problem stringlengths 10 5.15k | answer dict |
|---|---|
Given the point \( P \) on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \left(a>b>0, c=\sqrt{a^{2}-b^{2}}\right)\), and the equation of the line \( l \) is \(x=-\frac{a^{2}}{c}\), and the coordinate of the point \( F \) is \((-c, 0)\). Draw \( PQ \perp l \) at point \( Q \). If the points \( P \), \( Q \), and \( F \) form an isosceles triangle, what is the eccentricity of the ellipse? | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cars A and B simultaneously depart from locations $A$ and $B$, and travel uniformly towards each other. When car A has traveled 12 km past the midpoint of $A$ and $B$, the two cars meet. If car A departs 10 minutes later than car B, they meet exactly at the midpoint of $A$ and $B$. When car A reaches location $B$, car B is still 20 km away from location $A$. What is the distance between locations $A$ and $B$ in kilometers? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volume of a given sphere is \( 72\pi \) cubic inches. Calculate the surface area of the sphere in terms of \( \pi \). | {
"answer": "36\\pi 2^{2/3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the square roots of a positive number are $a+2$ and $2a-11$, find the positive number. | {
"answer": "225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, the equation of circle C is $\rho=4\sqrt{2}\cos(\theta-\frac{\pi}{4})$. A Cartesian coordinate system is established with the pole as the origin and the positive x-axis as the polar axis. The parametric equation of line $l$ is $\begin{cases} x=t+1 \\ y=t-1 \end{cases}$ (where $t$ is the parameter).
(1) Find the Cartesian equation of circle C and the standard equation of line $l$.
(2) Suppose line $l$ intersects circle C at points A and B. Find the area of triangle $\triangle ABC$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find \( g(2021) \) if for any real numbers \( x, y \) the following equation holds:
\[ g(x-y) = g(x) + g(y) - 2022(x + y) \] | {
"answer": "4086462",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the expansion of the binomial $({x+\frac{a}{{\sqrt{x}}}})^n$ where $n\in{N^*}$, in the expansion, ___, ___. Given the following conditions: ① the ratio of the binomial coefficients of the second term to the third term is $1:4$; ② the sum of all coefficients is $512$; ③ the $7$th term is a constant term. Choose two appropriate conditions from the above three conditions to fill in the blanks above, and complete the following questions.
$(1)$ Find the value of the real number $a$ and the term with the largest binomial coefficient in the expansion;
$(2)$ Find the constant term in the expansion of $({x\sqrt{x}-1}){({x+\frac{a}{{\sqrt{x}}}})^n}$. | {
"answer": "-48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function $y=\sin(\omega x+ \frac {5\pi}{6})$ where $0<\omega<\pi$ intersects with the coordinate axes at points closest to the origin, which are $(0, \frac {1}{2})$ and $( \frac {1}{2}, 0)$. Determine the axis of symmetry of this graph closest to the y-axis. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many squares of side at least $8$ have their four vertices in the set $H$, where $H$ is defined by the points $(x,y)$ with integer coordinates, $-8 \le x \le 8$ and $-8 \le y \le 8$? | {
"answer": "285",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of \[\cot(\cot^{-1}5 + \cot^{-1}11 + \cot^{-1}17 + \cot^{-1}23).\] | {
"answer": "\\frac{97}{40}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system (xOy), a pole coordinate system is established with O as the pole and the positive semi-axis of x as the polar axis. The shortest distance between a point on the curve C: ρcosθ - ρsinθ = 1 and a point on the curve M: x = -2 + cosφ, y = 1 + sinφ (φ is a parameter) can be calculated. | {
"answer": "2\\sqrt{2}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\displaystyle\prod_{i=6}^{2021} (1-\tan^2((2^i)^\circ))$ can be written in the form $a^b$ for positive integers $a,b$ with $a$ squarefree, find $a+b$ .
*Proposed by Deyuan Li and Andrew Milas* | {
"answer": "2018",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cagney can frost a cupcake every 25 seconds and Lacey can frost a cupcake every 35 seconds. They work together for 7 minutes, but there is a 1-minute period where only Cagney is frosting because Lacey takes a break. What is the number of cupcakes they can frost in these 7 minutes? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a line segment of length $6$ is divided into three line segments of lengths that are positive integers, calculate the probability that these three line segments can form a triangle. | {
"answer": "\\frac {1}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In 2019, our county built 4 million square meters of new housing, of which 2.5 million square meters are mid-to-low-priced houses. It is expected that in the coming years, the average annual increase in the area of new housing in our county will be 8% higher than the previous year. In addition, the area of mid-to-low-priced houses built each year will increase by 500,000 square meters compared to the previous year. So, by the end of which year:<br/>
$(1)$ The cumulative area of mid-to-low-priced houses built in our county over the years (with 2019 as the first cumulative year) will first exceed 22.5 million square meters?<br/>
$(2)$ The proportion of the area of mid-to-low-priced houses built in that year to the total area of housing built in that year will first exceed 85%? (Reference data: $1.08^{4}\approx 1.36$, $1.08^{5}\approx 1.47$) | {
"answer": "2024",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lateral surface area of a circular truncated cone is given by the formula, find the value for the lateral surface area of the cone where the radii of the upper and lower bases are $r=1$ and $R=4$ and the height is $4$. | {
"answer": "25\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \(12^{2}\) and \(13^{2}\)? | {
"answer": "165",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the radius of the base of a cone is $2$ and the area of the unfolded side of the cone is $8\pi$, find the volume of the inscribed sphere in the cone. | {
"answer": "\\frac{32\\sqrt{3}}{27}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a new function $\$N$ such that $\$N = 0.75N + 2$. Calculate $\$(\$(\$30))$. | {
"answer": "17.28125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The perimeter of triangle \(ABC\) is 1. A circle \(\omega\) touches side \(BC\), the extension of side \(AB\) at point \(P\), and the extension of side \(AC\) at point \(Q\). A line passing through the midpoints of \(AB\) and \(AC\) intersects the circumcircle of triangle \(APQ\) at points \(X\) and \(Y\). Find the length of segment \(XY\). | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Isaac repeatedly flips a fair coin. Whenever a particular face appears for the $2n+1$ th time, for any nonnegative integer $n$ , he earns a point. The expected number of flips it takes for Isaac to get $10$ points is $\tfrac ab$ for coprime positive integers $a$ and $b$ . Find $a + b$ .
*Proposed by Isaac Chen* | {
"answer": "201",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six chairs are placed in a row. Find the number of ways 3 people can sit randomly in these chairs such that no two people sit next to each other. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of $k$ such that the equation
\[\frac{x + 3}{kx - 2} = x\] has exactly one solution. | {
"answer": "-\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Pirate Bob shares his treasure with Pirate Sam in a peculiar manner. Bob first declares, ``One for me, one for you,'' keeping one coin for himself and starting Sam's pile with one coin. Then Bob says, ``Two for me, and two for you,'' adding two more coins to his pile but updating Sam's total to two coins. This continues until Bob says, ``$x$ for me, $x$ for you,'' at which he takes $x$ more coins and makes Sam's total $x$ coins in total. After all coins are distributed, Pirate Bob has exactly three times as many coins as Pirate Sam. Find out how many gold coins they have between them? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $\triangle ABC$ with $\cos C = \frac{2}{3}$, $AC = 4$, and $BC = 3$, calculate the value of $\tan B$. | {
"answer": "4\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $ ABC$ . Point $ O$ is the center of the excircle touching the side $ BC$ . Point $ O_1$ is the reflection of $ O$ in $ BC$ . Determine angle $ A$ if $ O_1$ lies on the circumcircle of $ ABC$ . | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Shift the graph of the function $y = \sin\left(\frac{\pi}{3} - x\right)$ to obtain the graph of the function $y = \cos\left(x + \frac{2\pi}{3}\right)$. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of solutions to:
\[\sin x = \left( \frac{1}{3} \right)^x\]
on the interval $(0,50 \pi)$. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the heights on the two equal sides of an isosceles triangle is equal to the height on the base. Find the sine of the base angle. | {
"answer": "$\\frac{\\sqrt{15}}{4}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two identical rectangular crates are packed with cylindrical pipes, using different methods. Each pipe has a diameter of 8 cm. In Crate A, the pipes are packed directly on top of each other in 25 rows of 8 pipes each across the width of the crate. In Crate B, pipes are packed in a staggered (hexagonal) pattern that results in 24 rows, with the rows alternating between 7 and 8 pipes.
After the crates have been packed with an equal number of 200 pipes each, what is the positive difference in the total heights (in cm) of the two packings? | {
"answer": "200 - 96\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A three-digit natural number $abc$ is termed a "convex number" if and only if the digits $a$, $b$, and $c$ (representing the hundreds, tens, and units place, respectively) satisfy $a < b$ and $c < b$. Given that $a$, $b$, and $c$ belong to the set $\{5, 6, 7, 8, 9\}$ and are distinct from one another, find the probability that a randomly chosen three-digit number $abc$ is a "convex number". | {
"answer": "\\frac {1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular garden needs to be enclosed on three sides using a 70-meter rock wall as one of the sides. Fence posts are placed every 10 meters along the fence, including at the ends where the fence meets the rock wall. If the area of the garden is 2100 square meters, calculate the fewest number of posts required. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can the number 5 be expressed as the sum of one or more positive integers? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the random variable $X$ follows a normal distribution $N(-1, \sigma^2)$, and $P(-3 \leq X \leq -1) = 0.4$, calculate the probability of $X$ being greater than or equal to $1$. | {
"answer": "0.1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the right focus of ellipse $I$: $\frac{{x}^{2}}{{a}^{2}}+ \frac{{y}^{2}}{{b}^{2}}=1 (a > 0,b > 0)$ is $(2 \sqrt{2},0)$, and ellipse $I$ passes through point $(3,1)$.
(1) Find the equation of ellipse $I$;
(2) Let line $l$ with slope $1$ intersect ellipse $I$ at two distinct points $A$ and $B$. Construct an isosceles triangle $PAB$ with base $AB$ and vertex $P$ at coordinates $(-3,2)$. Find the area of $∆PAB$. | {
"answer": "\\frac {9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere is inside a cube with an edge length of $3$, and it touches all $12$ edges of the cube. Find the volume of the sphere. | {
"answer": "9\\sqrt{2}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the total area of two overlapping circles where circle A has center at point $A(2, -1)$ and passes through point $B(5, 4)$, and circle B has center at point $C(3, 3)$ and passes through point $D(5, 8)$? Express your answer in terms of $\pi$. | {
"answer": "63\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the numerical coefficients in the binomial $(2a+2b)^7$ is $\boxed{32768}$. | {
"answer": "16384",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The congruent sides of an isosceles triangle are each 8 cm long, and the perimeter is 26 cm. In centimeters, what is the length of the base? Also, find the area of the triangle. | {
"answer": "5\\sqrt{39}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During a journey, the distance read on the odometer was 450 miles. On the return trip, using snow tires for the same distance, the reading was 440 miles. If the original wheel radius was 15 inches, find the increase in the wheel radius, correct to the nearest hundredth of an inch. | {
"answer": "0.34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A round-robin tennis tournament is organized where each player is supposed to play every other player exactly once. However, the tournament is scheduled to have one rest day during which no matches will be played. If there are 10 players in the tournament, and the tournament was originally scheduled for 9 days, but one day is now a rest day, how many matches will be completed? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of different arrangements of $6$ rescue teams to $3$ disaster sites, where site $A$ has at least $2$ teams and each site is assigned at least $1$ team. | {
"answer": "360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the vectors $\overrightarrow{a}=(\cos 40^{\circ},\sin 40^{\circ})$, $\overrightarrow{b}=(\sin 20^{\circ},\cos 20^{\circ})$, and $\overrightarrow{u}= \sqrt {3} \overrightarrow{a}+λ \overrightarrow{b}$ (where $λ∈R$), find the minimum value of $|\overrightarrow{u}|$. | {
"answer": "\\frac {\\sqrt {3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The angle that has the same terminal side as $- \frac{\pi}{3}$ is $\frac{\pi}{3}$. | {
"answer": "\\frac{5\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A die with faces showing the numbers $0,1,2,3,4,5$ is rolled until the total sum of the rolled numbers exceeds 12. What is the most likely value of this sum? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two vectors in the plane, $\mathbf{a} = (2m+1, 3)$ and $\mathbf{b} = (2, m)$, and $\mathbf{a}$ is in the opposite direction to $\mathbf{b}$, calculate the magnitude of $\mathbf{a} + \mathbf{b}$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence 1, 1+2, 2+3+4, 3+4+5+6, ..., the value of the 8th term in this sequence is: ______. | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What time is it 2017 minutes after 20:17? | {
"answer": "05:54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$g(x):\mathbb{Z}\rightarrow\mathbb{Z}$ is a function that satisfies $$ g(x)+g(y)=g(x+y)-xy. $$ If $g(23)=0$ , what is the sum of all possible values of $g(35)$ ? | {
"answer": "210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A note contains three two-digit numbers that are said to form a sequence with a fourth number under a cryptic condition. The numbers provided are 46, 19, and 63, but the fourth number is unreadable. You know that the sum of the digits of all four numbers is $\frac{1}{4}$ of the total sum of these four numbers. What is the fourth number? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(\alpha)=\frac{2\sin(2\pi-\alpha)\cos(2\pi+\alpha)-\cos(-\alpha)}{1+\sin^{2}\alpha+\sin(2\pi+\alpha)-\cos^{2}(4\pi-\alpha)}$, find the value of $f\left(-\frac{23}{6}\pi \right)$. | {
"answer": "-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a revised graph for Lambda Corp., the number of employees at different tenure periods is represented with the following marks:
- Less than 1 year: 3 marks
- 1 to less than 2 years: 6 marks
- 2 to less than 3 years: 5 marks
- 3 to less than 4 years: 4 marks
- 4 to less than 5 years: 2 marks
- 5 to less than 6 years: 2 marks
- 6 to less than 7 years: 3 marks
- 7 to less than 8 years: 2 marks
- 8 to less than 9 years: 1 mark
- 9 to less than 10 years: 1 mark
Determine what percent of the employees have worked there for $6$ years or more. | {
"answer": "24.14\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle?
*Author: Ray Li* | {
"answer": "256",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $|\sqrt{8}-2|+(\pi -2023)^{0}+(-\frac{1}{2})^{-2}-2\cos 60^{\circ}$. | {
"answer": "2\\sqrt{2}+2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=\sin (3x+ \frac {\pi}{3})\cos (x- \frac {\pi}{6})+\cos (3x+ \frac {\pi}{3})\sin (x- \frac {\pi}{6})$, find the equation of one of the axes of symmetry. | {
"answer": "\\frac {\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F$ is the left focus of the ellipse $C:\frac{{x}^{2}}{3}+\frac{{y}^{2}}{2}=1$, $M$ is a moving point on the ellipse $C$, and point $N(5,3)$, then the minimum value of $|MN|-|MF|$ is ______. | {
"answer": "5 - 2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the square root of $\dfrac{10!}{210}$. | {
"answer": "24\\sqrt{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the arithmetic sequence $\left\{a_{n}\right\}$, if $\frac{a_{11}}{a_{10}} < -1$ and the sum of its first $n$ terms $S_{n}$ has a maximum value, then when $S_{n}$ takes the smallest positive value, $n = (\quad$ ). | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there? | {
"answer": "115",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rhombus $ABCD$ is given with $\angle BAD = 60^o$ . Point $P$ lies inside the rhombus such that $BP = 1$ , $DP = 2$ , $CP = 3$ . Determine the length of the segment $AP$ . | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each segment with endpoints at the vertices of a regular 100-gon is colored red if there is an even number of vertices between its endpoints, and blue otherwise (in particular, all sides of the 100-gon are red). Numbers were placed at the vertices such that the sum of their squares equals 1, and at the segments, the products of the numbers at the endpoints were placed. Then, the sum of the numbers on the red segments was subtracted by the sum of the numbers on the blue segments. What is the largest possible value that could be obtained?
I. Bogdanov | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six cubes, each an inch on an edge, are fastened together, as shown. Find the total surface area in square inches. Include the top, bottom and sides.
[asy]/* AMC8 2002 #22 Problem */
draw((0,0)--(0,1)--(1,1)--(1,0)--cycle);
draw((0,1)--(0.5,1.5)--(1.5,1.5)--(1,1));
draw((1,0)--(1.5,0.5)--(1.5,1.5));
draw((0.5,1.5)--(1,2)--(1.5,2));
draw((1.5,1.5)--(1.5,3.5)--(2,4)--(3,4)--(2.5,3.5)--(2.5,0.5)--(1.5,.5));
draw((1.5,3.5)--(2.5,3.5));
draw((1.5,1.5)--(3.5,1.5)--(3.5,2.5)--(1.5,2.5));
draw((3,4)--(3,3)--(2.5,2.5));
draw((3,3)--(4,3)--(4,2)--(3.5,1.5));
draw((4,3)--(3.5,2.5));
draw((2.5,.5)--(3,1)--(3,1.5));[/asy] | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function f(x) = $\sqrt {2}$sin $\frac {x}{2}$cos $\frac {x}{2}$ - $\sqrt {2}$sin<sup>2</sup> $\frac {x}{2}$,
(1) Find the smallest positive period of f(x);
(2) Find the minimum value of f(x) in the interval [-π, 0]. | {
"answer": "-1 - \\frac { \\sqrt {2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=4\cos (2x+\frac{\pi}{4})$, determine the direction and magnitude of horizontal shift required to obtain the graph of the function $y=4\cos 2x$. | {
"answer": "\\frac{\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\tan α$ and $\tan β$ are the roots of the equation $x^{2}+3 \sqrt {3}x+4=0$, and $\(- \frac {π}{2} < α < \frac {π}{2}\)$, $\(- \frac {π}{2} < β < \frac {π}{2}\)$, find $α+β$. | {
"answer": "- \\frac {2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $(xOy)$, the focus of the parabola $y^{2}=2x$ is $F$. If $M$ is a moving point on the parabola, determine the maximum value of $\frac{|MO|}{|MF|}$. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"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{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Plane M is parallel to plane N. There are 3 different points on plane M and 4 different points on plane N. The maximum number of tetrahedrons with different volumes that can be determined by these 7 points is ____. | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \\(AB\\) is a chord passing through the focus of the parabola \\(y^{2} = 4\sqrt{3}x\\), and the midpoint \\(M\\) of \\(AB\\) has an x-coordinate of \\(2\\), calculate the length of \\(AB\\. | {
"answer": "4 + 2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, medians $\overline{AM}$ and $\overline{BN}$ are perpendicular. If $AM = 15$ and $BN = 20$, and the height from $C$ to line $AB$ is $12$, find the length of side $AB$. | {
"answer": "\\frac{50}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A hyperbola with its center shifted to $(1,1)$ passes through point $(4, 2)$. The hyperbola opens horizontally, with one of its vertices at $(3, 1)$. Determine $t^2$ if the hyperbola also passes through point $(t, 4)$. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square with a side length of one unit has one of its vertices separated from the other three by a line \( e \). The products of the distances of opposite vertex pairs from \( e \) are equal. What is the distance of the center of the square from \( e \)? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A belt is placed without slack around two non-crossing circular pulleys which have radii of $15$ inches and $5$ inches respectively. The distance between the points where the belt contacts the pulleys is $30$ inches. Determine the distance between the centers of the two pulleys.
A) $20$ inches
B) $10\sqrt{10}$ inches
C) $40$ inches
D) $50$ inches | {
"answer": "10\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square $ABCD$ with a side length of $2$ is rotated around $BC$ to form a cylinder. Find the surface area of the cylinder. | {
"answer": "16\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two non-collinear vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are given, and $\overrightarrow{a}+2\overrightarrow{b}$ is perpendicular to $2\overrightarrow{a}-\overrightarrow{b}$, $\overrightarrow{a}-\overrightarrow{b}$ is perpendicular to $\overrightarrow{a}$. The cosine of the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is __________. | {
"answer": "\\frac{\\sqrt{10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Find the value of $\cos\frac{5\pi}{3}$.
(2) Given that $\frac{\sin\alpha + 2\cos\alpha}{5\cos\alpha - \sin\alpha} = \frac{5}{16}$, find the value of $\tan\alpha$.
(3) Given that $\sin\theta = \frac{1}{3}$ and $\theta \in (0, \frac{\pi}{2})$, find the value of $\tan 2\theta$.
(4) In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Moreover, $a\sin A\cos C + c\sin A\cos A = \frac{1}{3}c$, $D$ is the midpoint of $AC$, and $\cos B = \frac{2\sqrt{5}}{5}$, $BD = \sqrt{26}$. Find the length of the shortest side of triangle $ABC$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle (right in $B$ ) inscribed in a semi-circumference of diameter $AC=10$ . Determine the distance of the vertice $B$ to the side $AC$ if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle. | {
"answer": "5/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The second hand on the clock is 6 cm long. How far in centimeters does the tip of this second hand travel during a period of 15 minutes? Express your answer in terms of $\pi$. | {
"answer": "180\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If physical education is not the first class, and Chinese class is not adjacent to physics class, calculate the total number of different scheduling arrangements for five subjects - mathematics, physics, history, Chinese, and physical education - on Tuesday morning. | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x$ and $y$ satisfy the equation $x^2 + y^2 - 4x - 6y + 12 = 0$, find the minimum value of $x^2 + y^2$. | {
"answer": "14 - 2\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine how many non-similar regular 500-pointed stars exist, given that a regular $n$-pointed star adheres to the rules set in the original problem description. | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $(a+1)x - 1 - \ln x \leqslant 0$ holds for any $x \in [\frac{1}{2}, 2]$, find the maximum value of $a$. | {
"answer": "1 - 2\\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $C:\frac{{x}^{2}}{4}+\frac{{y}^{2}}{3}=1$, let ${F}_{1}$ and ${F}_{2}$ be its left and right foci, respectively. A line $l$ passing through point ${F}_{2}$ with a slope of $1$ intersects ellipse $C$ at two distinct points $M$ and $N$. Calculate the area of triangle $MN{F}_{1}$. | {
"answer": "\\frac{12\\sqrt{2}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \begin{cases} a+\ln x,x > 0 \\ g(x)-x,x < 0\\ \end{cases}$, which is an odd function, and $g(-e)=0$, find the value of $a$. | {
"answer": "-1-e",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integer values of \( n \) between 1 and 999 inclusive does the decimal representation of \( \frac{n}{1000} \) terminate? | {
"answer": "999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sin (wx+\varphi+ \frac {\pi}{3})+1$ ($|\varphi| < \frac {\pi}{2},w > 0$) 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"
} |
A rectangular box $Q$ is inscribed in a sphere of radius $s$. The surface area of $Q$ is 576, and the sum of the lengths of its 12 edges is 168. Determine the radius $s$. | {
"answer": "3\\sqrt{33}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest constant \(C\) such that
\[ x^2 + y^2 + z^2 + 1 \ge C(x + y + z) \]
for all real numbers \(x, y,\) and \(z.\) | {
"answer": "\\sqrt{\\frac{4}{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Anh traveled 75 miles on the interstate and 15 miles on a mountain pass. The speed on the interstate was four times the speed on the mountain pass. If Anh spent 45 minutes driving on the mountain pass, determine the total time of his journey in minutes. | {
"answer": "101.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, if $a= \sqrt {5}$, $b= \sqrt {15}$, $A=30^{\circ}$, then $c=$ \_\_\_\_\_\_. | {
"answer": "2 \\sqrt {5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a pyramid $P-ABC$ where $PA=PB=2PC=2$, and $\triangle ABC$ is an equilateral triangle with side length $\sqrt{3}$, the radius of the circumscribed sphere of the pyramid $P-ABC$ is _______. | {
"answer": "\\dfrac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ellipse $5x^2 - ky^2 = 5$ has one of its foci at $(0, 2)$. Find the value of $k$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? | {
"answer": "\\sqrt{244}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In parallelogram $EFGH$, point $Q$ is on $\overline{EF}$ such that $\frac {EQ}{EF} = \frac {23}{1005}$, and point $R$ is on $\overline{EH}$ such that $\frac {ER}{EH} = \frac {23}{2011}$. Let $S$ be the point of intersection of $\overline{EG}$ and $\overline{QR}$. Find $\frac {EG}{ES}$. | {
"answer": "131",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $|\vec{a}|=4$, and $\vec{e}$ is a unit vector. When the angle between $\vec{a}$ and $\vec{e}$ is $\frac{2\pi}{3}$, the projection of $\vec{a} + \vec{e}$ on $\vec{a} - \vec{e}$ is ______. | {
"answer": "\\frac{5\\sqrt{21}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system xOy, curve $C_1: x^2+y^2=1$ is given. Taking the origin O of the Cartesian coordinate system xOy as the pole and the positive half-axis of x as the polar axis, a polar coordinate system is established with the same unit length. It is known that the line $l: \rho(2\cos\theta-\sin\theta)=6$.
(1) After stretching all the x-coordinates and y-coordinates of points on curve $C_1$ by $\sqrt{3}$ and 2 times respectively, curve $C_2$ is obtained. Please write down the Cartesian equation of line $l$ and the parametric equation of curve $C_2$;
(2) Find a point P on curve $C_2$ such that the distance from point P to line $l$ is maximized, and calculate this maximum value. | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A palindrome is an integer that reads the same forward and backward, such as 1221. What percent of the palindromes between 1000 and 2000 contain at least one digit 7? | {
"answer": "10\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin x\cos x-\sqrt{3}\cos^{2}x$.
(1) Find the smallest positive period of $f(x)$;
(2) Find the maximum and minimum values of $f(x)$ when $x\in[0,\frac{\pi }{2}]$. | {
"answer": "-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the sum of the squares of the first 10 base-6 numbers and express your answer in base 6. Specifically, find $1_6^2 + 2_6^2 + 3_6^2 + \cdots + 10_6^2$. | {
"answer": "231_6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the minimum value of the function $$y = \frac {4x^{2}+2x+5}{x^{2}+x+1}$$ for \(x > 1\). | {
"answer": "\\frac{16 - 2\\sqrt{7}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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