problem stringlengths 10 5.15k | answer dict |
|---|---|
Simplify the expression given by \(\frac{a^{-1} - b^{-1}}{a^{-3} + b^{-3}} : \frac{a^{2} b^{2}}{(a+b)^{2} - 3ab} \cdot \left(\frac{a^{2} - b^{2}}{ab}\right)^{-1}\) for \( a = 1 - \sqrt{2} \) and \( b = 1 + \sqrt{2} \). | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and the sizes of angles $A$, $B$, $C$ form an arithmetic sequence. Let vector $\overrightarrow{m}=(\sin \frac {A}{2},\cos \frac {A}{2})$, $\overrightarrow{n}=(\cos \frac {A}{2},- \sqrt {3}\cos \frac {A}{2})$, and $f(A)= \overrightarrow{m} \cdot \overrightarrow{n}$,
$(1)$ If $f(A)=- \frac { \sqrt {3}}{2}$, determine the shape of $\triangle ABC$;
$(2)$ If $b= \sqrt {3}$ and $a= \sqrt {2}$, find the length of side $c$ and the area $S_{\triangle ABC}$. | {
"answer": "\\frac {3+ \\sqrt {3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive integer $n$ not exceeding $120$ is chosen such that if $n\le 60$, then the probability of choosing $n$ is $p$, and if $n > 60$, then the probability of choosing $n$ is $2p$. The probability that a perfect square is chosen is?
A) $\frac{1}{180}$
B) $\frac{7}{180}$
C) $\frac{13}{180}$
D) $\frac{1}{120}$
E) $\frac{1}{60}$ | {
"answer": "\\frac{13}{180}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a trapezoid with bases 3 and 4, find the length of the segment parallel to the bases that divides the area of the trapezoid in the ratio $5:2$, counting from the shorter base. | {
"answer": "\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a circle, 103 natural numbers are written. It is known that among any 5 consecutive numbers, there will be at least two even numbers. What is the minimum number of even numbers that can be in the entire circle? | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
27 points P and a square with side length 1 lie on the same plane. Let the vertices of the square be \( A, B, C, \) and \( D \) (counterclockwise), and the distances from P to \( A, B, \) and \( C \) are \( u, v, \) and \( w \) respectively. If \( u^2 + v^2 = w^2 \), then the maximum distance from point P to point D is: | {
"answer": "2 + \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle \( \triangle ABC \) with interior angles \( \angle A, \angle B, \angle C \) and opposite sides \( a, b, c \) respectively, where \( \angle A - \angle C = \frac{\pi}{2} \) and \( a, b, c \) are in arithmetic progression, find the value of \( \cos B \). | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p,$ $q,$ $r$ be positive real numbers. Find the smallest possible value of
\[4p^3 + 6q^3 + 24r^3 + \frac{8}{3pqr}.\] | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of $x$ that satisfies $\sqrt[4]{x\sqrt{x^4}}=4$. | {
"answer": "2^{8/3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If we want to write down all the integers from 1 to 10,000, how many times do we have to write a digit, for example, the digit 5? | {
"answer": "4000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of $x$-intercepts on the graph of $y = \sin \frac{2}{x}$ (evaluated in terms of radians) in the interval $(0.0002, 0.002).$ | {
"answer": "2865",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\frac{137}{a}=0.1 \dot{2} 3 \dot{4}$, find the value of $a$. | {
"answer": "1110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a natural number $n$, if $n+6$ divides $n^3+1996$, then $n$ is called a lucky number of 1996. For example, since $4+6$ divides $4^3+1996$, 4 is a lucky number of 1996. Find the sum of all lucky numbers of 1996. | {
"answer": "3720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \begin{cases} \log_{2}x,x > 0 \\ x^{2}+4x+1,x\leqslant 0\\ \end{cases}$, if the real number $a$ satisfies $f(f(a))=1$, calculate the sum of all possible values of the real number $a$. | {
"answer": "-\\frac{15}{16} - \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In \\( \triangle ABC \\), \\( a \\), \\( b \\), and \\( c \\) are the sides opposite to angles \\( A \\), \\( B \\), and \\( C \\) respectively. The vectors \\( \overrightarrow{m} = (a, b+c) \\) and \\( \overrightarrow{n} = (1, \cos C + \sqrt{3} \sin C) \\) are given, and \\( \overrightarrow{m} \parallel \overrightarrow{n} \\).
\\((1)\\) Find angle \\( A \\).
\\((2)\\) If \\( 3bc = 16 - a^2 \\), find the maximum area of \\( \triangle ABC \\). | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $3 \times 3$ table is initially filled with zeros. In one move, any $2 \times 2$ square in the table is chosen, and all zeros in it are replaced with crosses, and all crosses with zeros. Let's call a "pattern" any arrangement of crosses and zeros in the table. How many different patterns can be obtained as a result of such moves? Patterns that can be transformed into each other by a $90^\circ$ or $180^\circ$ rotation are considered different. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify the expression \(\frac{\operatorname{tg}\left(\frac{5}{4} \pi - 4 \alpha\right) \sin^{2}\left(\frac{5}{4} \pi + 4 \alpha\right)}{1 - 2 \cos^{2} 4 \alpha}\). | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya is creating a 4-digit password for a combination lock. He dislikes the digit 2 and does not use it. Additionally, he does not like having two identical digits adjacent to each other. Moreover, he wants the first digit to be the same as the last digit. How many possible combinations need to be checked to guarantee guessing Vasya's password correctly? | {
"answer": "576",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the tetrahedron \( ABCD \), \( AC = 8 \), \( AB = CD = 7 \), \( BC = AD = 5 \), and \( BD = 6 \). Given a point \( P \) on \( AC \), find the minimum value of \( BP + PD \). | {
"answer": "2\\sqrt{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $(-2)^{23} + 2^{(2^4+5^2-7^2)}$. | {
"answer": "-8388607.99609375",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $y=2\sin(x+\frac{\pi}{3})$ has an axis of symmetry at $x=\frac{\pi}{6}$. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the sum \( S = \sum_{i=0}^{101} \frac{x_{i}^{3}}{1 - 3x_{i} + 3x_{i}^{2}} \) for \( x_{i} = \frac{i}{101} \). | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
PQR Entertainment wishes to divide their popular idol group PRIME, which consists of seven members, into three sub-units - PRIME-P, PRIME-Q, and PRIME-R - with each of these sub-units consisting of either two or three members. In how many different ways can they do this, if each member must belong to exactly one sub-unit? | {
"answer": "630",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On an island, there live knights, liars, and yes-men; each knows who is who among them. In a row, 2018 island inhabitants were asked the question: "Are there more knights than liars on the island?" Each inhabitant answered either "Yes" or "No" in turn such that everyone else could hear their response. Knights always tell the truth, liars always lie. Each yes-man answered according to the majority of responses given before their turn, or gave either response if there was a tie. It turned out that exactly 1009 answered "Yes." What is the maximum number of yes-men that could be among the island inhabitants? | {
"answer": "1009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given four circles with radii $2, 4, 6,$ and $8$ tangent to two perpendicular lines $\ell_1$ and $\ell_2$ intersecting at point $A$, and region $S$ consisting of all the points that lie inside exactly one of these four circles, find the maximum possible area of region $S$. | {
"answer": "120\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line in the plane is called strange if it passes through \((a, 0)\) and \((0, 10-a)\) for some \(a\) in the interval \([0,10]\). A point in the plane is called charming if it lies in the first quadrant and also lies below some strange line. What is the area of the set of all charming points? | {
"answer": "50/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let's call a word any finite sequence of letters of the Russian alphabet. How many different four-letter words can be made from the letters of the word КАША? And from the letters of the word ХЛЕБ? Indicate the sum of the found numbers in the answer. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Miki has 15 oranges and 15 pears. She extracts 12 ounces of orange juice from 3 oranges and 10 ounces of pear juice from 4 pears. Miki then makes a juice blend using 5 pears and 4 oranges. Calculate the percentage of the blend that is pear juice. | {
"answer": "43.86\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five points, no three of which are collinear, are given. Calculate the least possible value of the number of convex polygons whose some corners are formed by these five points. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{\pi / 4}^{\arccos (1 / \sqrt{26})} \frac{36 \, dx}{(6 - \tan x) \sin 2x}
$$ | {
"answer": "6 \\ln 5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, the equation of circle $C$ is $(x-2)^2 + y^2 = 4$, with the center at point $C$. Using the origin as the pole and the non-negative half of the $x$-axis as the initial ray, establish a polar coordinate system. The curve $C_1: \rho = -4\sqrt{3}\sin \theta$ intersects with circle $C$ at points $A$ and $B$.
(i) Find the polar equation of line $AB$.
(ii) If a line $C_2$ passing through point $C(2,0)$ with the parametric equations $\begin{cases} x=2+ \frac{\sqrt{3}}{2}t, \\ y=\frac{1}{2}t \end{cases}$ (where $t$ is the parameter) intersects line $AB$ at point $D$ and intersects the $y$-axis at point $E$, calculate the ratio $|CD| : |CE|$. | {
"answer": "1 : 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\left\{a_{n}\right\}$ that satisfies
$$
a_{n+1} = -\frac{1}{2} a_{n} + \frac{1}{3^{n}}\quad (n \in \mathbf{Z}_{+}),
$$
find all values of $a_{1}$ such that the sequence $\left\{a_{n}\right\}$ is monotonic, i.e., $\left\{a_{n}\right\}$ is either increasing or decreasing. | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using Green's theorem, evaluate the line integral \( I = \oint_{L} 2(x^{2}+y^{2}) \, dx + (x+y)^{2} \, dy \) along the contour \( L \) of triangle \( ABC \) with vertices \( A(1,1) \), \( B(2,2) \), \( C(1,3) \). | {
"answer": "-\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distance between Ivan's house and his grandmother's house is 12 km. Exactly at 12:00, Ivan left his house and walked along the straight road to his grandmother's house at a speed of 1 m/s. At 12:30, Ivan's parents called his grandmother, informed her that Ivan was coming to visit, and she released her dog Tuzik to meet him. Tuzik runs at a speed of 9 m/s. Determine the moment when Tuzik will reach Ivan. | {
"answer": "12:47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a hyperbola shares common foci $F_1$ and $F_2$ with the ellipse $\dfrac {x^{2}}{9}+ \dfrac {y^{2}}{25}=1$, and their sum of eccentricities is $2 \dfrac {4}{5}$.
$(1)$ Find the standard equation of the hyperbola;
$(2)$ Let $P$ be a point of intersection between the hyperbola and the ellipse, calculate $\cos \angle F_{1}PF_{2}$. | {
"answer": "- \\dfrac {1}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x_{i} \in \{-1,1\}, i=1,2,\cdots,2021 \), and \( x_{1}+x_{2}+\cdots+x_{k} \geq 0 \) for \( k=1,2,\cdots,2020 \), with \( x_{1}+x_{2}+\cdots+x_{2021}=-1 \). How many ordered arrays \( (x_{1}, x_{2}, \cdots, x_{2021}) \) are there? | {
"answer": "\\frac{1}{1011} \\binom{2020}{1010}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a geometric sequence $\left\{a_{n}\right\}$ with a common ratio $q \in (1,2)$, and that $a_{n}$ is a positive integer for $1 \leq n \leq 6$, find the minimum value of $a_{6}$. | {
"answer": "243",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Replace the symbols $\$ last\$ with the same numbers so that the equation becomes true: $\$ \$ |\frac{20}{x} - \frac{x}{15} = \frac{20}{15} \$ \$$ | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a, b, c$, and $d$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
a^2+b^2 &=& c^2+d^2 &=& 2016, \\
ac &=& bd &=& 1024.
\end{array}
\]
If $S = a + b + c + d$, compute the value of $\lfloor S \rfloor$. | {
"answer": "127",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\(f(x)\) is a linear function, and the equation \(f(f(x)) = x + 1\) has no solutions. Find all possible values of \(f(f(f(f(f(2022)))))-f(f(f(2022)))-f(f(2022))\). | {
"answer": "-2022",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the average of five-digit palindromic numbers? (A whole number is a palindrome if it reads the same backward as forward.) | {
"answer": "55000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the sides \(AC = 14\) and \(AB = 6\) are known. A circle with center \(O\) is constructed using side \(AC\) as the diameter, intersecting 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"
} |
When four lines intersect pairwise, and no three of them intersect at the same point, find the total number of corresponding angles formed by these lines. | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a positive integer \( n \) (\( n \geqslant 6 \)), let set \( A \) be composed of sums of any 5 consecutive positive integers not greater than \( n \), and let set \( B \) be composed of sums of any 6 consecutive positive integers not greater than \( n \). If the number of elements in the intersection of sets \( A \) and \( B \) is 2016, determine the maximum value of \( n \). | {
"answer": "12106",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $(2a-c)\cos B = b\cos C$.
(1) Find the size of angle $B$;
(2) If the area of $\triangle ABC$ is $T$ and $b=\sqrt{3}$, find the value of $a+c$; | {
"answer": "\\sqrt{ \\frac{30}{2} }",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the coefficient of the $x^{2}$ term in the expansion of $((1-ax)(1+2x)^{4})$ is $4$, then evaluate $\int_{\frac{e}{2}}^{a}{\frac{1}{x}dx}$ = \_\_\_\_\_\_. | {
"answer": "\\ln 5 - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At what angle to the x-axis is the tangent to the graph of the function \( g(x) = x^2 \ln x \) inclined at the point \( x_0 = 1 \)? | {
"answer": "\\frac{\\pi}{4}",
"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 $a=2b$. Also, $\sin A$, $\sin C$, and $\sin B$ form an arithmetic sequence.
$(1)$ Find the value of $\cos A$;
$(2)$ If the area of $\triangle ABC$ is $\frac{8\sqrt{15}}{3}$, find the value of $c$. | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Experimenters Glafira and Gavrila placed a triangle of thin wire with sides 30 mm, 40 mm, and 50 mm on a white flat surface. This wire is covered with millions of unknown microorganisms. Scientists found that when electric current is applied to the wire, these microorganisms start moving chaotically on this surface in different directions at an approximate speed of $\frac{1}{6}$ mm/sec. During their movement, the surface along their trajectory is painted red. Find the area of the painted surface 1 minute after the current is applied. Round the result to the nearest whole number of square millimeters. | {
"answer": "2114",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circular sheet of iron with a radius of 6 has a sector removed, which is $\frac{1}{6}$ of the original area. The remaining part is rolled into the lateral surface of a cone. The volume of the cone is \_\_\_\_\_\_. | {
"answer": "\\frac{25\\sqrt{11}}{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers from 1 to 100 are divisible by 3, but do not contain the digit 3? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What are the fractions that have the value $\frac{45}{56}$ and have a numerator that is a perfect square and a denominator that is a perfect cube? Among these fractions, which one is in the simplest form? | {
"answer": "\\frac{525^2}{70^3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the limit of the function:
\[
\lim _{x \rightarrow \frac{\pi}{4}}(\tan x)^{1 / \cos \left(\frac{3 \pi}{4}-x\right)}
\] | {
"answer": "e^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Billy and Bobbi each selected a positive integer less than 500. Billy's number is a multiple of 20, and Bobbi's number is a multiple of 30. What is the probability that they selected the same number? Express your answer as a common fraction. | {
"answer": "\\frac{1}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a modified SHORT BINGO, a $5\times5$ card uses a WILD square in the middle and places 24 other numbers in the remaining squares. Numbers are placed as follows: 5 distinct numbers from $1-15$ in the first column, 5 distinct numbers from $16-30$ in the second column, 4 distinct numbers from $31-40$ in the third column (skipping the WILD square in the middle), 5 distinct numbers from $41-55$ in the fourth column, and 5 distinct numbers from $56-70$ in the last column. Determine the number of distinct possibilities for the values in the second column of this SHORT BINGO card. | {
"answer": "360360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the natural numbers from 1 to 100, find the total number of numbers that contain a digit 7 or are multiples of 7. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a two-day math competition, Charlie and Delta both attempted a total of 600 points. Charlie scored 200 points out of 400 attempted on the first day, and 160 points out of 200 attempted on the second day. Delta, who did not attempt 400 points on the first day, scored a positive integer number of points each day, and Delta's daily success ratio was less than Charlie's on each day. Charlie's two-day success ratio was $\frac{360}{600} = \frac{3}{5}$.
Find the largest possible two-day success ratio that Delta could have achieved. | {
"answer": "\\frac{479}{600}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside a right circular cone with base radius $8$ and height $15$, there are three identical spheres. Each sphere is tangent to the others, the base, and the side of the cone. Determine the radius $r$ of each sphere.
A) $\frac{840 - 300\sqrt{3}}{121}$
B) $\frac{60}{19 + 5\sqrt{3}}$
C) $\frac{280 - 100\sqrt{3}}{121}$
D) $\frac{120}{19 + 5\sqrt{5}}$
E) $\frac{140 - 50\sqrt{3}}{61}$ | {
"answer": "\\frac{280 - 100\\sqrt{3}}{121}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Quadrilateral \(ABCD\) is such that \(\angle BAC = \angle CAD = 60^\circ\) and \(AB + AD = AC\). It is also known that \(\angle ACD = 23^\circ\). How many degrees is \(\angle ABC\)? | {
"answer": "83",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the sequence \( 5, 55, 555, 5555, 55555, \ldots \). Are any of the numbers in this sequence divisible by 495; if so, what is the smallest such number? | {
"answer": "555555555555555555",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A and B are playing a guessing game. First, A thinks of a number, denoted as $a$, then B guesses the number A was thinking of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\{0,1,2,…,9\}$. The probability that $|a-b|\leqslant 1$ is __________. | {
"answer": "\\dfrac{7}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides corresponding to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is known that $B= \frac{\pi}{4}$ and $\cos A-\cos 2A=0$.
$(1)$ Find the angle $C$.
$(2)$ If $b^{2}+c^{2}=a-bc+2$, find the area of $\triangle ABC$. | {
"answer": "1- \\frac{ \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Convex quadrilateral \(ABCD\) is such that \(\angle BAC = \angle BDA\) and \(\angle BAD = \angle ADC = 60^\circ\). Find the length of \(AD\) given that \(AB = 14\) and \(CD = 6\). | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the base \( AC \) of an isosceles triangle \( ABC \), a circle is constructed using it as a diameter. This circle intersects side \( BC \) at point \( N \) such that \( BN: NC = 7: 2 \). Find the ratio of the lengths of segments \( AN \) and \( BC \). | {
"answer": "\\frac{4 \\sqrt{2}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that 8 first-year high school students are divided evenly between two companies, A and B, with the condition that two students with excellent English grades cannot be assigned to the same company and three students with computer skills cannot be assigned to the same company, determine the number of different distribution schemes. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the integer solutions to the system of inequalities
\[
\begin{cases}
9x - a \geq 0, \\
8x - b < 0
\end{cases}
\]
are only 1, 2, and 3, find the number of ordered pairs \((a, b)\) of integers that satisfy this system. | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$, respectively, and $(2a-c)\cos B=b\cos C$.
(I) Find the measure of angle $B$;
(II) If $\cos A=\frac{\sqrt{2}}{2}$ and $a=2$, find the area of $\triangle ABC$. | {
"answer": "\\frac{3+\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sin ωx (ω > 0)$, if the minimum value of $f(x)$ in the interval $\left[ -\frac{\pi}{3}, \frac{\pi}{4} \right]$ is $-2$, find the minimum value of $ω$. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given four positive integers \(a, b, c,\) and \(d\) satisfying the equations \(a^2 = c(d + 20)\) and \(b^2 = c(d - 18)\). Find the value of \(d\). | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the arc lengths of curves given by the equations in polar coordinates.
$$
\rho = 3 \varphi, \quad 0 \leq \varphi \leq \frac{4}{3}
$$ | {
"answer": "\\frac{10}{3} + \\frac{3}{2} \\ln 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the minimum of $\sqrt{(x-1)^{2}+y^{2}}+\sqrt{(x+1)^{2}+y^{2}}+|2-y|$ for $x, y \in \mathbb{R}$. | {
"answer": "2 + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A tractor is dragging a very long pipe on sleds. Gavrila walked along the entire pipe in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction, the number of steps was 100. What is the length of the pipe if Gavrila's step is 80 cm? Round the answer to the nearest whole number of meters. | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a linear function \( f(x) \), it is known that the distance between the intersection points of the graphs \( y=x^{2}+2 \) and \( y=f(x) \) is \(\sqrt{10}\), and the distance between the intersection points of the graphs \( y=x^{2}-1 \) and \( y=f(x)+1 \) is \(\sqrt{42}\). Find the distance between the intersection points of the graphs \( y=x^{2} \) and \( y=f(x)+1\). | {
"answer": "\\sqrt{34}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points E and D are on sides AB and BC of triangle ABC, where AE:EB=1:3 and CD:DB=1:2, find the value of EF/FC + AF/FD. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\arccos (\sin 3)$. All functions are in radians. | {
"answer": "3 - \\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( ABCD \) be a square with side length 1. Points \( X \) and \( Y \) are on sides \( BC \) and \( CD \) respectively such that the areas of triangles \( ABX \), \( XCY \), and \( YDA \) are equal. Find the ratio of the area of \( \triangle AXY \) to the area of \( \triangle XCY \). | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f(x)=\sqrt{3} \sin 2x + 2 \cos^2 x + a \), if the minimum value of \( f(x) \) on the interval \(\left[ 0, \frac{\pi}{2} \right] \) is \(-1\), find the value of \( a \). | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a bag there are $1007$ black and $1007$ white balls, which are randomly numbered $1$ to $2014$ . In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absolute value of their differences. How many points can we guarantee to earn after $2014$ steps? | {
"answer": "1014049",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 1001 people sitting around a round table, each of whom is either a knight (always tells the truth) or a liar (always lies). It turned out that next to each knight there is exactly one liar, and next to each liar there is exactly one knight. What is the minimum number of knights that can be sitting at the table? | {
"answer": "501",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest $\ell > 0$ such that there exists a square of side $\ell$ contained in a cube with edge $1$ . | {
"answer": "\\frac{\\sqrt{6}}{2}",
"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 $b^2+c^2-a^2+bc=0$,
(1) Find the measure of angle $A$;
(2) If $a= \sqrt {3}$, find the maximum value of the area $S_{\triangle ABC}$ of triangle $ABC$. | {
"answer": "\\frac { \\sqrt {3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the lines $l_{1}$: $\left(3+a\right)x+4y=5-3a$ and $l_{2}$: $2x+\left(5+a\right)y=8$, if $l_{1}$ is parallel to $l_{2}$, determine the value of $a$. | {
"answer": "-7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate plane, there are four fixed points \(A(-3,0), B(1,-1), C(0,3), D(-1,3)\) and a moving point \(P\). What is the minimum value of \(|PA| + |PB| + |PC| + |PD|\)? | {
"answer": "3\\sqrt{2} + 2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\overrightarrow {BA} \cdot \overrightarrow {AC} = 6$, $b-c=2$, and $\tan A = -\sqrt {15}$, find the length of the altitude drawn from $A$ to side $BC$. | {
"answer": "\\frac{3\\sqrt{15}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle has dimensions $8 \times 12$, and a circle centered at one of its corners has a radius of 10. Calculate the area of the union of the regions enclosed by the rectangle and the circle. | {
"answer": "96 + 75\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a finite sequence $B = (b_1, b_2, \dots, b_n)$ of numbers, the Cesaro sum is defined as
\[\frac{T_1 + T_2 + \cdots + T_n}{n},\]
where $T_k = b_1 + b_2 + \cdots + b_k$ for $1 \leq k \leq n$.
If the Cesaro sum of the 100-term sequence $(b_1, b_2, \dots, b_{100})$ is 1200, where $b_1 = 2$, calculate the Cesaro sum of the 101-term sequence $(3, b_1, b_2, \dots, b_{100})$. | {
"answer": "1191",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $C(1,1)$ and $D(8,6)$ are the endpoints of a diameter of a circle on a coordinate plane. Calculate both the area and the circumference of the circle. Express your answers in terms of $\pi$. | {
"answer": "\\sqrt{74}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), let F be the right focus of the hyperbola. A perpendicular line from point F to the x-axis intersects the two asymptotes at points A and B, and intersects the hyperbola in the first quadrant at point P. Let O be the origin of the coordinate system. If $\vec{OP} = \lambda \vec{OA} + \mu \vec{OB}$ ($\lambda, \mu \in \mathbb{R}$), and $\lambda^2 + \mu^2 = \frac{5}{8}$, calculate the eccentricity of the hyperbola. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Charlie and Dana play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $g(n)$ points, where
\[g(n) = \left\{
\begin{array}{cl}
7 & \text{if } n \text{ is a multiple of 3 and 5}, \\
3 & \text{if } n \text{ is only a multiple of 3}, \\
0 & \text{if } n \text{ is not a multiple of 3}.
\end{array}
\right.\]
Charlie rolls the die four times and gets a 6, 2, 3, and 5. Dana rolls and gets 5, 3, 1, and 3. What is the product of Charlie's total points and Dana's total points? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f(x) = x^2 + 12x + 30 \). Find the largest real root of the equation \( f(f(f(x))) = 0 \). | {
"answer": "-6 + \\sqrt[8]{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
9 people are arranged in a 3×3 matrix (3 rows, 3 columns). Choose 3 people from them to serve as the team leader, deputy team leader, and discipline officer, respectively. The requirement is that at least two of these three people must be in the same row or column. The number of different methods to select these people is \_\_\_\_\_\_ . (Answer with a number) | {
"answer": "468",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A builder has two identical bricks. She places them side by side in three different ways, resulting in shapes with surface areas of 72, 96, and 102. What is the surface area of one original brick?
A) 36
B) 48
C) 52
D) 54
E) 60 | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$ . (Note: $n$ is written in the usual base ten notation.) | {
"answer": "9999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In 10 boxes, place ping-pong balls such that the number of balls in each box is at least 11, not equal to 17, not a multiple of 6, and all numbers are distinct. What is the minimum number of ping-pong balls needed? | {
"answer": "174",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( x \) and \( y \) are real numbers greater than 10, the leading digit of \( \lg x \) is \( a \) and the trailing digit is \( b \); the leading digit of \( \lg y \) is \( c \) and the trailing digit is \( d \). Additionally, it is known that \( |1 - a| + \sqrt{c - 4} = 1 \) and \( b + d = 1 \). Find the value of \( x \times y \). | {
"answer": "10^7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that $\tan\alpha$ and $\tan\beta$ are the two roots of the equation $x^2+6x+7=0$, and $\alpha, \beta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. What is the value of $\alpha + \beta$? | {
"answer": "- \\frac{3\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An infantry column stretched over 1 km. Sergeant Kim, riding a gyroscooter from the end of the column, reached its front and then returned to the end. During this time, the infantrymen covered 2 km 400 m. What distance did the sergeant travel during this time? | {
"answer": "3.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression $\frac{2020^3 - 3 \cdot 2020^2 \cdot 2021 + 5 \cdot 2020 \cdot 2021^2 - 2021^3 + 4}{2020 \cdot 2021}$. | {
"answer": "4042 + \\frac{3}{4080420}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the definite integral $\int_{-\sqrt{2}}^{\sqrt{2}} \sqrt{4-x^2}dx =$ \_\_\_\_\_\_. | {
"answer": "\\pi + 2",
"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 $\rho ^{2}-4\rho \sin \theta +3=0$.
$(1)$ Find the rectangular coordinate equation of the line $l$ and the general equation of the curve $C$;
$(2)$ 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"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.