problem stringlengths 10 5.15k | answer dict |
|---|---|
Let $a > 3$. Determine the value of $a$ given that $f(g(a)) = 16$, where $f(x) = x^2 + 10$ and $g(x) = x^2 - 6$. | {
"answer": "\\sqrt{\\sqrt{6} + 6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of distinct arrangements in a row of all natural numbers from 1 to 10 such that the sum of any three consecutive numbers is divisible by 3. | {
"answer": "1728",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two four-digit numbers \( M \) and \( N \) which are reverses of each other, and have \( q^{p}-1 \) identical positive divisors, \( M \) and \( N \) can be factorized into prime factors as \( p q^{q} r \) and \( q^{p+q} r \) respectively, where \( p \), \( q \), and \( r \) are prime numbers. Find the value of \( M \).
(Note: Numbers such as 7284 and 4827 are reverses of each other). | {
"answer": "1998",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a computer game, a player can choose to play as one of three factions: \( T \), \( Z \), or \( P \). There is an online mode where 8 players are divided into two teams of 4 players each. How many total different matches are possible, considering the sets of factions? The matches are considered different if there is a team in one match that is not in the other. The order of teams and the order of factions within a team do not matter. For example, the matches \((P Z P T ; T T Z P)\) and \((P Z T T ; T Z P P)\) are considered the same, while the matches \((P Z P Z ; T Z P Z)\) and \((P Z P T ; Z Z P Z)\) are different. | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation \(x^2 + y^2 = 2(|x| + |y|)\), calculate the area of the region enclosed by its graph. | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive angle \( x \) for which
\[
2^{\sin^2 x} \cdot 4^{\cos^2 x} \cdot 2^{\tan x} = 8
\] | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In different historical periods, the conversion between "jin" and "liang" was different. The idiom "ban jin ba liang" comes from the 16-based system. For convenience, we assume that in ancient times, 16 liang equaled 1 jin, with each jin being equivalent to 600 grams in today's terms. Currently, 10 liang equals 1 jin, with each jin equivalent to 500 grams today. There is a batch of medicine, with part weighed using the ancient system and the other part using the current system, and it was found that the sum of the number of "jin" is 5 and the sum of the number of "liang" is 68. How many grams is this batch of medicine in total? | {
"answer": "2800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2014$, and $a^2 - b^2 + c^2 - d^2 = 2014$. Find the number of possible values of $a$. | {
"answer": "502",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of the first $n$ terms of the positive arithmetic geometric sequence $\{a_n\}$ is $S_n$, and $\frac{a_{n+1}}{a_n} < 1$, if $a_3 + a_5 = 20$ and $a_2 \cdot a_6 = 64$, calculate $S_6$. | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid $PQRS$ with $PQ$ parallel to $RS$, the diagonals $PR$ and $QS$ intersect at $T$. If the area of triangle $PQT$ is 75 square units, and the area of triangle $PST$ is 30 square units, calculate the area of trapezoid $PQRS$. | {
"answer": "147",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At the first site, high-class equipment was used, while at the second site, first-class equipment was used, with the amount of high-class equipment being less than that of the first-class. Initially, 30% of the equipment from the first site was transferred to the second site. Then, 10% of the equipment that ended up at the second site was transferred back to the first site, with half of this transferred equipment being of the first class. After these transfers, the amount of high-class equipment at the first site was 6 units more than at the second site, and the total amount of equipment at the second site increased by more than 2% compared to the initial amount. Find the total amount of first-class equipment. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Natural numbers \( x, y, z \) are such that \( \operatorname{GCD}(\operatorname{LCM}(x, y), z) \cdot \operatorname{LCM}(\operatorname{GCD}(x, y), z) = 1400 \).
What is the maximum value that \( \operatorname{GCD}(\operatorname{LCM}(x, y), z) \) can take? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $ABC$ with vertices $A = (3,0)$, $B = (0,3)$, and $C$ lying on the line $x + 2y = 8$, find the area of triangle $ABC$. | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( O \) is the circumcenter of \(\triangle ABC\), and \( 3 \overrightarrow{OA} + 4 \overrightarrow{OB} + 5 \overrightarrow{OC} = \overrightarrow{0} \), find the value of \( \cos \angle BAC \). | {
"answer": "\\frac{\\sqrt{10}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Right triangles \(ABC\) and \(ABD\) share a common hypotenuse \(AB = 5\). Points \(C\) and \(D\) are located on opposite sides of the line passing through points \(A\) and \(B\), with \(BC = BD = 3\). Point \(E\) lies on \(AC\), and \(EC = 1\). Point \(F\) lies on \(AD\), and \(FD = 2\). Find the area of the pentagon \(ECBDF\). | {
"answer": "9.12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([1, 3]\). Find \( \underbrace{f(f(\ldots f}_{2017}\left(\frac{3+\sqrt{7}}{2}\right)) \ldots) \). If necessary, round your answer to two decimal places. | {
"answer": "0.18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cylinder with a volume of 9 is inscribed in a cone. The plane of the top base of this cylinder cuts off a frustum from the original cone, with a volume of 63. Find the volume of the original cone. | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, where $A > B > C$, if $2 \cos 2B - 8 \cos B + 5 = 0$, $\tan A + \tan C = 3 + \sqrt{3}$, and the height $CD$ from $C$ to $AB$ is $2\sqrt{3}$, then find the area of $\triangle ABC$. | {
"answer": "12 - 4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of books issued from the library to readers constitutes $\frac{1}{16}$ of the number of books on the shelves. After transferring 2000 books from the library to the reading room, the number of books absent from the shelves became $\frac{1}{15}$ of the number of books remaining on the shelves. How many books does the library have? | {
"answer": "544000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $2018 \times 2018$ square was cut into rectangles with integer side lengths. Some of these rectangles were used to form a $2000 \times 2000$ square, and the remaining rectangles were used to form a rectangle whose length differs from its width by less than 40. Find the perimeter of this rectangle. | {
"answer": "1076",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Car A and Car B are traveling in opposite directions on a road parallel to a railway. A 180-meter-long train is moving in the same direction as Car A at a speed of 60 km/h. The time from when the train catches up with Car A until it meets Car B is 5 minutes. If it takes the train 30 seconds to completely pass Car A and 6 seconds to completely pass Car B, after how many more minutes will Car A and Car B meet once Car B has passed the train? | {
"answer": "1.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the width and length of the pan be $w$ and $l$ respectively. If the number of interior pieces is twice the number of perimeter pieces, then find the greatest possible value of $w \cdot l$. | {
"answer": "294",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the roots of the polynomial $81x^3 - 162x^2 + 81x - 8 = 0$ are in arithmetic progression, find the difference between the largest and smallest roots. | {
"answer": "\\frac{4\\sqrt{6}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of ordered quadruples \((a,b,c,d)\) of nonnegative real numbers such that
\[
a^2 + b^2 + c^2 + d^2 = 9,
\]
\[
(a + b + c + d)(a^3 + b^3 + c^3 + d^3) = 81.
\] | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two people, Person A and Person B, start at the same time from point $A$ to point $B$: Person A is faster than Person B. After reaching point $B$, Person A doubles their speed and immediately returns to point $A$. They meet Person B at a point 240 meters from point $B$. After meeting, Person B also doubles their speed and turns back. When Person A returns to point $A$, Person B is still 120 meters away from point $A$. What is the distance between points $A$ and $B$ in meters? | {
"answer": "420",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), side \(BC\) is equal to 5. A circle passes through vertices \(B\) and \(C\) and intersects side \(AC\) at point \(K\), where \(CK = 3\) and \(KA = 1\). It is known that the cosine of angle \(ACB\) is \(\frac{4}{5}\). Find the ratio of the radius of this circle to the radius of the circle inscribed in triangle \(ABK\). | {
"answer": "\\frac{10\\sqrt{10} + 25}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many four-digit numbers, formed using the digits 0, 1, 2, 3, 4, 5 without repetition, are greater than 3410? | {
"answer": "132",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In parallelogram \(ABCD\), \(OE = EF = FD\). The area of the parallelogram is 240 square centimeters. The area of the shaded region is _______ square centimeters. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x \gt 0$, $y \gt 0$, $x+2y=1$, calculate the minimum value of $\frac{{(x+1)(y+1)}}{{xy}}$. | {
"answer": "8+4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of eight-digit numbers whose product of digits equals 1400. The answer must be presented as an integer. | {
"answer": "5880",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many factors are there in the product $1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$ if we know that it ends with 1981 zeros? | {
"answer": "7935",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum number of kings that can be placed on a chessboard so that no two of them attack each other? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many distinct four-digit even numbers can be formed using the digits 0, 1, 2, 3? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square field is enclosed by a wooden fence, which is made of 10-meter-long boards placed horizontally. The height of the fence is four boards. It is known that the number of boards in the fence is equal to the area of the field, expressed in hectares. Determine the dimensions of the field. | {
"answer": "16000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Kolya, an excellent student in the 7th-8th grade, found the sum of the digits of all the numbers from 0 to 2012 and added them all together. What number did he get? | {
"answer": "28077",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x) = |3\{x\} - 1.5|$, where $\{x\}$ denotes the fractional part of $x$. Find the smallest positive integer $n$ such that the equation \[nf(xf(x)) = 2x\] has at least $1000$ real solutions. | {
"answer": "250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( D \) lies on side \( BC \) of triangle \( ABC \), and point \( O \) is located on segment \( AD \) with \( AO : OD = 9 : 4 \). A line passing through vertex \( B \) and point \( O \) intersects side \( AC \) at point \( E \) with \( BO : OE = 5 : 6 \). Determine the ratio in which point \( E \) divides side \( AC \). | {
"answer": "21 : 44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a football tournament, 15 teams participated, each playing exactly once against every other team. A win awarded 3 points, a draw 1 point, and a loss 0 points.
After the tournament ended, it was found that some 6 teams each scored at least $N$ points. What is the maximum possible integer value of $N$? | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0, m]$ for some positive integer $m$. The probability that no two of $x$, $y$, and $z$ are within 2 units of each other is greater than $\frac{1}{2}$. Determine the smallest possible value of $m$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( p \) is a prime number, the decimal part of \( \sqrt{p} \) is \( x \). The decimal part of \( \frac{1}{x} \) is \( \frac{\sqrt{p} - 31}{75} \). Find all prime numbers \( p \) that satisfy these conditions. | {
"answer": "2011",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle \(ABCD\), points \(E\) and \(F\) lie on sides \(AB\) and \(CD\) respectively such that both \(AF\) and \(CE\) are perpendicular to diagonal \(BD\). Given that \(BF\) and \(DE\) separate \(ABCD\) into three polygons with equal area, and that \(EF = 1\), find the length of \(BD\). | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Please write an irrational number that is smaller than $3$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The brothers found a treasure of gold and silver. They divided it so that each got 100 kg. The eldest got the most gold - 25 kg - and one-eighth of all the silver. How much gold was in the treasure? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x \in [0, 2\pi] \), determine the maximum value of the function
\[
f(x) = \sqrt{4 \cos^2 x + 4 \sqrt{6} \cos x + 6} + \sqrt{4 \cos^2 x - 8 \sqrt{6} \cos x + 4 \sqrt{2} \sin x + 22}.
\] | {
"answer": "2(\\sqrt{6} + \\sqrt{2})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Steve's empty swimming pool holds 30,000 gallons of water when full and will be filled by 5 hoses, each supplying 2.5 gallons of water per minute, calculate the time required to fill the pool. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right triangle has integer side lengths. One of its legs is 1575 units shorter than its hypotenuse, and the other leg is less than 1991 units. Find the length of the hypotenuse of this right triangle. | {
"answer": "1799",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f(x) = x - \frac{x^3}{2} + \frac{x^5}{2 \cdot 4} - \frac{x^7}{2 \cdot 4 \cdot 6} + \cdots \), and \( g(x) = 1 + \frac{x^2}{2^2} + \frac{x^4}{2^2 \cdot 4^2} + \frac{x^6}{2^2 \cdot 4^2 \cdot 6^2} + \cdots \). Find \( \int_{0}^{\infty} f(x) g(x) \, dx \). | {
"answer": "\\sqrt{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Laura conducted a survey in her neighborhood about pest awareness. She found that $75.4\%$ of the people surveyed believed that mice caused electrical fires. Of these, $52.3\%$ incorrectly thought that mice commonly carried the Hantavirus. Given that these 31 people were misinformed, how many total people did Laura survey? | {
"answer": "78",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the diagram, in the square \(ABCD\), \(AB = 2\). Draw an arc with center \(C\) and radius equal to \(CD\), and another arc with center \(B\) and radius equal to \(BA\). The two arcs intersect at \(E\). What is the area of the sector \(BAE\)? | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a_{1}, a_{2}, \cdots, a_{n} \) be distinct positive integers such that \( a_{1} + a_{2} + \cdots + a_{n} = 2014 \), where \( n \) is some integer greater than 1. Let \( d \) be the greatest common divisor of \( a_{1}, a_{2}, \cdots, a_{n} \). For all values of \( n \) and \( a_{1}, a_{2}, \cdots, a_{n} \) that satisfy the above conditions, find the maximum value of \( n \cdot d \). | {
"answer": "530",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four students participate in a competition where each chooses one question from two options, A and B. The rules result in the following point system: 21 points for correct A, -21 points for incorrect A, 7 points for correct B, and -7 points for incorrect B. If the total score of the four students is 0, calculate the number of different scoring situations. | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vitya Perestukin always incorrectly calculates percentages during surveys: he divides the number of respondents who answered a certain way by the number of all remaining respondents. For instance, in the survey "What is your name?" conducted among 7 Annas, 9 Olgas, 8 Julias, Vitya calculated 50% Julias.
Vitya conducted a survey in his school: what type of triangle has sides of \(3, 4, 5\)? According to his calculations, 5% answered "acute", 5% "obtuse", and 5% "such a triangle does not exist", 50% "right", and the remaining \(a \%\) answered "depends on the geometry". What is \(a\)?
| {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cube \( A B C D A_{1} B_{1} C_{1} D_{1} \), \( M \) is the center of the face \( A B B_{1} A_{1} \), \( N \) is a point on the edge \( B_{1} C_{1} \), \( L \) is the midpoint of \( A_{1} B_{1} \); \( K \) is the foot of the perpendicular dropped from \( N \) to \( BC_{1} \). In what ratio does point \( N \) divide the edge \( B_{1} C_{1} \) if \( \widehat{L M K} = \widehat{M K N} \)? | {
"answer": "\\sqrt{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a rectangle \(ABCD\), a circle intersects the side \(AB\) at points \(K\) and \(L\), and the side \(CD\) at points \(M\) and \(N\). Find the length of segment \(MN\) if \(AK = 10\), \(KL = 17\), and \(DN = 7\). | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a point P $(x, y)$ on the circle $x^2 - 4x - 4 + y^2 = 0$, find the maximum value of $x^2 + y^2$. | {
"answer": "12 + 8\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function
$$
f(x)=\left(1-x^{2}\right)\left(x^{2}+b x+c\right) \text{ for } x \in [-1, 1].
$$
Let $\mid f(x) \mid$ have a maximum value of $M(b, c)$. As $b$ and $c$ vary, find the minimum value of $M(b, c)$. | {
"answer": "3 - 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 256 players in a tennis tournament who are ranked from 1 to 256, with 1 corresponding to the highest rank and 256 corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability \(\frac{3}{5}\).
In each round of the tournament, the player with the highest rank plays against the player with the second highest rank, the player with the third highest rank plays against the player with the fourth highest rank, and so on. At the end of the round, the players who win proceed to the next round and the players who lose exit the tournament. After eight rounds, there is one player remaining in the tournament, and they are declared the winner.
Determine the expected value of the rank of the winner. | {
"answer": "103",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the board is written the number 98. Every minute the number is erased and replaced with the product of its digits increased by 15. What number will be on the board in an hour? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the parallelogram \(KLMN\), side \(KL\) is equal to 8. A circle tangent to sides \(NK\) and \(NM\) passes through point \(L\) and intersects sides \(KL\) and \(ML\) at points \(C\) and \(D\) respectively. It is known that \(KC : LC = 4 : 5\) and \(LD : MD = 8 : 1\). Find the side \(KN\). | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f:\{1,2, \cdots, 10\} \rightarrow\{1,2,3,4,5\} \), and for each \( k=1,2, \cdots, 9 \), it is true that \( |f(k+1)-f(k)| \geq 3 \). Find the number of functions \( f \) that satisfy these conditions. | {
"answer": "288",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are \( n \) different positive integers, each one not greater than 2013, with the property that the sum of any three of them is divisible by 39. Find the greatest value of \( n \). | {
"answer": "52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let set $A=\{-1, 2, 3\}$, and set $B=\{a+2, a^2+2\}$. If $A \cap B = \{3\}$, then the real number $a=$ ___. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a trapezoid, the smaller base is 1 decimeter, and the angles adjacent to it are $135^{\circ}$. The angle between the diagonals, opposite to the base, is $150^{\circ}$. Find the area of the trapezoid. | {
"answer": "0.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in square ABCD, AE = 3EC and BF = 2FB, and G is the midpoint of CD, find the ratio of the area of triangle EFG to the area of square ABCD. | {
"answer": "\\frac{1}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
132009 students are taking a test which comprises ten true or false questions. Find the minimum number of answer scripts required to guarantee two scripts with at least nine identical answers. | {
"answer": "513",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The square root of a two-digit number is expressed as an infinite decimal fraction, the first four digits of which (including the integer part) are the same. Find this number without using tables. | {
"answer": "79",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the side \( AD \) of the rhombus \( ABCD \), a point \( M \) is taken such that \( MD = 0.3 \, AD \) and \( BM = MC = 11 \). Find the area of triangle \( BCM \). | {
"answer": "20\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve for $X$ if $\sqrt[4]{X^5} = 32\sqrt[16]{32}$. | {
"answer": "16\\sqrt[4]{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Billy Bones has two coins - a gold one and a silver one. One of them is symmetric, and the other is not. It is not known which coin is not symmetric, but it is given that the non-symmetric coin lands heads with a probability of $p = 0.6$.
Billy Bones flipped the gold coin, and it landed heads immediately. Then Billy Bones started flipping the silver coin, and heads came up only on the second flip. Find the probability that the gold coin is the non-symmetric one. | {
"answer": "0.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Factorize the number \( 989 \cdot 1001 \cdot 1007 + 320 \) into prime factors. | {
"answer": "991 * 997 * 1009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( A \) lies on the line \( y = \frac{15}{8} x - 4 \), and point \( B \) on the parabola \( y = x^{2} \). What is the minimum length of segment \( AB \)? | {
"answer": "47/32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the sides \( AB \) and \( AD \) of a square \( ABCD \) with side length 108, semicircles are constructed inward. Find the radius of a circle that touches one side of the square and the semicircles: one externally and the other internally. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five points are chosen on a sphere of radius 1. What is the maximum possible volume of their convex hull? | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of $b_1, b_2, \dots, b_{150}$ is equal to $2$ or $-2$. Find the minimum positive value of
\[\sum_{1 \le i < j \le 150} b_i b_j.\] | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence of numbers \(a_1, a_2, \cdots, a_n, \cdots\) is defined. Let \(S(a_i)\) be the sum of all the digits of \(a_i\). For example, \(S(22) = 2 + 2 = 4\). If \(a_1 = 2017\), \(a_2 = 22\), and \(a_n = S(a_{n-1}) + S(a_{n-2})\), what is the value of \(a_{2017}\)? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle \(A B C\) has side lengths \(A B = 65\), \(B C = 33\), and \(A C = 56\). Find the radius of the circle tangent to sides \(A C\) and \(B C\) and to the circumcircle of triangle \(A B C\). | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $e^{i \theta} = \frac{3 + i \sqrt{8}}{4},$ then find $\sin 6 \theta.$ | {
"answer": "-\\frac{855 \\sqrt{2}}{1024}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a, b, c, d, e, f, p, q$ are Arabic numerals and $b > c > d > a$, the difference between the four-digit numbers $\overline{c d a b}$ and $\overline{a b c d}$ is a four-digit number of the form $\overline{p q e f}$. If $\overline{e f}$ is a perfect square and $\overline{p q}$ is not divisible by 5, determine the four-digit number $\overline{a b c d}$ and explain the reasoning. | {
"answer": "1983",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Households A, B, and C plan to subscribe to newspapers. There are 5 different types of newspapers available. Each household subscribes to two different newspapers. It is known that each pair of households shares exactly one common newspaper. How many different subscription ways are there for the three households? | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the maximum number of different sets consisting of three terms that form arithmetic progressions and can be chosen from a sequence of real numbers \( a_1, a_2, \ldots, a_{101} \), where
\[
a_1 < a_2 < a_3 < \cdots < a_{101} .
\] | {
"answer": "2500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The bases of a trapezoid are 3 cm and 5 cm. One of the diagonals of the trapezoid is 8 cm, and the angle between the diagonals is $60^{\circ}$. Find the perimeter of the trapezoid. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of all roots of the equation:
$$
\begin{gathered}
\sqrt{2 x^{2}-2024 x+1023131} + \sqrt{3 x^{2}-2025 x+1023132} + \sqrt{4 x^{2}-2026 x+1023133} = \\
= \sqrt{x^{2}-x+1} + \sqrt{2 x^{2}-2 x+2} + \sqrt{3 x^{2}-3 x+3}
\end{gathered}
$$ | {
"answer": "2023",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five soccer teams play a match where each team plays every other team exactly once. Each match awards 3 points to the winner, 0 points to the loser, and 1 point to each team in the event of a draw. After all matches have been played, the total points of the five teams are found to be five consecutive natural numbers. Let the teams ranked 1st, 2nd, 3rd, 4th, and 5th have drawn $A$, $B$, $C$, $D$, and $E$ matches respectively. Determine the five-digit number $\overline{\mathrm{ABCDE}}$. | {
"answer": "13213",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest five-digit positive integer such that it is not a multiple of 11, and any number obtained by deleting some of its digits is also not divisible by 11. | {
"answer": "98765",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum and product of the distinct prime factors of 420? | {
"answer": "210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\sigma(n)$ be the number of positive divisors of $n$ , and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$ . By convention, $\operatorname{rad} 1 = 1$ . Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \]*Proposed by Michael Kural* | {
"answer": "164",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a round glass, the axial cross-section of which is the graph of the function \( y = x^4 \), a cherry (a sphere with radius \( r \)) is placed. For which maximum \( r \) will the sphere touch the bottom point of the glass? (In other words, what is the maximum radius \( r \) of the circle lying in the region \( y \geq x^4 \) and containing the origin?) | {
"answer": "\\frac{3 \\cdot 2^{1/3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 4 spheres in space with radii 2, 2, 3, and 3, respectively. Each sphere is externally tangent to the other 3 spheres. Additionally, there is a small sphere that is externally tangent to all 4 of these spheres. Find the radius of the small sphere. | {
"answer": "6/11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=x- \frac {1}{x}+2a\ln x$ $(a\in\mathbb{R})$.
$(1)$ Discuss the monotonicity of $f(x)$;
$(2)$ If $f(x)$ has two extreme values $x_{1}$ and $x_{2}$, where $x_{2}\in[e,+\infty)$, find the minimum value of $f(x_{1})-f(x_{2})$. | {
"answer": "\\frac {4}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the bisector \(BD\) is drawn, and in triangles \(ABD\) and \(CBD\), the bisectors \(DE\) and \(DF\) are drawn, respectively. It turns out that \(EF \parallel AC\). Find the angle \(\angle DEF\). | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regular octagon $ABCDEFGH$ is divided into eight smaller equilateral triangles, such as $\triangle ABJ$, where $J$ is the center of the octagon. By connecting every second vertex starting from $A$, we obtain a larger equilateral triangle $\triangle ACE$. Compute the ratio of the area of $\triangle ABJ$ to the area of $\triangle ACE$. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We draw the diagonals of the convex quadrilateral $ABCD$, then find the centroids of the 4 triangles formed. What fraction of the area of quadrilateral $ABCD$ is the area of the quadrilateral determined by the 4 centroids? | {
"answer": "\\frac{2}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The real numbers \(a, b, c\) satisfy the following system of equations:
$$
\left\{\begin{array}{l}
\frac{a b}{a+b}=4 \\
\frac{b c}{b+c}=5 \\
\frac{c a}{c+a}=7
\end{array}\right.
$$
Find the value of the expression \(\frac{a b c}{a b + b c + c a}\). | {
"answer": "280/83",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \((n+1)^{\alpha+1}-n^{\alpha+1} < n^{\alpha}(\alpha+1) < n^{\alpha+1}-(n-1)^{\alpha+1}, -1 < \alpha < 0\). Let \(x = \sum_{k=4}^{10^{6}} \frac{1}{\sqrt[3]{k}}\), find the integer part of \(x\). | {
"answer": "146",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We inscribed a regular hexagon $ABCDEF$ in a circle and then drew semicircles outward over the chords $AB$, $BD$, $DE$, and $EA$. Calculate the ratio of the combined area of the resulting 4 crescent-shaped regions (bounded by two arcs each) to the area of the hexagon. | {
"answer": "2:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Huanhuan and Lele are playing a game together. In the first round, they both gain the same amount of gold coins, and in the second round, they again gain the same amount of gold coins.
At the beginning, Huanhuan says: "The number of my gold coins is 7 times the number of your gold coins."
At the end of the first round, Lele says: "The number of your gold coins is 6 times the number of my gold coins now."
At the end of the second round, Huanhuan says: "The number of my gold coins is 5 times the number of your gold coins now."
Find the minimum number of gold coins Huanhuan had at the beginning. | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $C$ is an interior angle of $\triangle ABC$, and the vectors $\overrightarrow{m}=(2\cos C-1,-2)$, $\overrightarrow{n}=(\cos C,\cos C+1)$. If $\overrightarrow{m}\perp \overrightarrow{n}$, calculate the value of $\angle C$. | {
"answer": "\\dfrac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A farmer sold domestic rabbits. By the end of the market, he sold exactly one-tenth as many rabbits as the price per rabbit in forints. He then distributed the revenue between his two sons. Starting with the older son, the boys alternately received one-hundred forint bills, but at the end, the younger son received only a few ten-forint bills. The father then gave him his pocket knife and said that this made their shares equal in value. How much was the pocket knife worth? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given unit vectors $\vec{a}$ and $\vec{b}$ with an acute angle between them, for any $(x, y) \in \{(x, y) \mid | x \vec{a} + y \vec{b} | = 1, xy \geq 0 \}$, it holds that $|x + 2y| \leq \frac{8}{\sqrt{15}}$. Find the minimum possible value of $\vec{a} \cdot \vec{b}$. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, the same letters to the same digits). The result was the word "ГВАТЕМАЛА". How many different numbers could Egor have originally written if his number was divisible by 30? | {
"answer": "21600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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