problem stringlengths 10 5.15k | answer dict |
|---|---|
Given the function $f(x)=\cos 2x+2 \sqrt {3}\sin x\cos x$,
(1) Find the range of the function $f(x)$ and write out the interval where the function $f(x)$ is strictly increasing;
(2) If $0 < θ < \dfrac {π}{6}$ and $f(θ)= \dfrac {4}{3}$, compute the value of $\cos 2θ$. | {
"answer": "\\dfrac { \\sqrt {15}+2}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \ln x - ax$, where $a \in \mathbb{R}$.
(1) If the line $y = 3x - 1$ is a tangent line to the graph of the function $f(x)$, find the value of the real number $a$.
(2) If the maximum value of the function $f(x)$ on the interval $[1, e^2]$ is $1 - ae$ (where $e$ is the base of the natural logarithm), find the value of the real number $a$. | {
"answer": "\\frac{1}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a tetrahedron \( P-ABC \) with its four vertices on the surface of sphere \( O \), where \( PA = PB = PC \) and \( \triangle ABC \) is an equilateral triangle with side length 2. \( E \) and \( F \) are the midpoints of \( AC \) and \( BC \) respectively, and \( \angle EPF = 60^\circ \). Determine the surface area of sphere \( O \). | {
"answer": "6\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $A$ lies on the line $y = \frac{12}{5} x - 3$, and point $B$ lies on the parabola $y = x^2$. What is the minimum length of the segment $AB$? | {
"answer": "0.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the lengths of the arcs of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\left(t^{2}-2\right) \sin t+2 t \cos t \\
y=\left(2-t^{2}\right) \cos t+2 t \sin t
\end{array}\right. \\
& 0 \leq t \leq 2 \pi
\end{aligned}
$$ | {
"answer": "\\frac{8\\pi^3}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 8 cards; one side of each card is blank, and the other side has the letters И, Я, Л, З, Г, О, О, О written on them. The cards are placed on the table with the blank side up, shuffled, and then turned over one by one in sequence. What is the probability that the letters will appear in the order to form the word "ЗООЛОГИЯ"? | {
"answer": "1/6720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=ax^{2}+bx+c(a,b,c∈R)$, if there exists a real number $a∈[1,2]$, for any $x∈[1,2]$, such that $f(x)≤slant 1$, then the maximum value of $7b+5c$ is _____. | {
"answer": "-6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the sides of a unit square, points \( K, L, M, \) and \( N \) are marked such that line \( KM \) is parallel to two sides of the square, and line \( LN \) is parallel to the other two sides of the square. The segment \( KL \) cuts off a triangle from the square with a perimeter of 1. What is the area of the triangle cut off from the square by the segment \( MN \)? | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many solutions in natural numbers \( x, y \) does the system of equations have
\[
\left\{\begin{array}{l}
\gcd(x, y)=20! \\
\text{lcm}(x, y)=30!
\end{array} \quad (\text{where } n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n) ?\right.
\]
| {
"answer": "256",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the geometric sequence ${a_{n}}$, $a_{n}=9$, $a_{5}=243$, find the sum of the first 4 terms. | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle \( ABC \) with \( AC = 16 \) and \( BC = 12 \), a circle with center at \( B \) and radius \( BC \) is drawn. A tangent to this circle is constructed parallel to the hypotenuse \( AB \) (the tangent and the triangle lie on opposite sides of the hypotenuse). The leg \( BC \) is extended to intersect this tangent. Determine by how much the leg is extended. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
6 * cos(18°) + 2 * cos(36°) + 4 * cos(54°) + ... + 20 * cos(360°) = ? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point moving in the positive direction of the $OX$ axis has its horizontal coordinate given by $x(t) = 5(t + 1)^2 + \frac{a}{(t + 1)^5}$, where $a$ is a positive constant. Find the minimum value of $a$ such that $x(t) \geqslant 24$ for all $t \geqslant 0$. | {
"answer": "2 \\sqrt{\\left( \\frac{24}{7} \\right)^{7}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a $6\times 6\times h$ rectangular box containing a larger sphere of radius $3$ and four smaller spheres, each with radius $2$. The smaller spheres are placed at each corner of the bottom square face of the box and are tangent to two adjacent sides of the box. The larger sphere is tangent to all four smaller spheres. Determine the height $h$ of the box required to accommodate this configuration.
A) $5$
B) $5 + \sqrt{21}$
C) $5 + \sqrt{23}$
D) $5 + \sqrt{25}$
E) $5 + 2\sqrt{23}$ | {
"answer": "5 + \\sqrt{23}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A ball thrown vertically upwards has a height above the ground that is a quadratic function of its travel time. Xiaohong throws two balls vertically upwards at intervals of 1 second. Assuming the height above the ground is the same at the moment of release for both balls, and both balls reach the same maximum height 1.1 seconds after being thrown, find the time $t$ seconds after the first ball is thrown such that the height above the ground of the first ball is equal to the height of the second ball. Determine $t = \qquad$ . | {
"answer": "1.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 5 and 2. Find the length of the common chord, given that the radii of the circles are in the ratio \(4: 3\). | {
"answer": "2\\sqrt{23}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $ABCD$ is a rectangle with $AD = 10$ and the shaded area is $100, calculate the shortest distance between the semicircles. | {
"answer": "2.5 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Dima and Sergey were picking raspberries from a bush that had 900 berries. Dima alternated his actions while picking: he put one berry in the basket, and then he ate the next one. Sergey also alternated: he put two berries in the basket, and then he ate the next one. It is known that Dima picks berries twice as fast as Sergey. At some point, the boys collected all the raspberries from the bush.
Who ended up putting more berries in the basket? What will be the difference? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Pools $A$ and $B$ are both rectangular cuboids with a length of 3 meters, a width of 2 meters, and a depth of 1.2 meters. Valve 1 is used to fill pool $A$ with water and can fill an empty pool $A$ in 18 minutes. Valve 2 is used to transfer water from pool $A$ to pool $B$, taking 24 minutes to transfer a full pool $A$. If both Valve 1 and Valve 2 are opened simultaneously, how many cubic meters of water will be in pool $B$ when the water depth in pool $A$ is 0.4 meters? | {
"answer": "7.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two docks, $A$ and $B$, on a river, where dock $A$ is upstream and dock $B$ is downstream. There are two boats, Boat 1 and Boat 2. The speed of Boat 1 in still water is twice the speed of Boat 2. Both boats start simultaneously from docks $A$ and $B$, respectively, and move towards each other. When Boat 1 departs, it leaves a floating cargo box on the water. After 20 minutes, the two boats meet and Boat 1 leaves another identical cargo box on the water. A while later, Boat 1 notices that it is missing cargo and turns around to search for it. When Boat 1 finds the second cargo box, Boat 2 encounters the first cargo box. Determine how many minutes have passed since Boat 1 departed when it realizes its cargo is missing. (Assume the time for turning around is negligible.) | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(\omega\) denote the incircle of triangle \(ABC\). The segments \(BC, CA\), and \(AB\) are tangent to \(\omega\) at \(D, E\), and \(F\), respectively. Point \(P\) lies on \(EF\) such that segment \(PD\) is perpendicular to \(BC\). The line \(AP\) intersects \(BC\) at \(Q\). The circles \(\omega_1\) and \(\omega_2\) pass through \(B\) and \(C\), respectively, and are tangent to \(AQ\) at \(Q\); the former meets \(AB\) again at \(X\), and the latter meets \(AC\) again at \(Y\). The line \(XY\) intersects \(BC\) at \(Z\). Given that \(AB=15\), \(BC=14\), and \(CA=13\), find \(\lfloor XZ \cdot YZ \rfloor\). | {
"answer": "196",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 16 different cards, including 4 red, 4 yellow, 4 blue, and 4 green cards. If 3 cards are drawn at random, the requirement is that these 3 cards cannot all be of the same color, and at most 1 red card is allowed. The number of different ways to draw the cards is \_\_\_\_\_\_ . (Answer with a number) | {
"answer": "472",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$, the symmetric point $R$ of the left focus $F(-c,0)$ with respect to the line $bx + cy = 0$ is on the ellipse, find the eccentricity of the ellipse. | {
"answer": "\\dfrac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four mice: White, Gray, Fat, and Thin were dividing a cheese wheel. They cut it into 4 apparently identical slices. Some slices had more holes, so Thin's slice weighed 20 grams less than Fat's slice, and White's slice weighed 8 grams less than Gray's slice. However, White wasn't upset because his slice weighed exactly one-quarter of the total cheese weight.
Gray cut 8 grams from his piece, and Fat cut 20 grams from his piece. How should the 28 grams of removed cheese be divided so that all mice end up with equal amounts of cheese? Don't forget to explain your answer. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the side $BC$ of the triangle $ABC$, a point $D$ is chosen such that $\angle BAD = 50^\circ$, $\angle CAD = 20^\circ$, and $AD = BD$. Find $\cos \angle C$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, given the parabola $(E): y^2 = 2px (p > 0)$ with focus $F$, $P$ is an arbitrary point on the parabola $(E)$ in the first quadrant, and $Q$ is a point on the line segment $PF$ such that $\overrightarrow{OQ} = \frac{2}{3} \overrightarrow{OP} + \frac{1}{3} \overrightarrow{OF}$. Determine the maximum slope of the line $OQ$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Car $A$ departs from Station $J$ towards Station $Y$, while cars $B$ and $C$ depart from Station $Y$ towards Station $J$ simultaneously, and move in opposite directions towards car $A$. Car $A$ meets car $B$ first, then 20 minutes later it meets car $C$. Given the speeds of cars $A$, $B$, and $C$ are $90 \text{ km/h}$, $80 \text{ km/h}$, and $60 \text{ km/h}$ respectively, find the distance between stations $J$ and $Y$ in $\text{ km}$. | {
"answer": "425",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Aerith timed herself solving a contest and noted the time both as days:hours:minutes:seconds and in seconds. For example, if she spent 1,000,000 seconds, she recorded it as 11:13:46:40 and 1,000,000 seconds. Bob subtracts these numbers, ignoring punctuation. In this case, he computes:
\[ 11134640 - 1000000 = 10134640 \]
What is the largest number that always must divide his result? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A racer departs from point \( A \) along the highway, maintaining a constant speed of \( a \) km/h. After 30 minutes, a second racer starts from the same point with a constant speed of \( 1.25a \) km/h. How many minutes after the start of the first racer was a third racer sent from the same point, given that the third racer developed a speed of \( 1.5a \) km/h and simultaneously with the second racer caught up with the first racer? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To shift the graph of the function $y=\sin \left(2x- \frac{\pi}{4}\right)$ to the graph of the function $y = \sin(2x)$, determine the horizontal shift required. | {
"answer": "\\frac{\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse C: $$\frac {x^{2}}{4}+y^{2}=1,$$
(1) If a line $l$ passes through point Q $(1, \frac {1}{2})$ and intersects the ellipse C at points A and B, find the equation of line $l$ such that Q is the midpoint of AB;
(2) Given a fixed point M $(0, 2)$, and P is any point on the ellipse C, find the maximum value of the segment PM. | {
"answer": "\\frac {2 \\sqrt {21}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cubical cake with edge length 3 inches is iced on the sides and the top. It is cut vertically into four pieces, such that one of the cut starts at the midpoint of the top edge and ends at a corner on the opposite edge. The piece whose top is triangular contains an area $A$ and is labeled as triangle $C$. Calculate the volume $v$ and the total surface area $a$ of icing covering this triangular piece. Compute $v+a$.
A) $\frac{24}{5}$
B) $\frac{32}{5}$
C) $8+\sqrt{5}$
D) $5+\frac{16\sqrt{5}}{5}$
E) $22.5$ | {
"answer": "22.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=2t+1,\\ y=2t\end{array}\right.$ (where $t$ is a parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C$ is $\rho ^{2}-4\rho \sin \theta +3=0$. <br/>$(1)$ Find the rectangular coordinate equation of the line $l$ and the general equation of the curve $C$; <br/>$(2)$ A tangent line to the curve $C$ passes through a point $A$ on the line $l$, and the point of tangency is $B$. Find the minimum value of $|AB|$. | {
"answer": "\\frac{\\sqrt{14}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\[ y = x + \cos(2x) \] in the interval \((0, \pi / 4)\). | {
"answer": "\\frac{\\pi}{12} + \\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with a radius of 6 cm is tangent to three sides of a rectangle. The area of the circle is half the area of the rectangle. What is the length of the longer side of the rectangle, in centimeters? Express your answer in terms of $\pi$. | {
"answer": "6\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, it is known that $AB = 14$, $BC = 6$, and $AC = 10$. The angle bisectors $BD$ and $CE$ intersect at point $O$. Find $OD$. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a\) be a positive real number. Find the value of \(a\) such that the definite integral
\[
\int_{a}^{a^2} \frac{\mathrm{d} x}{x+\sqrt{x}}
\]
achieves its smallest possible value. | {
"answer": "3 - 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a math competition consisting of problems $A$, $B$, and $C$, among the 39 participants, each person answered at least one problem correctly. Among the people who answered $A$ correctly, the number of people who only answered $A$ is 5 more than the number of people who answered other problems as well. Among the people who did not answer $A$ correctly, the number of people who answered $B$ is twice the number of people who answered $C$. Additionally, the number of people who only answered $A$ equals the sum of the number of people who only answered $B$ and the number of people who only answered $C$. What is the maximum number of people who answered $A$ correctly? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When the product of $\underbrace{\frac{2}{3} \times \frac{2}{3} \times \cdots \times \frac{2}{3}}_{10}$ is written as a decimal, what are the first two decimal places? | {
"answer": "0.01",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a convex 1950-sided polygon, all the diagonals are drawn, dividing it into smaller polygons. Consider the polygon with the greatest number of sides among these smaller polygons. What is the maximum number of sides it can have? | {
"answer": "1949",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many 7-digit numbers divisible by 9 are there, where the second to last digit is 5? | {
"answer": "100000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p$, $q$, $r$, and $s$ be the roots of $x^4 - 24x^3 + 50x^2 - 26x + 7 = 0$. Compute \[(p+q)^2 + (q+r)^2 + (r+s)^2 + (s+p)^2 + (p+r)^2 + (q+s)^2.\] | {
"answer": "1052",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, sides a, b, and c are opposite to angles A, B, and C, respectively. Given $\vec{m} = (a-b, c)$ and $\vec{n} = (a-c, a+b)$, and that $\vec{m}$ and $\vec{n}$ are collinear, find the value of $2\sin(\pi+B) - 4\cos(-B)$. | {
"answer": "-\\sqrt{3} - 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a square with a side length of $1$, if two points are randomly selected with equal probability from the center and the vertices of the square, what is the probability that the distance between these two points is $\frac{\sqrt{2}}{2}$? | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Starting with the display "1," calculate the fewest number of keystrokes needed to reach "240" using only the keys [+1] and [x2]. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g$ be a function defined by $g(x) = \frac{px + q}{rx + s}$, where $p$, $q$, $r$ and $s$ are nonzero real numbers, and the function has the properties $g(31)=31$, $g(41)=41$, and $g(g(x))=x$ for all values except $\frac{-s}{r}$. Determine the unique number that is not in the range of $g$. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x \neq y \), and suppose the two sequences \( x, a_{1}, a_{2}, a_{3}, y \) and \( b_{1}, x, b_{2}, b_{3}, y, b_{1} \) are both arithmetic sequences. Determine the value of \( \frac{b_{4}-b_{3}}{a_{2}-a_{1}} \). | {
"answer": "8/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( p \) and \( q \) are prime numbers, the number of divisors \( d(a) \) of a natural number \( a = p^{\alpha} q^{\beta} \) is given by the formula
$$
d(a) = (\alpha+1)(\beta+1).
$$
For example, \( 12 = 2^2 \times 3^1 \), the number of divisors of 12 is
$$
d(12) = (2+1)(1+1) = 6,
$$
and the divisors are \( 1, 2, 3, 4, 6, \) and \( 12 \).
Using the given calculation formula, answer: Among the divisors of \( 20^{30} \) that are less than \( 20^{15} \), how many are not divisors of \( 20^{15} \)? | {
"answer": "450",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive integers \( a, b, c, \) and \( d \) such that \( a > b > c > d \) and \( a + b + c + d = 2004 \), as well as \( a^2 - b^2 + c^2 - d^2 = 2004 \), what is the minimum value of \( a \)? | {
"answer": "503",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $e = \dfrac{\sqrt{6}}{3}$, and the distance between the left focus and one endpoint of the minor axis is $\sqrt{3}$.
$(I)$ Find the standard equation of the ellipse;
$(II)$ Given the fixed point $E(-1, 0)$, if the line $y = kx + 2$ intersects the ellipse at points $A$ and $B$. Is there a real number $k$ such that the circle with diameter $AB$ passes through point $E$? Please explain your reasoning. | {
"answer": "\\dfrac{7}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( F_{1} \) and \( F_{2} \) be the two foci of an ellipse. A circle with center \( F_{2} \) is drawn, which passes through the center of the ellipse and intersects the ellipse at point \( M \). If the line \( ME_{1} \) is tangent to circle \( F_{2} \) at point \( M \), find the eccentricity \( e \) of the ellipse. | {
"answer": "\\sqrt{3}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $C:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ with its right vertex at point $A$, a circle $A$ is created with center at $A$ and radius $b$. Circle $A$ intersects with one of the asymptotes of hyperbola $C$ at points $M$ and $N$. If $\angle MAN = 60^{\circ}$, find the eccentricity of hyperbola $C$. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular box has a total surface area of 166 square inches and the sum of the lengths of all its edges is 64 inches. Find the sum of the lengths in inches of all of its interior diagonals. | {
"answer": "12\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)=\cos x+\cos (x+ \frac {π}{2}).$
(1) Find $f( \frac {π}{12})$;
(2) Suppose $α$ and $β∈(- \frac {π}{2},0)$, $f(α+ \frac {3π}{4})=- \frac {3 \sqrt {2}}{5}$, $f( \frac {π}{4}-β)=- \frac {5 \sqrt {2}}{13}$, find $\cos (α+β)$. | {
"answer": "\\frac {16}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point \( D \) is marked on the altitude \( BH \) of triangle \( ABC \). Line \( AD \) intersects side \( BC \) at point \( E \), and line \( CD \) intersects side \( AB \) at point \( F \). It is known that \( BH \) divides segment \( FE \) in the ratio \( 1:3 \), starting from point \( F \). Find the ratio \( FH:HE \). | {
"answer": "1:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCDE$ be a convex pentagon, and let $H_A,$ $H_B,$ $H_C,$ $H_D$ denote the centroids of triangles $BCD,$ $ACE,$ $ABD,$ and $ABC,$ respectively. Determine the ratio $\frac{[H_A H_B H_C H_D]}{[ABCDE]}.$ | {
"answer": "\\frac{1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cross sections of a right octagonal pyramid are obtained by cutting the pyramid with planes parallel to the octagonal base. The areas of the cross sections are $300\sqrt{2}$ square feet and $675\sqrt{2}$ square feet. The two planes are $10$ feet apart. How far from the apex of the pyramid is the larger cross section, in feet? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a > 0$, $b > 0$, and $2a+b=1$, find the maximum value of $2 \sqrt {ab}-4a^{2}-b^{2}$. | {
"answer": "\\dfrac { \\sqrt {2}-1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\triangle ABC$ is an acute triangle, vector $\overrightarrow{m} = (\cos(A + \frac{\pi}{3}), \sin(A + \frac{\pi}{3}))$, $\overrightarrow{n} = (\cos B, \sin B)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(Ⅰ) Find the value of $A-B$;
(Ⅱ) If $\cos B = \frac{3}{5}$ and $AC = 8$, find the length of $BC$. | {
"answer": "4\\sqrt{3} + 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the three sides $a, b, c$ form an arithmetic sequence, and $\angle A = 3 \angle C$. Find $\cos \angle C$. | {
"answer": "\\frac{1 + \\sqrt{33}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is randomly thrown on the segment [3, 8] and let \( k \) be the resulting value. Find the probability that the roots of the equation \((k^{2}-2k-3)x^{2}+(3k-5)x+2=0\) satisfy the condition \( x_{1} \leq 2x_{2} \). | {
"answer": "4/15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A B C D \) be a quadrilateral and \( P \) the intersection of \( (A C) \) and \( (B D) \). Assume that \( \widehat{C A D} = 50^\circ \), \( \widehat{B A C} = 70^\circ \), \( \widehat{D C A} = 40^\circ \), and \( \widehat{A C B} = 20^\circ \). Calculate the angle \( \widehat{C P D} \). | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers are in the list starting from $-48$, increasing by $7$ each time, up to and including $119$? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Katya is passing time while her parents are at work. On a piece of paper, she absentmindedly drew Cheburashkas in two rows (at least one Cheburashka was drawn in each row).
Afterwards, she drew a Crocodile Gena between every two adjacent Cheburashkas in both rows. Then she drew an Old Lady Shapoklyak to the left of each Cheburashka. Finally, she drew a Krakozyabra between each pair of characters in the rows.
Upon closer inspection of the drawing, she realized that only the Krakozyabras turned out nicely, so she angrily erased all the other characters. In the end, her parents saw two rows of Krakozyabras: a total of 29.
How many Cheburashkas were erased? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiaoming is riding a bicycle, while Xiaoming's father is walking. They start from locations $A$ and $B$ respectively, moving towards each other. After meeting, Xiaoming continues for another 18 minutes to reach $B$. It is known that Xiaoming's cycling speed is 4 times that of his father's walking speed, and it takes Xiaoming's father a certain number of minutes to walk from the meeting point to $A$. How long does Xiaoming's father need to walk from the meeting point to $A$? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse with a chord passing through the focus and perpendicular to the major axis of length $\sqrt{2}$, and a distance from the focus to the corresponding directrix of $1$, determine the eccentricity of the ellipse. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On every kilometer marker along the highway between the villages of Yolkino and Palkino, there is a post with a sign. On one side of the sign, the distance to Yolkino is indicated, and on the other side, the distance to Palkino is indicated. Borya noticed that the sum of all the digits on each sign equals 13. What is the distance from Yolkino to Palkino? | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An equilateral triangle is placed on top of a square with each side of the square equal to one side of the triangle, forming a pentagon. What percent of the area of the pentagon is the area of the equilateral triangle? | {
"answer": "\\frac{4\\sqrt{3} - 3}{13} \\times 100\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle has its center at $(2,0)$ with a radius of 2, and another circle has its center at $(5,0)$ with a radius of 1. A line is tangent to both circles in the first quadrant. The $y$-intercept of this line is closest to: | {
"answer": "$2 \\sqrt{2}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two lines passing through point \( M \), which lies outside the circle with center \( O \), touch the circle at points \( A \) and \( B \). Segment \( OM \) is divided in half by the circle. In what ratio is segment \( OM \) divided by line \( AB \)? | {
"answer": "1:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A = \{1, 2, \cdots, 2004\} \) and \( f: A \rightarrow A \) be a bijection satisfying \( f^{[2004]}(x) = f(x) \), where \( f^{[2004]}(x) \) denotes applying \( f \) 2004 times to \( x \). How many such functions \( f \) are there? | {
"answer": "1 + 2004!",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Billy Bones has two coins - one gold and one silver. One of them is symmetrical, and the other is not. It is unknown which coin is asymmetrical, but it is known that the asymmetrical coin lands heads with a probability of $p=0.6$.
Billy Bones tossed the gold coin, and it landed heads immediately. Then Billy Bones started tossing the silver coin, and heads appeared only on the second toss. Find the probability that the gold coin is the asymmetrical one. | {
"answer": "5/9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the expression \((\sqrt{2(1+\cos 2x)} - \sqrt{36 - 4\sqrt{5}} \sin x + 2) \cdot (3 + 2\sqrt{10 - \sqrt{5}} \cos y - \cos 2y)\). If the answer is not an integer, round it to the nearest whole number. | {
"answer": "-27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cone has base radius 1. The vertex is K. P is a point on the circumference of the base. The distance KP is 3. A particle travels from P around the cone and back by the shortest route. What is its minimum distance from K? | {
"answer": "1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ,$ $XY = 4,$ $XZ = 5,$ $YZ = 7,$ and $W$ lies on $\overline{YZ}$ such that $\overline{XW}$ bisects $\angle YXZ.$ Find $\cos \angle YXW.$ | {
"answer": "\\sqrt{\\frac{2}{5}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a triangle, one of the angles is less than $50^{\circ}$, and another is less than $70^{\circ}$. Find the cosine of the third angle if its sine is $\frac{4}{7}$. | {
"answer": "-\\frac{\\sqrt{33}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The car engine operates with a power of \( P = 60 \text{ kW} \). Determine the car's speed \( v_0 \) if, after turning off the engine, it stops after traveling a distance of \( s = 450 \text{ m} \). The force resisting the car's motion is proportional to its speed. The mass of the car is \( m = 1000 \text{ kg} \). | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the tetrahedron \(ABCD\), \(CD \perp BC\), \(AB \perp BC\), \(CD = AC\), \(AB = BC = 1\). The dihedral angle between the planes \(BCD\) and \(ABC\) is \(45^\circ\). Find the distance from point \(B\) to the plane \(ACD\). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$3-i(i)$ is a root of the equation $x^{2}+px+10=0(p∈R)$ with respect to $x$. Find the value of $p$. | {
"answer": "-6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A said: "I am 10 years old, 2 years younger than B, and 1 year older than C."
B said: "I am not the youngest, C and I have a 3-year difference, and C is 13 years old."
C said: "I am younger than A, A is 11 years old, and B is 3 years older than A."
Among the three statements made by each person, one of them is incorrect. Please determine A's age. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( ABC \), the sides \( AC = 14 \) and \( AB = 6 \) are known. A circle with center \( O \), constructed on side \( AC \) as the diameter, intersects side \( BC \) at point \( K \). It turns out that \( \angle BAK = \angle ACB \). Find the area of triangle \( BOC \). | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The eccentricity of the hyperbola defined by the equation $\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ given that a line with a slope of -1 passes through its right vertex A and intersects the two asymptotes of the hyperbola at points B and C, and if $\overrightarrow {AB}= \frac {1}{2} \overrightarrow {BC}$, determine the eccentricity of this hyperbola. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The first row of a triangle is given as:
$$
1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{1993}
$$
Each element of the following rows is calculated as the difference between two elements that are above it. The 1993rd row contains only one element. Find this element. | {
"answer": "\\frac{1}{1993}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the two-dimensional rectangular coordinate system, given the vector $\overrightarrow{a}=(-1,2)$, and points $A(8,0)$, $B(n,t)$, $C(k\sin θ,t)(0≤θ≤\frac {π}{2})$.
(1) If $\overrightarrow{AB} \perp \overrightarrow{a}$, and $|\overrightarrow{AB}|= \sqrt {5}|\overrightarrow{OA}|(O$ is the origin$)$, find the vector $\overrightarrow{OB}$;
(2) If the vector $\overrightarrow{AC}$ is collinear with the vector $\overrightarrow{a}$, when $k > 4$, and $t\sin θ$ takes the maximum value $4$, find $\overrightarrow{OA}\cdot \overrightarrow{OC}$. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the definite integral:
$$
\int_{0}^{2 \operatorname{arctg} \frac{1}{2}} \frac{(1-\sin x) dx}{\cos x(1+\cos x)}
$$ | {
"answer": "2 \\ln \\frac{3}{2} - \\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A normal line (a line that passes through a point of tangency and is perpendicular to the tangent line) is drawn to the parabola \( y = x^2 \) at point \( A \). The normal line intersects the parabola at another point \( B \). Let \( O \) be the origin of coordinates. When the area of triangle \( \triangle OAB \) is minimized, what is the y-coordinate of point \( A \)? | {
"answer": "\\frac{-3+\\sqrt{33}}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=4-\frac{{\sqrt{2}}}{2}t}\\{y=4+\frac{{\sqrt{2}}}{2}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive x-axis as the polar axis. The polar coordinate equation of curve $C$ is $\rho =8\sin \theta $, and $A$ is a point on curve $C$.
$(1)$ Find the maximum distance from $A$ to the line $l$;
$(2)$ If point $B$ is the intersection point of line $l$ and curve $C$ in the first quadrant, and $∠AOB=\frac{{7π}}{{12}}$, find the area of $\triangle AOB$. | {
"answer": "4 + 4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An environmental agency decides to expand its monitoring teams due to new regulations requiring more extensive testing. They estimate needing 120 new employees to monitor water pollution and 105 new employees to monitor air pollution. Additionally, they need 65 new employees capable of monitoring air and water pollution. On top of this, there should be another team where 40 of the new employees will also monitor soil pollution (including taking roles in air and water tasks). Determine the minimum number of new employees the agency must hire. | {
"answer": "160",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $d$ be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than $d$ degrees. What is the minimum possible value for $d$ ? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\omega$ be the unit circle centered at the origin of $R^2$ . Determine the largest possible value for the radius of the circle inscribed to the triangle $OAP$ where $ P$ lies the circle and $A$ is the projection of $P$ on the axis $OX$ . | {
"answer": "\\frac{\\sqrt{2} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A \) and \( B \) be two sets, and \((A, B)\) be called a "pair". If \( A \neq B \), then \((A, B)\) and \((B, A)\) are considered different "pairs". Find the number of different pairs \((A, B)\) that satisfy the condition \( A \cup B = \{1,2,3,4\} \). | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( d = \overline{xyz} \) be a three-digit number that cannot be divisible by 10. If the sum of \( \overline{xyz} \) and \( \overline{zyx} \) is divisible by \( c \), find the largest possible value of this integer \( d \). | {
"answer": "979",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid $PQRS$, the parallel sides $PQ$ and $RS$ have lengths of 10 and 30 units, respectively, and the altitude is 18 units. Points $T$ and $U$ are the midpoints of sides $PR$ and $QS$, respectively. What is the area of quadrilateral $TURS$? | {
"answer": "225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
With all angles measured in degrees, consider the product $\prod_{k=1}^{22} \sec^2(4k)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Density is defined as the ratio of mass to volume. There are two cubes. The second cube is made from a material with twice the density of the first, and the side length of the second cube is 100% greater than the side length of the first. By what percentage is the mass of the second cube greater than the mass of the first? | {
"answer": "1500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mário completed 30 hours in an extra math course. On days when he had class, it lasted only 1 hour, and it took place exclusively in the morning or in the afternoon. Additionally, there were 20 afternoons and 18 mornings without classes during the course period.
a) On how many days were there no classes?
b) How many days did the course last? | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equation \( x^{[x]} = \frac{9}{2} \) for real numbers \( x \), where \( [x] \) represents the greatest integer less than or equal to \( x \). | {
"answer": "\\frac{3\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the planar vectors $\overrightarrow {a}$ and $\overrightarrow {b}$, with $|\overrightarrow {a}|=1$, $|\overrightarrow {b}|=2$, and $\overrightarrow {a} \cdot \overrightarrow {b} = 1$, find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sqrt{3}\sin{2x}-\cos{2x}$.
$(1)$ Find the smallest positive period and the interval of monotonic increase of the function $f(x)$.
$(2)$ In triangle $ABC$, where the internal angles $A$, $B$, $C$ are opposite to sides $a$, $b$, $c$ respectively, if $f\left(\frac{A}{2}\right)=2$, $b=1$, $c=2$, find the value of $a$. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the perimeter of triangle \( \triangle ABC \) is 20, the radius of the inscribed circle is \( \sqrt{3} \), and \( BC = 7 \). Find the value of \( \tan A \). | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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