problem stringlengths 10 5.15k | answer dict |
|---|---|
Given that $a$, $b$, $c$ form an arithmetic sequence in triangle $ABC$, $\angle B=30^{\circ}$, and the area of $\triangle ABC$ is $\frac{1}{2}$, determine the value of $b$. | {
"answer": "\\frac{3+ \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite to angles A, B, and C are denoted by $a$, $b$, and $c$ respectively, with $A+C=\frac{2\pi}{3}$ and $b=1$.
(1) If we let angle A be $x$ and define $f(x)=a+c$, find the range of $f(x)$ when triangle $ABC$ is an acute triangle;
(2) Determine the maximum area of triangle $ABC$. | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an equilateral triangle $ABC$ with side length of 10, a similar process of division by midpoints and shading of one of these triangles occurs. If this dividing and shading process is repeated indefinitely, and the first triangle to be shaded is the triangle involving vertex $A$, the total shaded area will converge towards?
A) $15\sqrt{3}$
B) $18\sqrt{3}$
C) $\frac{25\sqrt{3}}{3}$
D) $25$
E) $30\sqrt{3}$ | {
"answer": "\\frac{25\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $(x,y)$ satisfy the constraints $\begin{cases} 8x - y - 4 \leqslant 0 \\ x + y + 1 \geqslant 0 \\ y - 4x \leqslant 0 \end{cases}$, and the maximum value of the objective function $z = ax + by (a > 0, b > 0)$ is $2$. Find the minimum value of $\frac{1}{a} + \frac{1}{b}$. | {
"answer": "\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of an infinite geometric series is 64 times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $x$ and $y$ satisfying $x^2+4y^2=4$, find the maximum value of $\frac {xy}{x+2y-2}$. | {
"answer": "\\frac {1+ \\sqrt {2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The greatest common divisor of two integers is $(x+3)$ and their least common multiple is $x(x+3)$, where $x$ is a positive integer. If one of the integers is 30, what is the smallest possible value of the other one? | {
"answer": "162",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jane Doe invested some amount of money into a savings account and mutual funds. The total amount she invested was \$320,000. If she invested 6 times as much in mutual funds as she did in the savings account, what was her total investment in mutual funds? | {
"answer": "274,285.74",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = |x^2 + bx|$ ($b \in \mathbb{R}$), when $x \in [0, 1]$, find the minimum value of the maximum value of $f(x)$. | {
"answer": "3-2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a given list of three numbers, the operation "changesum" replaces each number in the list with the sum of the other two. For example, applying "changesum" to \(3,11,7\) gives \(18,10,14\). Arav starts with the list \(20,2,3\) and applies the operation "changesum" 2023 times. What is the largest difference between two of the three numbers in his final list?
A 17
B 18
C 20
D 2021
E 2023 | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the price of a stamp is 50 cents, what is the maximum number of stamps that could be purchased with $50? Furthermore, if a customer buys more than 80 stamps, they receive a discount of 5 cents per stamp. How many stamps would then be purchased at maximum? | {
"answer": "111",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point P lies on the curve represented by the equation $$\sqrt {(x-5)^{2}+y^{2}}- \sqrt {(x+5)^{2}+y^{2}}=6$$. If the y-coordinate of point P is 4, then its x-coordinate is ______. | {
"answer": "x = -3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest possible area, in square units, of a right triangle with two sides measuring $7$ units and $8$ units? | {
"answer": "\\frac{7\\sqrt{15}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of all two-digit positive integers whose squares end with the digits 25? | {
"answer": "495",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $f(x)$ is a function defined on $[1,+\infty)$, and
$$
f(x)=\begin{cases}
1-|2x-3|, & 1\leqslant x < 2, \\
\frac{1}{2}f\left(\frac{1}{2}x\right), & x\geqslant 2,
\end{cases}
$$
then the number of zeros of the function $y=2xf(x)-3$ in the interval $(1,2015)$ is ______. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A national team needs to select 4 out of 6 sprinters to participate in the 4×100 m relay at the Asian Games. If sprinter A cannot run the first leg and sprinter B cannot run the fourth leg, there are a total of ______ ways to participate. | {
"answer": "252",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive two-digit number is odd and is a multiple of 9. The product of its digits is a perfect square. What is this two-digit number? | {
"answer": "99",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$ | {
"answer": "1011",
"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 $$bsin(C- \frac {π}{3})-csinB=0$$
(I) Find the value of angle C;
(II) If $a=4$, $c=2 \sqrt {7}$, find the area of $\triangle ABC$. | {
"answer": "2 \\sqrt {3}",
"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 $\triangle ABC$, if $\sin B= \sqrt {3}\sin A$, $BC= \sqrt {2}$, and $C= \frac {\pi}{6}$, then the height to side $AC$ is ______. | {
"answer": "\\frac { \\sqrt {2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive $x$-axis as the polar axis, the parametric equation of curve $C_1$ is
$$
\begin{cases}
x=2+ \sqrt {3}\cos \theta \\
y= \sqrt {3}\sin \theta
\end{cases}
(\theta \text{ is the parameter}),
$$
and the polar equation of curve $C_2$ is $\theta= \frac {\pi}{6} (\rho \in \mathbb{R})$.
$(1)$ Find the general equation of curve $C_1$ and the Cartesian coordinate equation of curve $C_2$;
$(2)$ Curves $C_1$ and $C_2$ intersect at points $A$ and $B$. Given point $P(3, \sqrt {3})$, find the value of $||PA|-|PB||$. | {
"answer": "2 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of rhombus \( EFGH \) are the hypotenuses of the isosceles right triangles \( EAF, FDG, GCH, \) and \( HBE \), and all these triangles have common interior points with the rhombus \( EFGH \). The sum of the areas of quadrilateral \( ABCD \) and rhombus \( EFGH \) is 12. Find \( GH \). | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\tan 2\alpha= \frac {3}{4}$, $\alpha\in(- \frac {\pi}{2}, \frac {\pi}{2})$, $f(x)=\sin (x+\alpha)+\sin (\alpha-x)-2\sin \alpha$, and for any $x\in\mathbb{R}$, it always holds that $f(x)\geqslant 0$, find the value of $\sin (\alpha- \frac {\pi}{4})$. | {
"answer": "- \\frac {2 \\sqrt {5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T = (2+i)^{20} - (2-i)^{20}$, where $i = \sqrt{-1}$. Find $|T|$. | {
"answer": "19531250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin\theta + \cos\theta = \frac{3}{4}$, where $\theta$ is an angle of a triangle, find the value of $\sin\theta - \cos\theta$. | {
"answer": "\\frac{\\sqrt{23}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathbf{v}$ be a vector such that
\[\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.\]
Find the smallest possible value of $\|\mathbf{v}\|$. | {
"answer": "10 - 2 \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a point P $(x, y)$ on the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$, find the minimum distance from point P to the line $2x + y - 10 = 0$. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle has a radius of 6. What is the area of the smallest square that can entirely contain this circle, and what is the circumference of the circle? | {
"answer": "12\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $4 \in \{a^2-3a, a\}$, then the value of $a$ equals ____. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\tan \alpha = 2$, where $\alpha$ is an angle in the first quadrant, find the value of $\sin 2\alpha + \cos \alpha$. | {
"answer": "\\dfrac{4 + \\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cara is sitting at a circular table with her seven friends. How many different possible pairs of people could Cara be sitting between? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that three balls are randomly and independently tossed into bins numbered with the positive integers such that for each ball, the probability that it is tossed into bin i is $3^{-i}$ for i = 1,2,3,..., find the probability that all balls end up in consecutive bins. | {
"answer": "1/702",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $c\sin\frac{A+C}{2}=b\sin C$.
$(1)$ Find angle $B$;
$(2)$ Let $BD$ be the altitude from $B$ to side $AC$, and $BD=1$, $b=\sqrt{3}$. Find the perimeter of $\triangle ABC$. | {
"answer": "3 + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $( \sqrt {x}+ \dfrac {2}{x^{2}})^{n}$, the ratio of the coefficient of the fifth term to the coefficient of the third term in its expansion is $56:3$.
(Ⅰ) Find the constant term in the expansion;
(Ⅱ) When $x=4$, find the term with the maximum binomial coefficient in the expansion. | {
"answer": "\\dfrac {63}{256}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Estimate the number of primes among the first thousand primes divide some term of the sequence
\[2^0+1,2^1+1,2^2+1,2^3+1,\ldots.\]
An estimate of $E$ earns $2^{1-0.02|A-E|}$ points, where $A$ is the actual answer.
*2021 CCA Math Bonanza Lightning Round #5.4* | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation of the parabola $y^{2}=4x$, and a line $l$ passing through its focus $F$ intersecting the parabola at points $A$ and $B$. If $S_{\triangle AOF}=3S_{\triangle BOF}$ (where $O$ is the origin), calculate the length of $|AB|$. | {
"answer": "\\dfrac {16}{3}",
"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, and it is given that $2c-2a\cos B=b$.
$(1)$ Find the size of angle $A$;
$(2)$ If the area of $\triangle ABC$ is $\frac{\sqrt{3}}{4}$, and $c^{2}+ab\cos C+a^{2}=4$, find $a$. | {
"answer": "\\frac{\\sqrt{7}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mark rolls 5 fair 8-sided dice. What is the probability that at least three of the dice show the same number? | {
"answer": "\\frac{1052}{8192}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\tan(\alpha-\beta) = \frac{1}{2}$ and $\tan(\alpha+\beta) = \frac{1}{3}$, calculate the value of $\tan 2\beta$. | {
"answer": "- \\frac{1}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse with foci at $(1,0)$ and $(-1,0)$, it intersects with the line $y=x-2$. Determine the maximum value of the eccentricity of this ellipse. | {
"answer": "\\frac{\\sqrt{10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A \) and \( B \) be points on the curve \( xy = 1 \) (where \( x > 0 \) and \( y > 0 \)) in the Cartesian coordinate system \( xOy \). Given the vector \( \vec{m} = (1, |OA|) \), find the minimum value of the dot product \( \vec{m} \cdot \overrightarrow{OB} \). | {
"answer": "2 \\sqrt[4]{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the power function $y = (m^2 - m - 1)x^{2m+1}$, if it is a decreasing function for $x \in (0, +\infty)$, then the value of the real number $m$ is ______. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The inclination angle of the line $x+ \sqrt {3}y+c=0$ is \_\_\_\_\_\_. | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A store offers an initial discount of 25% on all its merchandise. Subsequently, an additional discount of 10% is applied to the already reduced prices. The store advertises that the final price of the goods is 35% less than the original price. What is the actual percentage difference between the store's claimed discount and the true discount from the original price? | {
"answer": "2.5\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line $l$ passing through the focus of the parabola $y=4x^2$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $y_1+y_2=2$, then the length of segment $AB$ equals \_\_\_\_\_\_. | {
"answer": "\\frac{17}{8}",
"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. Given vectors $\overrightarrow{m}=(\sin B+\sin C, \sin A-\sin B)$ and $\overrightarrow{n}=(\sin B-\sin C, \sin(B+C))$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(1) Find the magnitude of angle $C$;
(2) If $\sin A= \frac{4}{5}$, find the value of $\cos B$. | {
"answer": "\\frac{4\\sqrt{3}-3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the greatest possible value of $pq + r$ , where p, q, and r are (not necessarily distinct) prime numbers satisfying $pq + qr + rp = 2016$ .
| {
"answer": "1008",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that Xiao Li has a probability of hitting the bullseye of $40\%$ each time he shoots. Now, the method of random simulation is used to estimate the probability that Xiao Li hits the bullseye exactly twice in three shots. First, a calculator is used to generate random integers between $0$ and $9$, with $0$, $1$, $2$, $3$ representing hitting the bullseye, and $4$, $5$, $6$, $7$, $8$, $9$ representing missing the bullseye. Then, every three random numbers are grouped together to represent the results of three shots. After random simulation, the following $20$ groups of random numbers were generated:
$321$ $421$ $191$ $925$ $271$ $932$ $800$ $478$ $589$ $663$
$531$ $297$ $396$ $021$ $546$ $388$ $230$ $113$ $507$ $965$
Based on this, estimate the probability that Xiao Li hits the bullseye exactly twice in three shots. | {
"answer": "0.30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $A(-2,α)$ lies on the terminal side of angle $α$ and $\sin α=- \dfrac{ \sqrt{5}}{5}$.
(1) Find the values of $α$, $\cos α$, and $\tan α$.
(2) Find the value of $\dfrac{\cos ( \dfrac{π}{2}+α)\sin (-π-α)}{\cos ( \dfrac{11π}{2}-α)\sin ( \dfrac{9π}{2}+α)}$. | {
"answer": "\\dfrac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A farmer buys 600 cows. He sells 500 of them for the price he paid for all 600 cows. The remaining 100 cows are sold for 10% more per cow than the price of the 500 cows. Calculate the percentage gain on the entire transaction. | {
"answer": "22\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | {
"answer": "\\sqrt{55}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the curve $y=\sin x$ ($0 \leq x \leq \pi$) and the line $y= \frac {1}{2}$, calculate the area of the enclosed shape formed by these two functions. | {
"answer": "\\sqrt {3}- \\frac {\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pawn moves on a $6 \times 6$ chessboard. It starts from the bottom-left square. At each step, it can either jump to the square directly to its right or the square directly above it. It must reach the top-right square, such that it never lies strictly above the diagonal connecting the starting square and the destination square. Determine the number of possible paths. | {
"answer": "132",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Arnaldo claimed that one billion is the same as one million millions. Professor Piraldo corrected him and said, correctly, that one billion is the same as one thousand millions. What is the difference between the correct value of one billion and Arnaldo's assertion?
(a) 1000
(b) 999000
(c) 1000000
(d) 999000000
(e) 999000000000 | {
"answer": "999000000000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose (4-4): Coordinate Systems and Parametric Equations
In the rectangular coordinate system $xOy$, the parametric equations of the curve $C$ are $\begin{cases} x=3\cos \alpha \\ y=\sin \alpha \end{cases}$, where $\alpha$ is the parameter. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of the line $l$ is $\rho \sin (\theta - \frac{\mathrm{\pi }}{4})=\sqrt{2}$.
(I) Find the ordinary equation of $C$ and the slope angle of the line $l$;
(II) Let point $P(0,2)$, and $l$ intersects $C$ at points $A$ and $B$. Find $|PA|+|PB|$. | {
"answer": "\\frac{18\\sqrt{2}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of $\sqrt{64 \times \sqrt{49}}$? | {
"answer": "8\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An isosceles triangle with a base of $\sqrt{2}$ has medians intersecting at a right angle. Calculate the area of this triangle. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the 2016 art exam of a certain high school, there were 6 contestants, including 3 females and 3 males. Now, these six contestants are to perform their talents in sequence. If any two of the three males cannot perform consecutively, and the female contestant A cannot be the first to perform, then calculate the number of possible sequences for the contestants to perform. | {
"answer": "132",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Griffin and Hailey run for $45$ minutes on a circular track. Griffin runs counterclockwise at $260 m/min$ and uses the outer lane with a radius of $50$ meters. Hailey runs clockwise at $310 m/min$ and uses the inner lane with a radius of $45$ meters, starting on the same radial line as Griffin. Determine how many times do they pass each other. | {
"answer": "86",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ is a moving point on the parabola $y^{2}=4x$, the minimum value of the sum of the distance from point $P$ to line $l$: $2x-y+3=0$ and the $y$-axis is ___. | {
"answer": "\\sqrt{5}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ is a moving point on circle $C$: $x^{2}+y^{2}-2x-4y+1=0$, the maximum distance from point $P$ to a certain line $l$ is $6$. If a point $A$ is taken arbitrarily on line $l$ to form a tangent line $AB$ to circle $C$, with $B$ being the point of tangency, then the minimum value of $AB$ is _______. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f\left(x\right)$ is an even function on $R$, and $f\left(x+2\right)$ is an odd function. If $f\left(0\right)=1$, then $f\left(1\right)+f\left(2\right)+\ldots +f\left(2023\right)=\_\_\_\_\_\_$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Samantha has 10 green marbles and 5 purple marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 7 marbles. What is the probability that exactly four of the marbles that she removes are green? Express your answer as a decimal. | {
"answer": "0.256",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of the first eight prime numbers that have a units digit of 3. | {
"answer": "404",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T$ be the sum of all the real coefficients of the expansion of $(1 + ix)^{2018}$. What is $\log_2(T)$? | {
"answer": "1009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of $y = ax^2 + bx + c$ has a maximum value of 75, and passes through the points $(-3,0)$ and $(3,0)$. Find the value of $a + b + c$ at $x = 2$. | {
"answer": "\\frac{125}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( x_{i}=\frac{i}{101} \), then the value of \( S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \) is | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A dodecahedron consists of two pentagonal-based pyramids glued together along their pentagonal bases, forming a polyhedron with 12 faces. Consider an ant at the top vertex of one of the pyramids, selecting randomly one of the five adjacent vertices, designated as vertex A. From vertex A, the ant then randomly selects one of the five adjacent vertices, called vertex B. What is the probability that vertex B is the bottom vertex opposite the top one? | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given non-zero plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}-\overrightarrow{c}|=1$. If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{π}{3}$, calculate the minimum value of $|\overrightarrow{a}-\overrightarrow{c}|$. | {
"answer": "\\sqrt{3} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $xOy$, the parametric equations of line $l$ are given by $\begin{cases} x=2t \\ y=-2-t \end{cases}$ (where $t$ is the parameter). In the polar coordinate system (using the same length unit as the rectangular coordinate system and with the origin $O$ as the pole and the polar axis coinciding with the non-negative half of the $x$-axis), the equation of circle $C$ is given by $ρ=4\sqrt{2}\cos(θ+\frac{π}{4})$. Find the length of the chord cut off by line $l$ on circle $C$. | {
"answer": "\\frac{12\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sqrt{3}\sin x\cos x+\cos^{2}x$.
(1) Find the value of $f(\frac{\pi }{24})$;
(2) If the function $f(x)$ is monotonically increasing in the interval $[-m,m]$, find the maximum value of the real number $m$. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $A = 3009 \div 3$, $B = A \div 3$, and $Y = A - 2B$, then what is the value of $Y$? | {
"answer": "335",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of $\frac{3}{5} + \frac{2}{3} + 1\frac{1}{15}$? | {
"answer": "2\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On Arbor Day, 20 students in a class planted trees along one side of a straight road, with each person planting one tree, and the distance between two adjacent trees being 10 meters. Initially, it is required to place all the saplings next to one of the tree pits so that the total distance traveled back and forth by each student from their respective tree pits to collect the saplings is minimized. The minimum value of this total distance is (meters). | {
"answer": "2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be an isosceles right triangle with $\angle A=90^o$ . Point $D$ is the midpoint of the side $[AC]$ , and point $E \in [AC]$ is so that $EC = 2AE$ . Calculate $\angle AEB + \angle ADB$ . | {
"answer": "135",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mathematical operation refers to the ability to solve mathematical problems based on clear operation objects and operation rules. Because of operations, the power of numbers is infinite; without operations, numbers are just symbols. Logarithmic operation and exponential operation are two important types of operations.
$(1)$ Try to calculate the value of $\frac{{\log 3}}{{\log 4}}\left(\frac{{\log 8}}{{\log 9}+\frac{{\log 16}}{{\log 27}}\right)$ using the properties of logarithmic operations.
$(2)$ Given that $x$, $y$, and $z$ are positive numbers, if $3^{x}=4^{y}=6^{z}$, find the value of $\frac{y}{z}-\frac{y}{x}$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $0 < α < \frac {π}{2}$, and $\cos (2π-α)-\sin (π-α)=- \frac { \sqrt {5}}{5}$.
(1) Find the value of $\sin α+\cos α$
(2) Find the value of $\frac {2\sin α\cos α-\sin ( \frac {π}{2}+α)+1}{1-\cot ( \frac {3π}{2}-α)}$. | {
"answer": "\\frac {\\sqrt {5}-9}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive multiple of $225$ that can be written using
digits $0$ and $1$ only? | {
"answer": "11111111100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point? | {
"answer": "$\\sqrt{13}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers $\sqrt{3v-2}$, $\sqrt{3v+1}$, and $2\sqrt{v}$ are the side lengths of a triangle. What is the measure of the largest angle? | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\binom{24}{5}=42504$, and $\binom{24}{6}=134596$, find $\binom{26}{6}$. | {
"answer": "230230",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_n\}$ that satisfies $3a_{n+1}+a_n=0$ and $a_2=-\frac{4}{3}$, find the sum of the first $10$ terms of $\{a_n\}$. | {
"answer": "3(1-3^{-10})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer \( k \) such that \( (k-10)^{5026} \geq 2013^{2013} \). | {
"answer": "55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $F$, $G$, $H$, $I$, and $J$ are positioned in 3-dimensional space with $FG = GH = HI = IJ = JF = 3$ and $\angle FGH = \angle HIJ = \angle JIF = 90^\circ$. The plane of triangle $FGH$ is parallel to $\overline{IJ}$. Determine the area of triangle $GIJ$. | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum $\frac{3}{50} + \frac{5}{500} + \frac{7}{5000}$.
A) $0.0714$
B) $0.00714$
C) $0.714$
D) $0.0357$
E) $0.00143$ | {
"answer": "0.0714",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The positive integers $A, B$, and $C$ form an arithmetic sequence, while the integers $B, C$, and $D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{3},$ what is the smallest possible value of $A + B + C + D$? | {
"answer": "76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x+1) = x^2 - 1$,
(1) Find $f(x)$.
(2) Find the maximum or minimum value of $f(x)$ and the corresponding value of $x$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{8^k : k \text{ is an integer}, 0 \le k \le 3000\} \). Given that \( 8^{3000} \) has 2712 digits and that its first (leftmost) digit is 8, how many elements of \( S \) have 8 as their leftmost digit? | {
"answer": "153",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If real numbers \( x \) and \( y \) satisfy \( x^{3} + y^{3} + 3xy = 1 \), then the minimum value of \( x^{2} + y^{2} \) is ____. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_n\}$ satisfying $a_1=0$, for any $k\in N^*$, $a_{2k-1}$, $a_{2k}$, $a_{2k+1}$ form an arithmetic sequence with a common difference of $k$. If $b_n= \dfrac {(2n+1)^{2}}{a_{2n+1}}$, calculate the sum of the first $10$ terms of the sequence $\{b_n\}$. | {
"answer": "\\dfrac {450}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\frac{ax^{2}+bx+c}{e^{x}} (a > 0)$ whose derivative $y=f′(x)$ has two zeros at $-3$ and $0$.
1. Find the monotonic intervals of $f(x)$;
2. If the minimum value of $f(x)$ is $-e^{3}$, find the maximum value of $f(x)$ on the interval $[-5,+\infty)$. | {
"answer": "5e^{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Starting with $10,000,000$, Esha forms a sequence by alternatively dividing by 2 and multiplying by 3. If she continues this process, what is the form of her sequence after 8 steps? Express your answer in the form $a^b$, where $a$ and $b$ are integers and $a$ is as small as possible. | {
"answer": "(2^3)(3^4)(5^7)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system \(xOy\), there is a point \(P(0, \sqrt{3})\) and a line \(l\) with the parametric equations \(\begin{cases} x = \dfrac{1}{2}t \\ y = \sqrt{3} + \dfrac{\sqrt{3}}{2}t \end{cases}\) (where \(t\) is the parameter). Using the origin as the pole and the non-negative half-axis of \(x\) to establish a polar coordinate system, the polar equation of curve \(C\) is \(\rho^2 = \dfrac{4}{1+\cos^2\theta}\).
(1) Find the general equation of line \(l\) and the Cartesian equation of curve \(C\).
(2) Suppose line \(l\) intersects curve \(C\) at points \(A\) and \(B\). Calculate the value of \(\dfrac{1}{|PA|} + \dfrac{1}{|PB|}\). | {
"answer": "\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the following propositions, the true one is __________
(1) In a plane, the locus of points whose sum of distances from two fixed points $F_{1}$ and $F_{2}$ is a constant is an ellipse;
(2) If vectors $\overrightarrow{e_{1}}$, $\overrightarrow{e_{2}}$, $\overrightarrow{e_{3}}$ are three non-collinear vectors, and $\overrightarrow{a}$ is any vector in space, then there exists a unique set of real numbers $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$ such that $\overrightarrow{a}=\lambda_{1} \overrightarrow{e_{1}}+\lambda_{2} \overrightarrow{e_{2}}+\lambda_{3} \overrightarrow{e_{3}}$;
(3) If proposition $p$ is a sufficient but not necessary condition for proposition $q$, then $\neg p$ is a necessary but not sufficient condition for $\neg q$. | {
"answer": "(3)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles \( C_{1} \) and \( C_{2} \) have their centers at the point \( (3, 4) \) and touch a third circle, \( C_{3} \). The center of \( C_{3} \) is at the point \( (0, 0) \) and its radius is 2. What is the sum of the radii of the two circles \( C_{1} \) and \( C_{2} \)? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Source: 1976 Euclid Part B Problem 1
-----
Triangle $ABC$ has $\angle{B}=30^{\circ}$ , $AB=150$ , and $AC=50\sqrt{3}$ . Determine the length of $BC$ . | {
"answer": "50\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is known that $A=\frac{\pi}{4}$ and $b=\frac{\sqrt{2}}{2}a$.
(Ⅰ) Find the magnitude of $B$;
(Ⅱ) If $a=\sqrt{2}$, find the area of $\Delta ABC$. | {
"answer": "\\frac{\\sqrt{3}+1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive numbers $m$ and $n$ that satisfy $m^2 + n^2 = 100$, find the maximum or minimum value of $m + n$. | {
"answer": "10\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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