problem stringlengths 10 5.15k | answer dict |
|---|---|
Given vectors $\overrightarrow{m}=( \sqrt {3}\sin x-\cos x,1)$ and $\overrightarrow{n}=(\cos x, \frac {1}{2})$, and the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$,
(1) Find the interval(s) where the function $f(x)$ is monotonically increasing;
(2) If $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ of $\triangle ABC$, $a=2 \sqrt {3}$, $c=4$, and $f(A)=1$, find the area of $\triangle ABC$. | {
"answer": "2 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain middle school, 500 eighth-grade students took the biology and geography exam. There were a total of 180 students who scored between 80 and 100 points. What is the frequency of this score range? | {
"answer": "0.36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two particles move along the sides of a right $\triangle ABC$ with $\angle B = 90^\circ$ in the direction \[A\Rightarrow B\Rightarrow C\Rightarrow A,\] starting simultaneously. One starts at $A$ moving at speed $v$, the other starts at $C$ moving at speed $2v$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $R$. What is the ratio of the area of $R$ to the area of $\triangle ABC$?
A) $\frac{1}{16}$
B) $\frac{1}{12}$
C) $\frac{1}{4}$
D) $\frac{1}{2}$ | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $\left\{a_{n}\right\}$ consists of 9 terms, where $a_{1} = a_{9} = 1$, and for each $i \in \{1,2, \cdots, 8\}$, we have $\frac{a_{i+1}}{a_{i}} \in \left\{2,1,-\frac{1}{2}\right\}$. Find the number of such sequences. | {
"answer": "491",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the positive real numbers \(a_{1}, a_{2}, a_{3}, a_{4}\) satisfy the conditions \(a_{1} \geqslant a_{2} a_{3}^{2}, a_{2} \geqslant a_{3} a_{4}^{2}, a_{3} \geqslant a_{4} a_{1}^{2}, a_{4} \geqslant a_{1} a_{2}^{2}\), find the maximum value of \(a_{1} a_{2} a_{3} a_{4}\left(a_{1}-a_{2} a_{3}^{2}\right)\left(a_{2}-a_{3} a_{4}^{2}\right)\left(a_{3}-a_{4} a_{1}^{2}\right)\left(a_{4}-a_{1} a_{2}^{2}\right)\). | {
"answer": "1/256",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a magical tree with 58 fruits. On the first day, 1 fruit falls from the tree. From the second day onwards, the number of fruits falling each day increases by 1 compared to the previous day. However, if on any given day the number of fruits on the tree is less than the number of fruits that should fall on that day, then the tree restarts by dropping 1 fruit and continues this new sequence. Given this process, on which day will all the fruits have fallen from the tree? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the side face $A A^{\prime} B^{\prime} B$ of a unit cube $A B C D - A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, there is a point $M$ such that its distances to the two lines $A B$ and $B^{\prime} C^{\prime}$ are equal. What is the minimum distance from a point on the trajectory of $M$ to $C^{\prime}$? | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the lengths of the arcs of the curves given by the equations in polar coordinates.
$$
\rho = 2 \varphi, \; 0 \leq \varphi \leq \frac{4}{3}
$$ | {
"answer": "\\frac{20}{9} + \\ln 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $$\frac{\tan 7.5^\circ \cdot \tan 15^\circ}{\tan 15^\circ - \tan 7.5^\circ}$$ + $$\sqrt{3}(\sin^2 7.5^\circ - \cos^2 7.5^\circ)$$. | {
"answer": "-\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Circles $A$ and $B$ each have a radius of 1 and are tangent to each other. Circle $C$ has a radius of 2 and is tangent to the midpoint of $\overline{AB}.$ What is the area inside circle $C$ but outside circle $A$ and circle $B?$
A) $1.16$
B) $3 \pi - 2.456$
C) $4 \pi - 4.912$
D) $2 \pi$
E) $\pi + 4.912$ | {
"answer": "4 \\pi - 4.912",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a=1$, $B=45^{\circ}$, $S_{\triangle ABC}=2$, find the diameter of the circumcircle of $\triangle ABC$. | {
"answer": "5 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, angle $B$ equals $120^\circ$, and $AB = 2 BC$. The perpendicular bisector of side $AB$ intersects $AC$ at point $D$. Find the ratio $CD: DA$. | {
"answer": "3:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $(x_n)$ is defined by $x_1 = 150$ and $x_k = x_{k - 1}^2 - x_{k - 1}$ for all $k \ge 2.$ Compute
\[\frac{1}{x_1 + 1} + \frac{1}{x_2 + 1} + \frac{1}{x_3 + 1} + \dots.\] | {
"answer": "\\frac{1}{150}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $a=(\cos α, \sin α)$ and $b=(\cos β, \sin β)$, with $|a-b|= \frac{2 \sqrt{5}}{5}$, find the value of $\cos (α-β)$.
(2) Suppose $α∈(0,\frac{π}{2})$, $β∈(-\frac{π}{2},0)$, and $\cos (\frac{5π}{2}-β) = -\frac{5}{13}$, find the value of $\sin α$. | {
"answer": "\\frac{33}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence ${a_n}$, where $a_1=1$, and $P(a_n,a_{n+1})(n∈N^{+})$ is on the line $x-y+1=0$. If the function $f(n)= \frac {1}{n+a_{1}}+ \frac {1}{n+a_{2}}+ \frac {1}{n+a_{3}}+…+ \frac {1}{n+a_{n}}(n∈N^{\*})$, and $n\geqslant 2$, find the minimum value of the function $f(n)$. | {
"answer": "\\frac {7}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \sqrt {x^{2}-4x+4}-|x-1|$:
1. Solve the inequality $f(x) > \frac {1}{2}$;
2. If positive numbers $a$, $b$, $c$ satisfy $a+2b+4c=f(\frac {1}{2})+2$, find the minimum value of $\sqrt { \frac {1}{a}+ \frac {2}{b}+ \frac {4}{c}}$. | {
"answer": "\\frac {7}{3} \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( a > b \), the quadratic inequality \( ax^{2}+2x+b \geqslant 0 \) holds for all real numbers \( x \), and there exists \( x_{0} \in \mathbb{R} \) such that \( ax_{0}^{2}+2x_{0}+b=0 \) is satisfied. Find the minimum value of \( 2a^{2}+b^{2} \). | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute: $\frac{\cos 10^{\circ} - 2\sin 20^{\circ}}{\sin 10^{\circ}} = \_\_\_\_\_\_ \text{.}$ | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Given $\cos (α+ \frac {π}{6})- \sin α= \frac {3 \sqrt {3}}{5}$, find the value of $\sin (α+ \frac {5π}{6})$;
(2) Given $\sin α+ \sin β= \frac {1}{2}, \cos α+ \cos β= \frac { \sqrt {2}}{2}$, find the value of $\cos (α-β)$. | {
"answer": "-\\frac {5}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fill in the table with the numbers $0, 1, 2, \cdots, 14, 15$ so that for each row and each column, the remainders when divided by 4 are exactly $0, 1, 2, 3$ each, and the quotients when divided by 4 are also exactly $0, 1, 2, 3$ each, and determine the product of the four numbers in the bottom row of the table. | {
"answer": "32760",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If you set the clock back by 10 minutes, the number of radians the minute hand has turned is \_\_\_\_\_\_. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the polar coordinate equation of curve $C\_1$ is $ρ=2\sin θ$, and the polar coordinate equation of curve $C\_2$ is $θ =\dfrac{π }{3}(ρ \in R)$, curves $C\_1$ and $C\_2$ intersect at points $M$ and $N$. The length of chord $MN$ is _______. | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the principal (smallest positive) period of the function
$$
y=(\arcsin (\sin (\arccos (\cos 3 x))))^{-5}
$$ | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $(xOy)$, the parametric equations of the curve $C$ are given by $\begin{cases} x=3\cos \alpha \\ y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of the line $l$ is given by $\rho \sin \left( \theta -\dfrac{\pi }{4} \right)=\sqrt{2}$.
(1) Find the Cartesian equation of $C$ and the angle of inclination of $l$;
(2) Let $P$ be the point $(0,2)$, and suppose $l$ intersects $C$ at points $A$ and $B$. Find $|PA|+|PB|$. | {
"answer": "\\dfrac{18\\sqrt{2}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function f(x) = 2x^3 - ax^2 + 1, where a ∈ R.
(I) When a = 6, the line y = -6x + m is tangent to f(x). Find the value of m.
(II) If the function f(x) has exactly one zero in the interval (0, +∞), find the monotonic intervals of the function.
(III) When a > 0, if the sum of the maximum and minimum values of the function f(x) on the interval [-1, 1] is 1, find the value of the real number a. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest integer $n$ such that $2007^{1024}-1$ is divisible by $2^n$. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all real numbers \( x \) that satisfy the equation
\[
\frac{x-2020}{1}+\frac{x-2019}{2}+\cdots+\frac{x-2000}{21}=\frac{x-1}{2020}+\frac{x-2}{2019}+\cdots+\frac{x-21}{2000},
\]
and simplify your answer(s) as much as possible. Justify your solution. | {
"answer": "2021",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, if the area of $\triangle ABC$ is 36 where $A(3, 15)$, $B(15, 0)$, and $C(0, q)$ lie on a Cartesian plane. Determine the value of $q$.
[asy]
size(5cm);defaultpen(fontsize(9));
pair a = (3, 15); pair b = (15, 0); pair c = (0, 12);pair d= (3, 0);
draw(a--b--c--cycle);
label("$A(3, 15)$", a, N);
label("$B(15, 0)$", b, S);
label("$C(0, q)$", c, W);
label("$x$", (17, 0), E);
label("$y$", (0, 17), N);
draw((-2,0)--(17,0), Arrow);
draw((0,-2)--(0,17), Arrow);
[/asy] | {
"answer": "12.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle $ABC$ (right angle at $C$), the bisector $BK$ is drawn. Point $L$ is on side $BC$ such that $\angle C K L = \angle A B C / 2$. Find $KB$ if $AB = 18$ and $BL = 8$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the quadrilateral pyramid \( P-ABCD \), \( BC \parallel AD \), \( AD \perp AB \), \( AB=2\sqrt{3} \), \( AD=6 \), \( BC=4 \), \( PA = PB = PD = 4\sqrt{3} \). Find the surface area of the circumscribed sphere of the triangular pyramid \( P-BCD \). | {
"answer": "80\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\(\cos \frac{\pi}{15} - \cos \frac{2\pi}{15} - \cos \frac{4\pi}{15} + \cos \frac{7\pi}{15} =\) | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $\sin \frac{\pi}{7} \sin \frac{2\pi}{7} \sin \frac{3\pi}{7}$. | {
"answer": "\\frac{\\sqrt{7}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product of two positive integers plus their sum is 119. The integers are relatively prime and each is less than 30. What is the sum of the two integers? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a cube PQRSTUVW with a side length s. Let M and N be the midpoints of edges PU and RW, and let K be the midpoint of QT. Find the ratio of the area of triangle MNK to the area of one of the faces of the cube. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two integers \( m \) and \( n \) which are coprime, calculate the GCD of \( 5^m + 7^m \) and \( 5^n + 7^n \). | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cut a 12cm long thin iron wire into three segments with lengths a, b, and c,
(1) Find the maximum volume of the rectangular solid with lengths a, b, and c as its dimensions;
(2) If these three segments each form an equilateral triangle, find the minimum sum of the areas of these three equilateral triangles. | {
"answer": "\\frac {4 \\sqrt {3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the probability, expressed as a decimal, of drawing one marble which is either green or white from a bag containing 4 green, 3 white, and 8 black marbles? | {
"answer": "0.4667",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $$f(x)=\sin^{2}x+ \sqrt {3}\sin x\cos x+2\cos^{2}x，x∈R$$.
(I) Find the smallest positive period and the interval where the function is monotonically increasing;
(II) Find the maximum value of the function on the interval $$[- \frac {π}{3}， \frac {π}{12}]$$. | {
"answer": "\\frac { \\sqrt {3}+3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Choose one of the following three conditions:①$a_{2}=60$, ②the sum of binomial coefficients is $64$, ③the maximum term of the binomial coefficients is the $4$th term. Fill in the blank below. Given ${(1-2x)}^{n}={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+…+{a}_{n}{x}^{n}(n∈{N}_{+})$,_____, find:<br/>$(1)$ the value of $n$;<br/>$(2)$ the value of $-\frac{{a}_{1}}{2}+\frac{{a}_{2}}{{2}^{2}}-\frac{{a}_{3}}{{2}^{3}}+…+(-1)^{n}\frac{{a}_{n}}{{2}^{n}}$. | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two parallel chords of a circle have lengths 24 and 32 respectively, and the distance between them is 14. What is the length of another parallel chord midway between the two chords? | {
"answer": "2\\sqrt{249}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are three kinds of saltwater solutions: A, B, and C, with concentrations of 5%, 8%, and 9% respectively, and their weights are 60 grams, 60 grams, and 47 grams. Now, we want to prepare 100 grams of 7% saltwater solution. What is the maximum and minimum amount of solution A that can be used? | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A total of $960$ people are randomly numbered from $1$ to $960$. Using systematic sampling, $32$ people are selected for a survey. Find the number of people to be selected from those with numbers falling within $[450,750]$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, draw one at random. The probability of drawing a prime number is ____, and the probability of drawing a composite number is ____. | {
"answer": "\\frac{4}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$ are distinct positive integers whose sum equals 159, find the maximum value of the smallest number $a_1$. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 4 different points \( A, B, C, D \) on two non-perpendicular skew lines \( a \) and \( b \), where \( A \in a \), \( B \in a \), \( C \in b \), and \( D \in b \). Consider the following two propositions:
(1) Line \( AC \) and line \( BD \) are always skew lines.
(2) Points \( A, B, C, D \) can never be the four vertices of a regular tetrahedron.
Which of the following is correct? | {
"answer": "(1)(2)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An Ultraman is fighting a group of monsters. It is known that Ultraman has one head and two legs. Initially, each monster has two heads and five legs. During the battle, some monsters split, with each splitting monster creating two new monsters, each with one head and six legs (they cannot split again). At a certain moment, there are 21 heads and 73 legs on the battlefield. How many monsters are there at this moment? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each term of a geometric sequence $\left\{a_{n}\right\}$ is a real number, and the sum of the first $n$ terms is $S_{n}$. If $S_{10} = 10$ and $S_{30} = 70$, then find $S_{4n}$. | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice and Bob are playing a game where Alice declares, "My number is 36." Bob has to choose a number such that all the prime factors of Alice's number are also prime factors of his, but with the condition that the exponent of at least one prime factor in Bob's number is strictly greater than in Alice's. What is the smallest possible number Bob can choose? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $sinα + cosα = \frac{\sqrt{2}}{3}, α ∈ (0, π)$, find the value of $sin(α + \frac{π}{12})$. | {
"answer": "\\frac{2\\sqrt{2} + \\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCD$ is a trapezium such that $\angle ADC=\angle BCD=60^{\circ}$ and $AB=BC=AD=\frac{1}{2}CD$. If this trapezium is divided into $P$ equal portions $(P>1)$ and each portion is similar to trapezium $ABCD$ itself, find the minimum value of $P$.
The sum of tens and unit digits of $(P+1)^{2001}$ is $Q$. Find the value of $Q$.
If $\sin 30^{\circ}+\sin ^{2} 30^{\circ}+\ldots+\sin Q 30^{\circ}=1-\cos ^{R} 45^{\circ}$, find the value of $R$.
Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}-8x+(R+1)=0$. If $\frac{1}{\alpha^{2}}$ and $\frac{1}{\beta^{2}}$ are the roots of the equation $225x^{2}-Sx+1=0$, find the value of $S$. | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest divisor by which $29 \cdot 14$ leaves the same remainder when divided by $13511, 13903,$ and $14589$? | {
"answer": "98",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose the curve C has the polar coordinate equation $ρ\sin^2θ - 8\cos θ = 0$. Establish a rectangular coordinate system $xoy$ with the pole as the origin and the non-negative semi-axis of the polar axis as the $x$-axis. A line $l$, with an inclination angle of $α$, passes through point $P(2, 0)$.
(1) Write the rectangular coordinate equation of curve C and the parametric equation of line $l$.
(2) Suppose points $Q$ and $G$ have polar coordinates $\left(2, \dfrac{3π}{2}\right)$ and $\left(2, π\right)$, respectively. If line $l$ passes through point $Q$ and intersects curve $C$ at points $A$ and $B$, find the area of triangle $GAB$. | {
"answer": "16\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a shelf, there are 4 different comic books, 5 different fairy tale books, and 3 different story books, all lined up in a row. If the fairy tale books cannot be separated from each other, and the comic books also cannot be separated from each other, how many different arrangements are there? | {
"answer": "345600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A box contains 5 white balls and 5 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors, starting and ending with the same color? | {
"answer": "\\frac{1}{126}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Pigsy picked a bag of wild fruits from the mountain to bring back to Tang Monk. As he walked, he got hungry and took out half of them, hesitated, put back two, and then ate the rest. This happened four times: taking out half, putting back two, and then eating. By the time he reached Tang Monk, there were 5 wild fruits left in the bag. How many wild fruits did Pigsy originally pick? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In acute triangle $\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, $c$, and $\sqrt{3}b=2asinBcosC+2csinBcosA$.
$(1)$ Find the measure of angle $B$;
$(2)$ Given $a=3$, $c=4$,
① Find $b$,
② Find the value of $\cos \left(2A+B\right)$. | {
"answer": "-\\frac{23}{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers \(a, b, c\) are such that \(a + \frac{1}{b} = 9\), \(b + \frac{1}{c} = 10\), \(c + \frac{1}{a} = 11\). Find the value of the expression \(abc + \frac{1}{abc}\). | {
"answer": "960",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $$\overrightarrow {a} = (sinx, \frac {3}{4})$$, $$\overrightarrow {b} = (\frac {1}{3}, \frac {1}{2}cosx)$$, and $$\overrightarrow {a}$$ is parallel to $$\overrightarrow {b}$$. Find the acute angle $x$. | {
"answer": "\\frac {\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The cube of $a$ and the fourth root of $b$ vary inversely. If $a=3$ when $b=256$, then find $b$ when $ab=81$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $11 = x^6 + \frac{1}{x^6}$, find the value of $x^3 + \frac{1}{x^3}$. | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(A, B, C\), and \(D\) be four points that are not coplanar. A plane passes through the centroid of triangle \(ABC\) that is parallel to the lines \(AB\) and \(CD\). In what ratio does this plane divide the median drawn to the side \(CD\) of triangle \(ACD\)? | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least natural number that can be added to 71,382 to create a palindrome? | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The height of a trapezoid, whose diagonals are mutually perpendicular, is 4. Find the area of the trapezoid if one of its diagonals is 5. | {
"answer": "\\frac{50}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a table, there are 2004 boxes, each containing one ball. It is known that some of the balls are white, and their number is even. You are allowed to point to any two boxes and ask if there is at least one white ball in them. What is the minimum number of questions needed to guarantee the identification of a box that contains a white ball? | {
"answer": "2003",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number that starts with the digit 5, which, when this 5 is removed from the beginning of its decimal representation and appended to its end, becomes four times smaller. | {
"answer": "512820",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(\sin \alpha + \sin (\alpha + \beta) + \cos (\alpha + \beta) = \sqrt{3}\), where \(\beta \in \left[\frac{\pi}{4}, \pi\right]\), find the value of \(\beta\). | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ten football teams played each other exactly once. As a result, each team ended up with exactly $x$ points.
What is the largest possible value of $x$? (A win earns 3 points, a draw earns 1 point, and a loss earns 0 points.) | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A game board is constructed by shading three of the regions formed by the diagonals of a regular pentagon. What is the probability that the tip of the spinner will come to rest in a shaded region? Assume the spinner can land in any region with equal likelihood. | {
"answer": "\\frac{3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence ${a_n}$ is an arithmetic sequence, with $a_1 \geq 1$, $a_2 \leq 5$, $a_5 \geq 8$, let the sum of the first n terms of the sequence be $S_n$. The maximum value of $S_{15}$ is $M$, and the minimum value is $m$. Determine $M+m$. | {
"answer": "600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( M \) is the midpoint of the hypotenuse \( AC \) of right triangle \( ABC \). Points \( P \) and \( Q \) on lines \( AB \) and \( BC \) respectively are such that \( AP = PM \) and \( CQ = QM \). Find the measure of angle \( \angle PQM \) if \( \angle BAC = 17^{\circ} \). | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular hexagon has a side length of 8 cm. Calculate the area of the shaded region formed by connecting two non-adjacent vertices to the center of the hexagon, creating a kite-shaped region.
[asy]
size(100);
pair A,B,C,D,E,F,O;
A = dir(0); B = dir(60); C = dir(120); D = dir(180); E = dir(240); F = dir(300); O = (0,0);
fill(A--C--O--cycle,heavycyan);
draw(A--B--C--D--E--F--A);
draw(A--C--O);
[/asy] | {
"answer": "16\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The difference of the logarithms of the hundreds digit and the tens digit of a three-digit number is equal to the logarithm of the difference of the same digits, and the sum of the logarithms of the hundreds digit and the tens digit is equal to the logarithm of the sum of the same digits, increased by 4/3. If you subtract the number, having the reverse order of digits, from this three-digit number, their difference will be a positive number, in which the hundreds digit coincides with the tens digit of the given number. Find this number. | {
"answer": "421",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two identical cylindrical sheets are cut open along the dotted lines and glued together to form one bigger cylindrical sheet. The smaller sheets each enclose a volume of 100. What volume is enclosed by the larger sheet? | {
"answer": "400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Juan is measuring the diameter of a large rather ornamental plate to cover it with a decorative film. Its actual diameter is 30cm, but his measurement tool has an error of up to $30\%$. Compute the largest possible percent error, in percent, in Juan's calculated area of the ornamental plate. | {
"answer": "69",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system \(x O y\), the equation of the ellipse \(C\) is \(\frac{x^{2}}{9}+\frac{y^{2}}{10}=1\). Let \(F\) be the upper focus of \(C\), \(A\) be the right vertex of \(C\), and \(P\) be a moving point on \(C\) located in the first quadrant. Find the maximum area of the quadrilateral \(O A P F\). | {
"answer": "\\frac{3}{2} \\sqrt{11}",
"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, and $\frac{\cos B}{b} + \frac{\cos C}{c} = \frac{\sin A}{\sqrt{3} \sin C}$.
(1) Find the value of $b$.
(2) If $\cos B + \sqrt{3} \sin B = 2$, find the maximum area of triangle $ABC$. | {
"answer": "\\frac{3\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
31 people attended a class club afternoon event, and after the program, they danced. Ági danced with 7 boys, Anikó with 8, Zsuzsa with 9, and so on, with each subsequent girl dancing with one more boy than the previously mentioned one. Finally, Márta danced with all but 3 boys. How many boys were at the club afternoon? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The greatest prime number that is a divisor of $16,385$ can be deduced similarly, find the sum of the digits of this greatest prime number. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A club has 99 members. Find the smallest positive integer $n$ such that if the number of acquaintances of each person is greater than $n$, there must exist 4 people who all know each other mutually (here it is assumed that if $A$ knows $B$, then $B$ also knows $A$). | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya, Petya, and Kolya are in the same class. Vasya always lies in response to any question, Petya alternates between lying and telling the truth, and Kolya lies in response to every third question but tells the truth otherwise. One day, each of them was asked six consecutive times how many students are in their class. The responses were "Twenty-five" five times, "Twenty-six" six times, and "Twenty-seven" seven times. Can we determine the actual number of students in their class based on their answers? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the equation \(2x^3 - 7x^2 + 7x + p = 0\) has three distinct roots, and these roots form a geometric progression. Find \(p\) and solve this equation. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangle $\triangle ABC$, $\angle A = 60^{\circ}$ and $\angle B = 45^{\circ}$. A line $DE$, with $D$ on $AB$ and $E$ on $BC$, such that $\angle ADE =75^{\circ}$, divides $\triangle ABC$ into two pieces of equal area. Determine the ratio $\frac{AD}{AB}$.
A) $\frac{1}{2}$
B) $\frac{1}{\sqrt{3}}$
C) $\frac{1}{\sqrt{6}}$
D) $\frac{1}{3}$
E) $\frac{1}{4}$ | {
"answer": "\\frac{1}{\\sqrt{6}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A father and son were walking one after the other along a snow-covered road. The father's step length is $80 \mathrm{~cm}$, and the son's step length is $60 \mathrm{~cm}$. Their steps coincided 601 times, including at the very beginning and at the end of the journey. What distance did they travel? | {
"answer": "1440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given complex numbers \( x \) and \( y \), find the maximum value of \(\frac{|3x+4y|}{\sqrt{|x|^{2} + |y|^{2} + \left|x^{2}+y^{2}\right|}}\). | {
"answer": "\\frac{5\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the complex plane, consider the parallelogram formed by the points 0, $z,$ $\frac{1}{z},$ and $z + \frac{1}{z}$ where the area of the parallelogram is $\frac{24}{25}.$ If the real part of $z$ is positive, determine the smallest possible value of $\left| z + \frac{1}{z} \right|.$ Compute the square of this value. | {
"answer": "\\frac{36}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. If $c\cos B + b\cos C = 2a\cos A$, $M$ is the midpoint of $BC$, and $AM=1$, find the maximum value of $b+c$. | {
"answer": "\\frac{4\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$48n$ is the smallest positive integer that satisfies the following conditions:
1. $n$ is a multiple of 75;
2. $n$ has exactly 75 positive divisors (including 1 and itself).
Find $\frac{n}{75}$. | {
"answer": "432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the graph of the function $f(x) = \sin \omega x \cos \omega x + \sqrt{3} \sin^2 \omega x - \frac{\sqrt{3}}{2}$ ($\omega > 0$) is tangent to the line $y = m$ ($m$ is a constant), and the abscissas of the tangent points form an arithmetic sequence with a common difference of $\pi$.
(Ⅰ) Find the values of $\omega$ and $m$;
(Ⅱ) Find the sum of all zeros of the function $y = f(x)$ in the interval $x \in [0, 2\pi]$. | {
"answer": "\\frac{11\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere with radius $r$ is inside a cone, whose axial section is an equilateral triangle with the sphere inscribed in it. The ratio of the total surface area of the cone to the surface area of the sphere is \_\_\_\_\_\_. | {
"answer": "9:4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A group of 8 boys and 8 girls was paired up randomly. Find the probability that there is at least one pair with two girls. Round your answer to the nearest hundredth. | {
"answer": "0.98",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The calculation result of the expression \(143 \times 21 \times 4 \times 37 \times 2\) is $\qquad$. | {
"answer": "888888",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the least possible value of $ABCD - AB \times CD$ , where $ABCD$ is a 4-digit positive integer, and $AB$ and $CD$ are 2-digit positive integers. (Here $A$ , $B$ , $C$ , and $D$ are digits, possibly equal. Neither $A$ nor $C$ can be zero.) | {
"answer": "109",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a football tournament, each team is supposed to play one match against each of the other teams. However, during the tournament, half of the teams were disqualified and did not participate further. As a result, a total of 77 matches were played, and the disqualified teams managed to play all their matches against each other, with each disqualified team having played the same number of matches. How many teams were there at the beginning of the tournament? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with a side length of 1, points $E$ and $F$ are located on $A A_{1}$ and $C C_{1}$ respectively, such that $A E = C_{1} F$. Determine the minimum area of the quadrilateral $E B F D_{1}$. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. The sequence satisfies the conditions $a\_1=23$, $a\_2=-9$, and $a_{n+2}=a\_n+6\times(-1)^{n+1}-2$ for all $n \in \mathbb{N}^*$.
(1) Find the general formula for the terms of the sequence $\{a\_n\}$;
(2) Find the value of $n$ when $S\_n$ reaches its maximum. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{5}}$ with an accuracy of $10^{-3}$. | {
"answer": "0.973",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1$$ ($a>0$, $b>0$) with its right focus at $F(c, 0)$.
(1) If one of the asymptotes of the hyperbola is $y=x$ and $c=2$, find the equation of the hyperbola;
(2) With the origin $O$ as the center and $c$ as the radius, draw a circle. Let the intersection of the circle and the hyperbola in the first quadrant be $A$. Draw the tangent line to the circle at $A$, with a slope of $-\sqrt{3}$. Find the eccentricity of the hyperbola. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The reciprocal of $\frac{2}{3}$ is ______, the opposite of $-2.5$ is ______. | {
"answer": "2.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=e^{x}$, for real numbers $m$, $n$, $p$, it is known that $f(m+n)=f(m)+f(n)$ and $f(m+n+p)=f(m)+f(n)+f(p)$. Determine the maximum value of $p$. | {
"answer": "2\\ln2-\\ln3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that m > 0, p: 0 < x < m, q: x(x - 1) < 0, if p is a sufficient but not necessary condition for q, then the value of m can be _______. (Only one value of m that satisfies the condition is needed) | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.