1}{2}\right) \pi\)
D) 2 \(\pi\)
E) \((3 \sqrt{2} - 2) \pi\) | {
"answer": "(2\\sqrt{2} - 1)\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two distinct numbers a and b are chosen randomly from the set $\{3, 3^2, 3^3, ..., 3^{20}\}$. What is the probability that $\mathrm{log}_a b$ is an integer?
A) $\frac{12}{19}$
B) $\frac{1}{4}$
C) $\frac{24}{95}$
D) $\frac{1}{5}$ | {
"answer": "\\frac{24}{95}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCDEF$ be a regular hexagon, and let $G$ , $H$ , $I$ , $J$ , $K$ , and $L$ be the midpoints of sides $AB$ , $BC$ , $CD$ , $DE$ , $EF$ , and $FA$ , respectively. The intersection of lines $\overline{AH}$ , $\overline{BI}$ , $\overline{CJ}$ , $\overline{DK}$ , $\overline{EL}$ , and $\overline{FG}$ bound a smaller regular hexagon. Find the ratio of the area of the smaller hexagon to the area of $ABCDEF$ . | {
"answer": "4/7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Aces are playing the Kings in a playoff series, where the first team to win 5 games wins the series. Each game's outcome leads to the Aces winning with a probability of $\dfrac{7}{10}$, and there are no draws. Calculate the probability that the Aces win the series. | {
"answer": "90\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a circle, points $A, B, C, D, E, F, G$ are located clockwise as shown in the diagram. It is known that $AE$ is the diameter of the circle. Additionally, it is known that $\angle ABF = 81^\circ$ and $\angle EDG = 76^\circ$. How many degrees is the angle $FCG$? | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a function $f(x)$ with domain $I$, if there exists an interval $\left[m,n\right]\subseteq I$ such that $f(x)$ is a monotonic function on the interval $\left[m,n\right]$, and the range of the function $y=f(x)$ for $x\in \left[m,n\right]$ is $\left[m,n\right]$, then the interval $\left[m,n\right]$ is called a "beautiful interval" of the function $f(x)$;
$(1)$ Determine whether the functions $y=x^{2}$ ($x\in R$) and $y=3-\frac{4}{x}$ ($x \gt 0$) have a "beautiful interval". If they exist, write down one "beautiful interval" that satisfies the condition. (Provide the conclusion directly without requiring a proof)
$(2)$ If $\left[m,n\right]$ is a "beautiful interval" of the function $f(x)=\frac{{({{a^2}+a})x-1}}{{{a^2}x}}$ ($a\neq 0$), find the maximum value of $n-m$. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three people, A, B, and C, visit three tourist spots, with each person visiting only one spot. Let event $A$ be "the three people visit different spots," and event $B$ be "person A visits a spot alone." Then, the probability $P(A|B)=$ ______. | {
"answer": "\\dfrac{1}{2}",
"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. If $a\cos B - b\cos A = c$, and $C = \frac{π}{5}$, calculate the measure of $\angle B$. | {
"answer": "\\frac{3\\pi}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An object in the plane moves from the origin and takes a ten-step path, where at each step the object may move one unit to the right, one unit to the left, one unit up, or one unit down. How many different points could be the final point? | {
"answer": "221",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sarah baked 4 dozen pies for a community fair. Out of these pies:
- One-third contained chocolate,
- One-half contained marshmallows,
- Three-fourths contained cayenne pepper,
- One-eighth contained walnuts.
What is the largest possible number of pies that had none of these ingredients? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | {
"answer": "2028",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line connects points $(2,1)$ and $(7,3)$ on a square that has vertices at $(2,1)$, $(7,1)$, $(7,6)$, and $(2,6)$. What fraction of the area of the square is above this line? | {
"answer": "\\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $1-2+3-4+\dots+100-101$. | {
"answer": "-151",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the curve $C:\begin{cases}x=2\cos a \\ y= \sqrt{3}\sin a\end{cases} (a$ is the parameter) and the fixed point $A(0,\sqrt{3})$, ${F}_1,{F}_2$ are the left and right foci of this curve, respectively. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established.
$(1)$ Find the polar equation of the line $AF_{2}$;
$(2)$ A line passing through point ${F}_1$ and perpendicular to the line $AF_{2}$ intersects this conic curve at points $M$, $N$, find the value of $||MF_{1}|-|NF_{1}||$. | {
"answer": "\\dfrac{12\\sqrt{3}}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $m$ for which the quadratic equation $3x^2 + mx + 36 = 0$ has exactly one solution in $x$. | {
"answer": "12\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given angles $α$ and $β$ whose vertices are at the origin of coordinates, and their initial sides coincide with the positive half-axis of $x$, $α$, $β$ $\in(0,\pi)$, the terminal side of angle $β$ intersects the unit circle at a point whose x-coordinate is $- \dfrac{5}{13}$, and the terminal side of angle $α+β$ intersects the unit circle at a point whose y-coordinate is $ \dfrac{3}{5}$, then $\cos α=$ ______. | {
"answer": "\\dfrac{56}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two teachers, A and B, and four students stand in a row. (Explain the process, list the expressions, and calculate the results, expressing the results in numbers)<br/>$(1)$ The two teachers cannot be adjacent. How many ways are there to arrange them?<br/>$(2)$ A is to the left of B. How many ways are there to arrange them?<br/>$(3)$ A must be at the far left or B must be at the far left, and A cannot be at the far right. How many ways are there to arrange them?<br/>$(4)$ The two teachers are in the middle, with two students at each end. If the students are of different heights and must be arranged from tallest to shortest from the middle to the ends, how many ways are there to arrange them? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$ . How many passcodes satisfy these conditions? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider numbers of the form \(10n + 1\), where \(n\) is a positive integer. We shall call such a number 'grime' if it cannot be expressed as the product of two smaller numbers, possibly equal, both of which are of the form \(10k + 1\), where \(k\) is a positive integer. How many 'grime numbers' are there in the sequence \(11, 21, 31, 41, \ldots, 981, 991\)?
A) 0
B) 8
C) 87
D) 92
E) 99 | {
"answer": "87",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C:\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\left( a > b > 0 \right)$ with left and right focal points ${F}_{1},{F}_{2}$, and a point $P\left( 1,\frac{\sqrt{2}}{2} \right)$ on the ellipse such that $\vert P{F}_{1}\vert+\vert P{F}_{2}\vert=2 \sqrt{2}$.
(1) Find the standard equation of the ellipse $C$.
(2) A line $l$ passes through ${F}_{2}$ and intersects the ellipse at two points $A$ and $B$. Find the maximum area of $\triangle AOB$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, where $AB = 50$, $BC = 36$, and $AC = 42$. A line $CX$ from $C$ is perpendicular to $AB$ and intersects $AB$ at point $X$. Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$. Express your answer as a simplified common fraction. | {
"answer": "\\frac{6}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = 3x^4 + 6$, find the value of $f^{-1}(150)$. | {
"answer": "\\sqrt[4]{48}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, given that $A=60^{\circ}$ and $BC=4$, the diameter of the circumcircle of $\triangle ABC$ is ____. | {
"answer": "\\frac{8\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The minimum value of the function $y = \sin 2 \cos 2x$ is ______. | {
"answer": "- \\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the integer \( n \), \( -180 \le n \le 180 \), such that \( \cos n^\circ = \cos 745^\circ \). | {
"answer": "-25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a ball with a diameter of 6 inches rolling along the path consisting of four semicircular arcs, with radii $R_1 = 100$ inches, $R_2 = 60$ inches, $R_3 = 80$ inches, and $R_4 = 40$ inches, calculate the distance traveled by the center of the ball from the start to the end of the track. | {
"answer": "280\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive integer is *bold* iff it has $8$ positive divisors that sum up to $3240$ . For example, $2006$ is bold because its $8$ positive divisors, $1$ , $2$ , $17$ , $34$ , $59$ , $118$ , $1003$ and $2006$ , sum up to $3240$ . Find the smallest positive bold number. | {
"answer": "1614",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f(x) = |x-1| + |x-3| + \mathrm{e}^x \) (where \( x \in \mathbf{R} \)), find the minimum value of the function. | {
"answer": "6-2\\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) If the terminal side of angle $\theta$ passes through $P(-4t, 3t)$ ($t>0$), find the value of $2\sin\theta + \cos\theta$.
(2) Given that a point $P$ on the terminal side of angle $\alpha$ has coordinates $(x, -\sqrt{3})$ ($x\neq 0$), and $\cos\alpha = \frac{\sqrt{2}}{4}x$, find $\sin\alpha$ and $\tan\alpha$. | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins or indefinitely. If Nathaniel goes first, determine the probability that he ends up winning. | {
"answer": "5/11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Set \( S \) satisfies the following conditions:
1. The elements of \( S \) are positive integers not exceeding 100.
2. For any \( a, b \in S \) where \( a \neq b \), there exists \( c \in S \) different from \( a \) and \( b \) such that \(\gcd(a + b, c) = 1\).
3. For any \( a, b \in S \) where \( a \neq b \), there exists \( c \in S \) different from \( a \) and \( b \) such that \(\gcd(a + b, c) > 1\).
Determine the maximum value of \( |S| \). | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ is an intersection point of the ellipse $\frac{x^{2}}{a_{1}^{2}} + \frac{y^{2}}{b_{1}^{2}} = 1 (a_{1} > b_{1} > 0)$ and the hyperbola $\frac{x^{2}}{a_{2}^{2}} - \frac{y^{2}}{b_{2}^{2}} = 1 (a_{2} > 0, b_{2} > 0)$, $F_{1}$, $F_{2}$ are the common foci of the ellipse and hyperbola, $e_{1}$, $e_{2}$ are the eccentricities of the ellipse and hyperbola respectively, and $\angle F_{1}PF_{2} = \frac{2\pi}{3}$, find the maximum value of $\frac{1}{e_{1}} + \frac{1}{e_{2}}$. | {
"answer": "\\frac{4 \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Complex numbers $p, q, r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48,$ find $|pq + pr + qr|.$ | {
"answer": "768",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A Martian traffic light consists of six identical bulbs arranged in two horizontal rows (one below the other) with three bulbs in each row. A rover driver in foggy conditions can distinguish the number and relative positions of the lit bulbs on the traffic light (for example, if two bulbs are lit, whether they are in the same horizontal row or different ones, whether they are in the same vertical column, adjacent vertical columns, or in the two outermost vertical columns). However, the driver cannot distinguish unlit bulbs and the traffic light's frame. Therefore, if only one bulb is lit, it is impossible to determine which of the six bulbs it is. How many different signals from the Martian traffic light can the rover driver distinguish in the fog? If none of the bulbs are lit, the driver cannot see the traffic light. | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α∈( \dfrac {π}{2},π)$, and $\sin \dfrac {α}{2}+\cos \dfrac {α}{2}= \dfrac {2 \sqrt {3}}{3}$.
(1) Find the values of $\sin α$ and $\cos α$;
(2) If $\sin (α+β)=- \dfrac {3}{5},β∈(0, \dfrac {π}{2})$, find the value of $\sin β$. | {
"answer": "\\dfrac {6 \\sqrt {2}+4}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle has length $AC = 40$ and width $AE = 24$. Point $B$ is positioned one-third of the way along $AC$ from $A$ to $C$, and point $F$ is halfway along $AE$. Find the total area enclosed by rectangle $ACDE$ and the semicircle with diameter $AC$ minus the area of quadrilateral $ABDF$. | {
"answer": "800 + 200\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Crystal Products Factory in the East Sea had an annual output of 100,000 pieces last year, with each crystal product selling for 100 yuan and a fixed cost of 80 yuan. Starting this year, the factory invested 1 million yuan in technology costs and plans to invest an additional 1 million yuan in technology costs each year thereafter. It is expected that the output will increase by 10,000 pieces each year, and the relationship between the fixed cost $g(n)$ of each crystal product and the number of times $n$ technology costs are invested is $g(n) = \frac{80}{\sqrt{n+1}}$. If the selling price of the crystal products remains unchanged, the annual profit after the $n$th investment is $f(n)$ million yuan.
(1) Find the expression for $f(n)$;
(2) Starting from this year, in which year will the profit be the highest? What is the highest profit in million yuan? | {
"answer": "520",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $0 < x < \frac{1}{2}$, find the minimum and maximum value of the function $x^{2}(1-2x)$. | {
"answer": "\\frac{1}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the center of the ellipse C is at the origin, its eccentricity is equal to $\frac{1}{2}$, and one of its minor axis endpoints is the focus of the parabola x²=$4\sqrt{3}y$.
1. Find the standard equation of the ellipse C.
2. The left and right foci of the ellipse are $F_1$ and $F_2$, respectively. A line $l$ passes through $F_2$ and intersects the ellipse at two distinct points A and B. Determine whether the area of the inscribed circle of $\triangle{F_1AB}$ has a maximum value. If it exists, find this maximum value and the equation of the line at this time; if not, explain the reason. | {
"answer": "\\frac{9}{16}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rhombus \(ABCD\), the angle \(\angle ABC = 60^{\circ}\). A circle is tangent to the line \(AD\) at point \(A\), and the center of the circle lies inside the rhombus. Tangents to the circle, drawn from point \(C\), are perpendicular. Find the ratio of the perimeter of the rhombus to the circumference of the circle. | {
"answer": "\\frac{\\sqrt{3} + \\sqrt{7}}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the common ratio of the infinite geometric series: $$\frac{-4}{7}+\frac{14}{3}+\frac{-98}{9} + \dots$$ | {
"answer": "-\\frac{49}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle \(ABCD\), \(A B: AD = 1: 2\). Point \(M\) is the midpoint of \(AB\), and point \(K\) lies on \(AD\), dividing it in the ratio \(3:1\) starting from point \(A\). Find the sum of \(\angle CAD\) and \(\angle AKM\). | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parabola has focus $F$ and vertex $V$ , where $VF = 1$ 0. Let $AB$ be a chord of length $100$ that passes through $F$ . Determine the area of $\vartriangle VAB$ .
| {
"answer": "100\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any positive integer $n$, let
\[f(n) =\left\{\begin{matrix}\log_{4}{n}, &\text{if }\log_{4}{n}\text{ is rational,}\\ 0, &\text{otherwise.}\end{matrix}\right.\]
What is $\sum_{n = 1}^{1023}{f(n)}$?
A) $\frac{40}{2}$
B) $\frac{45}{2}$
C) $\frac{21}{2}$
D) $\frac{36}{2}$ | {
"answer": "\\frac{45}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the number of increasing sequences of positive integers $b_1 \le b_2 \le b_3 \le \cdots \le b_{15} \le 3005$ such that $b_i-i$ is odd for $1\le i \le 15$. Express your answer as ${p \choose q}$ for some integers $p > q$ and find the remainder when $p$ is divided by 1000. | {
"answer": "509",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $ a,b,c,d$ are rational numbers with $ a>0$ , find the minimal value of $ a$ such that the number $ an^{3} + bn^{2} + cn + d$ is an integer for all integers $ n \ge 0$ . | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiaoming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, due to some reasons, Xiaoming first took the subway and then transferred to the bus, taking 40 minutes to reach the school. The transfer process took 6 minutes. How many minutes did Xiaoming spend on the bus that day? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a plane, there is a point set \( M \) and seven distinct circles \( C_{1}, C_{2}, \ldots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \), circle \( C_{6} \) passes through exactly 6 points in \( M \), and so on, until circle \( C_{1} \) passes through exactly 1 point in \( M \). What is the minimum number of points in \( M \)? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a finite sequence \( B = (b_1, b_2, \dots, b_{150}) \) of numbers, the Cesaro sum of \( B \) is defined to be
\[
\frac{S_1 + \cdots + S_{150}}{150},
\]
where \( S_k = b_1 + \cdots + b_k \) and \( 1 \leq k \leq 150 \).
If the Cesaro sum of the 150-term sequence \( (b_1, \dots, b_{150}) \) is 1200, what is the Cesaro sum of the 151-term sequence \( (2, b_1, \dots, b_{150}) \)? | {
"answer": "1194",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$\triangle PQR$ is similar to $\triangle STU$. The length of $\overline{PQ}$ is 10 cm, $\overline{QR}$ is 12 cm, and the length of $\overline{ST}$ is 5 cm. Determine the length of $\overline{TU}$ and the perimeter of $\triangle STU$. Express your answer as a decimal. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle{ABC}$ with side lengths $AB = 15$, $AC = 8$, and $BC = 17$, let $O$ and $I$ denote the circumcenter and incenter, respectively. A circle with center $M$ is tangent to the legs $AC$ and $BC$ and to the circumcircle of $\triangle{ABC}$. What is the area of $\triangle{MOI}$? | {
"answer": "3.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a hyperbola with the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, its asymptotes intersect the parabola $y^2 = 4x$ at two points A and B, distinct from the origin O. Let F be the focus of the parabola $y^2 = 4x$. If $\angle AFB = \frac{2\pi}{3}$, then the eccentricity of the hyperbola is ________. | {
"answer": "\\frac{\\sqrt{21}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two circles intersecting at points A(1, 3) and B(m, -1), where the centers of both circles lie on the line $x - y + c = 0$, find the value of $m + c$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the sum of the squares of cosine for the angles progressing by 10 degrees starting from 0 degrees up to 180 degrees:
\[\cos^2 0^\circ + \cos^2 10^\circ + \cos^2 20^\circ + \dots + \cos^2 180^\circ.\] | {
"answer": "\\frac{19}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are 21 students in Dr. Smith's physics class, the average score before including Simon's project score was 86. After including Simon's project score, the average for the class rose to 88. Calculate Simon's score on the project. | {
"answer": "128",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{(x, y) \mid x, y \in \mathbb{Z}, 0 \leq x, y \leq 2016\} \). Given points \( A = (x_1, y_1), B = (x_2, y_2) \) in \( S \), define
\[ d_{2017}(A, B) = (x_1 - x_2)^2 + (y_1 - y_2)^2 \pmod{2017} \]
The points \( A = (5, 5) \), \( B = (2, 6) \), and \( C = (7, 11) \) all lie in \( S \). There is also a point \( O \in S \) that satisfies
\[ d_{2017}(O, A) = d_{2017}(O, B) = d_{2017}(O, C) \]
Find \( d_{2017}(O, A) \). | {
"answer": "1021",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | {
"answer": "0.293",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right-angled geometric setup, $\angle ABC$ and $\angle ADB$ are both right angles. The lengths of segments are given as $AC = 25$ units and $AD = 7$ units. Determine the length of segment $DB$. | {
"answer": "3\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of $\sqrt[4]{2^3 + 2^4 + 2^5 + 2^6}$? | {
"answer": "2^{3/4} \\cdot 15^{1/4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the frequency distribution histogram of a sample, there are a total of $m(m\geqslant 3)$ rectangles, and the sum of the areas of the first $3$ groups of rectangles is equal to $\frac{1}{4}$ of the sum of the areas of the remaining $m-3$ rectangles. The sample size is $120$. If the areas of the first $3$ groups of rectangles, $S_1, S_2, S_3$, form an arithmetic sequence and $S_1=\frac{1}{20}$, then the frequency of the third group is ______. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the greatest integer less than or equal to $\frac{4^{50}+3^{50}}{4^{47}+3^{47}}$? | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a geometric sequence $\{a_n\}$ with the first term $\frac{3}{2}$ and common ratio $- \frac{1}{2}$, the sum of the first $n$ terms is $S_n$. If for any $n \in N^*$, it holds that $S_n - \frac{1}{S_n} \in [s, t]$, then the minimum value of $t-s$ is \_\_\_\_\_\_. | {
"answer": "\\frac{17}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A string has 150 beads of red, blue, and green colors. It is known that among any six consecutive beads, there is at least one green bead, and among any eleven consecutive beads, there is at least one blue bead. What is the maximum number of red beads that can be on the string? | {
"answer": "112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a$, $b$, $c \in \{1, 2, 3, 4, 5, 6\}$, if the lengths $a$, $b$, and $c$ can form an isosceles (including equilateral) triangle, then there are \_\_\_\_\_\_ such triangles. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Divide 6 volunteers into 4 groups for service at four different venues of the 2012 London Olympics. Among these groups, 2 groups will have 2 people each, and the other 2 groups will have 1 person each. How many different allocation schemes are there? (Answer with a number) | {
"answer": "540",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the expansion of $(x^2+ \frac{4}{x^2}-4)^3(x+3)$, find the constant term. | {
"answer": "-240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Divers extracted a certain number of pearls, not exceeding 1000. The distribution of the pearls happens as follows: each diver in turn approaches the heap of pearls and takes either exactly half or exactly one-third of the remaining pearls. After all divers have taken their share, the remainder of the pearls is offered to the sea god. What is the maximum number of divers that could have participated in the pearl extraction? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Form a six-digit number using the digits 1, 2, 3, 4, 5, 6 without repetition, where both 5 and 6 are on the same side of 3. How many such six-digit numbers are there? | {
"answer": "480",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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