problem stringlengths 10 5.15k | answer dict |
|---|---|
Petya can draw only 4 things: a sun, a ball, a tomato, and a banana. Today he drew several things, including exactly 15 yellow items, 18 round items, and 13 edible items. What is the maximum number of balls he could have drawn?
Petya believes that all tomatoes are round and red, all balls are round and can be of any color, and all bananas are yellow and not round. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f(x) = \left\{ \begin{array}{cc} 1 & 1 \leqslant x \leqslant 2 \\ x-1 & 2 < x \leqslant 3 \end{array} \right. \). For any \( a \,(a \in \mathbb{R}) \), define \( v(a) = \max \{ f(x) - a x \mid x \in [1,3] \} - \min \{ f(x) - a x \mid x \in [1,3] \} \). Draw the graph of \( v(a) \) and find the minimum value of \( v(a) \). | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the flux of the vector field \(\mathbf{a} = y^2 \mathbf{j} + z \mathbf{k}\) through the part of the surface \(z = x^2 + y^2\), cut off by the plane \(z=2\). The normal vector is taken to be outward with respect to the region bounded by the paraboloid. | {
"answer": "-2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A production team in a factory is manufacturing a batch of parts. Initially, when each worker is on their own original position, the task can be completed in 9 hours. If the positions of workers $A$ and $B$ are swapped, and other workers' efficiency remains the same, the task can be completed one hour earlier. Similarly, if the positions of workers $C$ and $D$ are swapped, the task can also be completed one hour earlier. How many minutes earlier can the task be completed if the positions of $A$ and $B$ as well as $C$ and $D$ are swapped at the same time, assuming other workers' efficiency remains unchanged? | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The common ratio of the geometric sequence \( a+\log _{2} 3, a+\log _{1} 3, a+\log _{8} 3 \) is ______. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer \( m \) such that the equation regarding \( x, y, \) and \( z \):
\[ 2^x + 3^y - 5^z = 2m \]
has no positive integer solutions. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles trapezoid \(ABCD\), the larger base \(AD = 12\) and \(AB = 6\). Find the distance from point \(O\), the intersection of the diagonals, to point \(K\), the intersection of the extensions of the lateral sides, given that the extensions of the lateral sides intersect at a right angle. | {
"answer": "\\frac{12(3 - \\sqrt{2})}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the minimum number of points that can be chosen on a circle with a circumference of 1956 so that for each of these points there is exactly one chosen point at a distance of 1 and exactly one at a distance of 2 (distances are measured along the circle)? | {
"answer": "1304",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Positive numbers \(a\), \(b\), and \(c\) satisfy the following equations:
\[ a^{2} + a b + b^{2} = 1 \]
\[ b^{2} + b c + c^{2} = 3 \]
\[ c^{2} + c a + a^{2} = 4 \]
Find \(a + b + c\). | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each positive integer $n$, an associated non-negative integer $f(n)$ is defined to satisfy the following three rules:
i) $f(a b)=f(a)+f(b)$.
ii) $f(n)=0$ if $n$ is a prime greater than 10.
iii) $f(1)<f(243)<f(2)<11$.
Given that $f(2106)<11$, determine the value of $f(96)$. | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two types of tables in a restaurant: a square table can seat 4 people, and a round table can seat 9 people. The restaurant manager calls a number a "wealth number" if the total number of diners can exactly fill a certain number of tables. How many "wealth numbers" are there among the numbers from 1 to 100? | {
"answer": "88",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product of three consecutive even numbers has the form $87XXXXX8$. Provide the 5 missing digits. ($X$ does not necessarily represent the same digits.) | {
"answer": "52660",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given ten 0's and ten 1's, how many 0-1 binary sequences can be formed such that no three or more consecutive 0's are together? For example, 01001001010011101011 is such a sequence, but the sequence 01001000101001110111 does not satisfy this condition. | {
"answer": "24068",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equations of the sides of a quadrilateral are:
$$
y=-x+7, \quad y=\frac{x}{2}+1, \quad y=-\frac{3}{2} x+2 \quad \text {and} \quad y=\frac{7}{4} x+\frac{3}{2}.
$$
Determine the area of the quadrilateral. | {
"answer": "\\frac{327}{52}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the set \( \{x \mid a \leqslant x \leqslant b\} \), we define \( b-a \) as its length. Let the set \( A=\{x \mid a \leqslant x \leqslant a+1981\} \), \( B=\{x \mid b-1014 \leqslant x \leqslant b\} \), and both \( A \) and \( B \) are subsets of the set \( U=\{x \mid 0 \leqslant x \leqslant 2012\} \). The minimum length of the set \( A \cap B \) is ______. | {
"answer": "983",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Investment funds A, B, and C claim that they can earn profits of 200%, 300%, and 500% respectively in one year. Tommy has $90,000 and plans to invest in these funds. However, he knows that only one of these funds can achieve its claim while the other two will close down. He has thought of an investment plan which can guarantee a profit of at least \$n in one year. Find the greatest possible value of $n$. | {
"answer": "30000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 2006 positive integers \( a_{1}, a_{2}, \cdots, a_{2006} \) (which can be the same) such that the ratios \( \frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \cdots, \frac{a_{2005}}{a_{2006}} \) are all distinct. What is the minimum number of distinct numbers among \( a_{1}, a_{2}, \cdots, a_{2006} \)? | {
"answer": "46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(CD\) be a chord of a circle \(\Gamma_{1}\) and \(AB\) a diameter of \(\Gamma_{1}\) perpendicular to \(CD\) at \(N\) with \(AN > NB\). A circle \(\Gamma_{2}\) centered at \(C\) with radius \(CN\) intersects \(\Gamma_{1}\) at points \(P\) and \(Q\), and the segments \(PQ\) and \(CD\) intersect at \(M\). Given that the radii of \(\Gamma_{1}\) and \(\Gamma_{2}\) are 61 and 60 respectively, find the length of \(AM\). | {
"answer": "78",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is given that \( a = 103 \times 97 \times 10009 \). Find \( a \). | {
"answer": "99999919",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Numbers $1,2,3,4,5,6,7,$ and $8$ are placed at the vertices of a cube such that the sum of any three numbers belonging to any face of the cube is not less than 10. Find the minimum possible sum of four numbers belonging to one face. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a $10 \times 5$ grid, an ant starts from point $A$ and can only move right or up along the grid lines but is not allowed to pass through point $C$. How many different paths are there from point $A$ to point $B$? | {
"answer": "1827",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two identical cylindrical vessels are connected at the bottom with a small-diameter tube with a valve. While the valve was closed, water was poured into the first vessel and oil was poured into the second vessel, so that the liquid levels were identical and equal to $h = 40 \text{ cm}$. At what level will the water establish in the first vessel if the valve is opened? The density of water is 1000 kg/m³, the density of oil is 700 kg/m³. Neglect the volume of the connecting tube. Provide the answer in centimeters. | {
"answer": "32.94",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a taxi, a passenger can sit in the front and three passengers can sit in the back. In how many ways can four passengers sit in a taxi if one of these passengers wants to sit by the window? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number $ABCC$ (written in decimal system) given that
$$
ABCC = (DD - E) \cdot 100 + DD \cdot E
$$
where $A, B, C, D,$ and $E$ are distinct digits. | {
"answer": "1966",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve mayoral candidates each made a statement about how many times lies had been told before their turn. The first candidate said, "Before me, one lie was told." The second candidate said, "Now, two lies have been told." The third candidate said, "Now, three lies have been told," and so on, until the twelfth candidate who said, "Now, twelve lies have been told." It was later revealed that at least one candidate correctly counted how many times lies had been told up to their turn. How many lies in total were told by the candidates? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{-1}^{0}(x+2)^{3} \cdot \ln ^{2}(x+2) \, dx
$$ | {
"answer": "4 \\ln^{2} 2 - 2 \\ln 2 + \\frac{15}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f(n) \) defined on the set of natural numbers \(\mathbf{N}\), and satisfies:
\[
\begin{array}{l}
f(1) = f(2) = 1, \\
f(3n) = 3 f(n) - 2, \\
f(3n+1) = 3 f(n) + 1, \\
f(3n+2) = 3 f(n) + 4 \quad (n \in \mathbf{N}).
\end{array}
\]
Determine the largest positive integer \( n \) less than or equal to 1992 for which \( f(n) = n \). | {
"answer": "1093",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The real numbers \( x, y, z \) satisfy the equations \( x + y + z = 2 \) and \( xy + yz + zx = 1 \). Find the maximum possible value of \( x - y \). | {
"answer": "\\frac{2 \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Subset \( S \subseteq \{1, 2, 3, \ldots, 1000\} \) is such that if \( m \) and \( n \) are distinct elements of \( S \), then \( m + n \) does not belong to \( S \). What is the largest possible number of elements in \( S \)? | {
"answer": "501",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the volume of the tetrahedron \(ABCD\) is 40, \(AB = 7\), \(BC = 8\), \(AC = 9\), and the orthogonal projection of vertex \(D\) onto the plane \(ABC\) is precisely the incenter \(H\) of triangle \(ABC\), what is the surface area of the tetrahedron? | {
"answer": "60 + 12 \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the first 539 positive integers, we select some such that their sum is at least one-third of the sum of the original numbers. What is the minimum number of integers we need to select for this condition to be satisfied? | {
"answer": "99",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cashier, upon checking the account before leaving work, finds that the cash is 153 yuan less than the account book. She knows the actual amount collected cannot be wrong, so it must be due to a decimal point error during bookkeeping. What is the actual amount of the cash that was recorded incorrectly? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each vertex of the parallelogram $ABCD$ lies on the same side of the plane $S$ such that the distances of the vertices $A, B$, and $C$ from the plane $S$ are 4 cm, 6 cm, and 8 cm, respectively. The area of the projection of the parallelogram onto the plane $S$, which forms the quadrilateral $A'B'C'D'$, is $10 \text{ cm}^2$. What is the volume of the solid $ABCD A'B'C'D'$? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many five-digit numbers are there that are divisible by 5 and do not contain repeating digits? | {
"answer": "5712",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the 20 numbers 11, 12, 13, 14, ... 30, how many numbers must be chosen to ensure that there are at least two numbers whose sum is a multiple of 10? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Before the lesson, Nestor Petrovich wrote several words on the board. When the bell rang for the lesson, he noticed a mistake in the first word. If he corrects the mistake in the first word, the words with mistakes will constitute $24\%$, and if he erases the first word from the board, the words with mistakes will constitute $25\%$. What percentage of the total number of written words were words with mistakes before the bell rang for the lesson? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>b>0)$ have its right focus at $F$ and eccentricity $e$. A line passing through $F$ with a slope of 1 intersects the asymptotes of the hyperbola at points $A$ and $B$. If the midpoint of $A$ and $B$ is $M$ and $|FM|=c$, find $e$. | {
"answer": "\\sqrt[4]{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle, the bisector of an acute angle divides the opposite leg into segments of lengths 4 cm and 5 cm. Determine the area of the triangle. | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular hexagon \(ABCDEF\) is inscribed in a circle with a radius of \(3+\sqrt{3}\). Find the radius of the inscribed circle of triangle \(ACD\). | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{\pi}\left(9 x^{2}+9 x+11\right) \cos 3 x \, dx
$$ | {
"answer": "-2\\pi - 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many nine-digit integers of the form 'pqrpqrpqr' are multiples of 24? (Note that p, q, and r need not be different.) | {
"answer": "112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular tetrahedron \(ABCD\), points \(E\) and \(F\) are on edges \(AB\) and \(AC\) respectively, satisfying \(BE=3\) and \(EF=4\), and \(EF \parallel\) plane \(BCD\). Find the area of \(\triangle DEF\). | {
"answer": "2 \\sqrt{33}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a point inside an equilateral triangle with side length \( d \) whose distances from the vertices are 3, 4, and 5 units. Find the side length \( d \). | {
"answer": "\\sqrt{25 + 12 \\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive real number \( x \) is such that
$$
\sqrt[3]{1-x^{3}} + \sqrt[3]{1+x^{3}} = 1.
$$
Find \( x^{2} \). | {
"answer": "\\frac{\\sqrt[3]{28}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Li and Xiao Hua are racing up the stairs. When Xiao Li reaches the 5th floor, Xiao Hua has reached the 3rd floor. At this rate, how many floors will Xiao Hua have reached when Xiao Li reaches the 25th floor? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sets
$$
\begin{array}{l}
M=\{x, x y, \lg (x y)\} \\
N=\{0,|x|, y\},
\end{array}
$$
and $M=N$, determine the value of:
$$\left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\frac{1}{y^{3}}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right).$$ | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of the digits of all numbers from one to one billion? | {
"answer": "40500000001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the radius of the circumscribed circle around an isosceles trapezoid with bases 2 and 14 and a lateral side of 10. | {
"answer": "5\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On an island of Liars and Knights, a circular arrangement is called correct if each person in the circle can say that among their two neighbors, there is at least one member of their tribe. One day, 2019 natives formed a correct circle. A liar approached them and said, "Now we too can form a correct circle." How many knights could have been in the initial arrangement? | {
"answer": "1346",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
2002 is a palindromic year, meaning it reads the same backward and forward. The previous palindromic year was 11 years ago (1991). What is the maximum number of non-palindromic years that can occur consecutively (between the years 1000 and 9999)? | {
"answer": "109",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At the intersection of perpendicular roads, a highway from Moscow to Kazan intersects with a road from Vladimir to Ryazan. Dima and Tolya are traveling at constant speeds from Moscow to Kazan and from Vladimir to Ryazan, respectively.
When Dima crossed the intersection, Tolya was 900 meters away from it. When Tolya crossed the intersection, Dima was 600 meters away from it. How many meters will be between the boys when Tolya travels 900 meters from the moment he crosses the intersection? | {
"answer": "1500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A company offers its employees a salary increase, provided they increase their work productivity by 2% per week. If the company operates 5 days a week, by what percentage per day must employees increase their productivity to achieve the desired goal? | {
"answer": "0.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the Cartesian plane, find the number of integer coordinate points (points where both x and y are integers) that satisfy the following system of inequalities:
\[
\begin{cases}
y \leq 3x, \\
y \geq \frac{1}{3}x, \\
x + y \leq 100.
\end{cases}
\] | {
"answer": "2551",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pedestrian is moving in a straight line towards a crosswalk at a constant speed of 3.6 km/h. Initially, the distance from the pedestrian to the crosswalk is 40 meters. The length of the crosswalk is 6 meters. What distance from the crosswalk will the pedestrian be after two minutes? | {
"answer": "74",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A necklace has a total of 99 beads. Among them, the first bead is white, the 2nd and 3rd beads are red, the 4th bead is white, the 5th, 6th, 7th, and 8th beads are red, the 9th bead is white, and so on. Determine the total number of red beads on this necklace. | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In square \( A B C D \), \( P \) and \( Q \) are points on sides \( C D \) and \( B C \), respectively, such that \( \angle A P Q = 90^\circ \). If \( A P = 4 \) and \( P Q = 3 \), find the area of \( A B C D \). | {
"answer": "\\frac{256}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In two days, 50 financiers raised funds to combat a new virus. Each of them contributed a one-time whole number amount in thousands of rubles, not exceeding 100. Each contribution on the first day did not exceed 50 thousand rubles, while each on the second day was greater than this amount; and no pair of all 50 contributions differed by exactly 50 thousand rubles. What was the total amount collected? | {
"answer": "2525",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ana and Luíza train every day for the Big Race that will take place at the end of the year at school, each running at the same speed. The training starts at point $A$ and ends at point $B$, which are $3000 \mathrm{~m}$ apart. They start at the same time, but when Luíza finishes the race, Ana still has $120 \mathrm{~m}$ to reach point $B$. Yesterday, Luíza gave Ana a chance: "We will start at the same time, but I will start some meters before point $A$ so that we arrive together." How many meters before point $A$ should Luíza start? | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways can the king reach from C5 to H2 on a chessboard using the shortest path (with the fewest moves)? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One morning at 9:00 AM, a pedestrian named Fedya left the village of Fedino and headed towards the village of Novoverandovo. At the same time, a cyclist named Vera left Novoverandovo heading towards Fedya. It is known that by the time they met, Fedya had covered one-third of the distance between the villages. However, if Fedya had left an hour earlier, he would have covered half the distance by the time they met. At what time did Fedya and Vera meet? Assume that the speeds of Vera and Fedya are constant. | {
"answer": "11:00",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Several girls (all of different ages) were picking white mushrooms in the forest. They distributed the mushrooms as follows: the youngest received 20 mushrooms and 0.04 of the remainder. The next oldest received 21 mushrooms and 0.04 of the remainder, and so on. It turned out that they all received the same amount. How many mushrooms were collected, and how many girls were there? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\(ABCD\) is a square and \(X\) is a point on the side \(DA\) such that the semicircle with diameter \(CX\) touches the side \(AB\). Find the ratio \(AX: XD\). | {
"answer": "1 : 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f(x)=\frac{x^{3}}{1+x^{3}} \), find the value of the sum
\[
f\left(\frac{1}{1}\right)+f\left(\frac{2}{1}\right)+\ldots+f\left(\frac{2007}{1}\right) + f\left(\frac{1}{2}\right)+f\left(\frac{2}{2}\right)+\ldots+f\left(\frac{2007}{2}\right) + \ldots + f\left(\frac{1}{2007}\right)+f\left(\frac{2}{2007}\right)+\ldots+f\left(\frac{2007}{2007}\right)
\] | {
"answer": "\\frac{2007^2}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular hexagonal pyramid \( M A B C D E F \). Point \( K \) bisects edge \( B M \). Find the ratio in which the plane \( F E K \) divides edge \( A M \) (at point \( X \)). | {
"answer": "2:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 5 integers written on a board. By summing them pairwise, the following set of 10 numbers is obtained: \( 5, 9, 10, 11, 12, 16, 16, 17, 21, 23 \). Determine which numbers are written on the board. Provide the product of these numbers in your answer. | {
"answer": "5292",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangular pyramid \( S-ABC \) with a base that is an isosceles right triangle with \( AB \) as the hypotenuse, \( SA = SB = SC = 2 \), and \( AB = 2 \), suppose points \( S, A, B, C \) are all on a sphere with center \( Q \). Find the distance from point \( O \) to the plane \( ABC \). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all values of \( x \) for which the greater of the numbers \( \sqrt{\frac{x}{2}} \) and \( \operatorname{tg} x \) is not greater than 1. Provide the total length of the intervals on the number line that satisfy this condition, rounding the result to the nearest hundredth if necessary. | {
"answer": "1.21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A \) be a set with 225 elements, and \( A_{1}, A_{2}, \cdots, A_{11} \) be 11 subsets of \( A \) each containing 45 elements, such that for any \( 1 \leq i < j \leq 11 \), \(|A_{i} \cap A_{j}| = 9\). Find the minimum value of \(|A_{1} \cup A_{2} \cup \cdots \cup A_{11}|\). | {
"answer": "165",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Brothers Lyosha and Sasha decided to get from home to the skate park. They left at the same time, but Lyosha walked with the skateboard in his hands, while Sasha rode the skateboard. It is known that Sasha rides the skateboard 3 times faster than Lyosha walks with the skateboard. After some time, they simultaneously changed their mode of transportation: Lyosha started riding the skateboard, and Sasha started walking. As a result, the speed of each of them changed by a factor of 2: Lyosha's speed increased, and Sasha's speed decreased. They both arrived at the skate park at the same time. How many meters did Sasha ride on the skateboard if the distance from home to the skate park is 3300 meters? | {
"answer": "1100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cyclist rode 96 km 2 hours faster than expected. At the same time, he covered 1 km more per hour than he expected to cover in 1 hour 15 minutes. What was his speed? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are exactly 120 ways to color five cells in a $5 \times 5$ grid such that each row and each column contains exactly one colored cell.
There are exactly 96 ways to color five cells in a $5 \times 5$ grid without the corner cell, such that each row and each column contains exactly one colored cell.
How many ways are there to color five cells in a $5 \times 5$ grid without two corner cells, such that each row and each column contains exactly one colored cell? | {
"answer": "78",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), \(\angle A = 45^\circ\) and \(M\) is the midpoint of \(\overline{BC}\). \(\overline{AM}\) intersects the circumcircle of \(ABC\) for the second time at \(D\), and \(AM = 2MD\). Find \(\cos \angle AOD\), where \(O\) is the circumcenter of \(ABC\). | {
"answer": "-\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 2019 bags, each bag containing 2019 beads with a total weight of 1 kilogram. The beads in each bag are numbered \(1, 2, \cdots, 2019\). A selection is considered proper if and only if beads with different numbers are selected from different bags (at most one bead from each bag), and the total weight is at least 1 kilogram. Find the maximum value of \(k\) such that there always exist at least \(k\) different proper selections. | {
"answer": "2018!",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The school plans to organize a movie viewing for the students either on January 4th or January 10th. After finalizing the date, the teacher informs the class leader. However, due to the similarity in pronunciation between "four" and "ten," there is a 10% chance that the class leader hears it incorrectly (mistaking 4 for 10 or 10 for 4). The class leader then informs Xiaoming, who also has a 10% chance of hearing it incorrectly. What is the probability that Xiaoming correctly believes the movie date? | {
"answer": "0.82",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a ring road, there are three cities: $A$, $B$, and $C$. It is known that the path from $A$ to $C$ along the arc not containing $B$ is three times longer than the path through $B$. The path from $B$ to $C$ along the arc not containing $A$ is four times shorter than the path through $A$. By what factor is the path from $A$ to $B$ shorter along the arc not containing $C$ than the path through $C$? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a general triangle \(ABC\) with points \(K, L, M, N, U\) on its sides:
- Point \(K\) is the midpoint of side \(AC\).
- Point \(U\) is the midpoint of side \(BC\).
- Points \(L\) and \(M\) lie on segments \(CK\) and \(CU\) respectively, such that \(LM \parallel KU\).
- Point \(N\) lies on segment \(AB\) such that \(|AN| : |AB| = 3 : 7\).
- The ratio of the areas of polygons \(UMLK\) and \(MLKNU\) is 3 : 7.
Determine the length ratio of segments \(LM\) and \(KU\). | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Junior and Carlson ate a barrel of jam and a basket of cookies, starting and finishing at the same time. Initially, Junior ate the cookies and Carlson ate the jam, then (at some point) they switched. Carlson ate both the jam and the cookies three times faster than Junior. What fraction of the jam did Carlson eat, given that they ate an equal amount of cookies? | {
"answer": "9/10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all natural numbers whose own divisors can be paired such that the numbers in each pair differ by 545. An own divisor of a natural number is a natural divisor different from one and the number itself. | {
"answer": "1094",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a nursery group, there are two small Christmas trees and five children. The caregivers want to split the children into two dance circles around each tree, with at least one child in each circle. The caregivers distinguish between the children but not between the trees: two such groupings are considered identical if one can be obtained from the other by swapping the trees (along with the corresponding circles) and rotating each circle around its tree. How many ways can the children be divided into dance circles? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are numbers $1, 2, \cdots, 36$ to be filled into a $6 \times 6$ grid, with each cell containing one number. Each row must be in increasing order from left to right. What is the minimum sum of the six numbers in the third column? | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their place to the spectator (there were no draws). As a result, Egor participated in 13 games, and Nikita participated in 27 games. How many games did Innokentiy play? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequences \(\{a_{n}\}\) and \(\{b_{n}\}\) with general terms \(a_{n} = 2^{n}\) and \(b_{n} = 5n - 2\), find the sum of all elements in the set \(\{a_{1}, a_{2}, \cdots, a_{2019}\} \cap \{b_{1}, b_{2}, \cdots, b_{2019}\}\). | {
"answer": "2184",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer \( n > 1 \) such that the arithmetic mean of the squares of the integers \( 1^2, 2^2, 3^2, \ldots, n^2 \) is a perfect square. | {
"answer": "337",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A thief on a bus gets off at a bus stop and walks in the direction opposite to the bus’s travel direction. The bus continues its journey, and a passenger realizes they have been robbed. The passenger gets off at the next stop and starts chasing the thief. If the passenger's speed is twice that of the thief, the bus's speed is ten times the speed of the thief, and the bus takes 40 seconds to travel between two stops, how many seconds will it take for the passenger to catch up with the thief after getting off the bus? | {
"answer": "440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that the numbers \( x, y, z \) form an arithmetic progression with the common difference \( \alpha=\arccos \frac{2}{3} \), and the numbers \( \frac{1}{\sin x}, \frac{6}{\sin y}, \frac{1}{\sin z} \) also form an arithmetic progression in the given order. Find \( \sin ^{2} y \). | {
"answer": "\\frac{5}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( n \) be the smallest positive integer that satisfies the following conditions: (1) \( n \) is a multiple of 75; (2) \( n \) has exactly 75 positive integer factors (including 1 and itself). Find \(\frac{n}{75}\). | {
"answer": "432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid \(ABCD\), \(\overrightarrow{AB} = 2 \overrightarrow{DC}\), \(|\overrightarrow{BC}| = 6\). Point \(P\) is a point in the plane of trapezoid \(ABCD\) and satisfies \(\overrightarrow{AP} + \overrightarrow{BP} + 4 \overrightarrow{DP} = 0\). Additionally, \(\overrightarrow{DA} \cdot \overrightarrow{CB} = |\overrightarrow{DA}| \cdot |\overrightarrow{DP}|\). Point \(Q\) is a variable point on side \(AD\). Find the minimum value of \(|\overrightarrow{PQ}|\). | {
"answer": "\\frac{4 \\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sasha wrote down numbers from one to one hundred, and Misha erased some of them. Among the remaining numbers, 20 have the digit one in their recording, 19 have the digit two in their recording, and 30 numbers have neither the digit one nor the digit two. How many numbers did Misha erase? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point $Q$ is randomly chosen inside an equilateral triangle $DEF$. What is the probability that the area of $\triangle DEQ$ is greater than both $\triangle DFQ$ and $\triangle EFQ$? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The first digit on the left of a six-digit number is 1. If this digit is moved to the last place, the resulting number is 64 times greater than the original number. Find the original number. | {
"answer": "142857",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Natural numbers are arranged in a spiral, turning the first bend at 2, the second bend at 3, the third bend at 5, and so on. What is the number at the twentieth bend? | {
"answer": "71",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parabola \( y^2 = 6x \) with two variable points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), where \( x_1 \neq x_2 \) and \( x_1 + x_2 = 4 \). The perpendicular bisector of segment \( AB \) intersects the x-axis at point \( C \). Find the maximum area of triangle \( \triangle ABC \). | {
"answer": "\\frac{14}{3}\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area in the plane contained by the graph of
\[|2x + 3y| + |2x - 3y| \le 12.\] | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a set of sample data with $8$ numbers, the average is $8$, and the variance is $12$. Two unknown numbers are added to this set of sample data to form a new set of sample data. It is known that the average of the new sample data is $9$. Find the minimum value of the variance of the new sample data. | {
"answer": "13.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a course conducted by Professor Jones, each student is on average absent for one day out of a 40-day course. What is the probability that out of any two randomly selected students, one student will be absent while the other is present? Express your answer as a percent rounded to the nearest tenth. | {
"answer": "4.9\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \overrightarrow{m} \cdot \overrightarrow{n} - \frac{1}{2}$, where $\overrightarrow{m}=( \sqrt{3}\sin x,\cos x)$ and $\overrightarrow{n}=(\cos x,-\cos x)$.
1. Find the range of the function $y=f(x)$ when $x\in[0, \frac{\pi}{2}]$.
2. In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $c=2$, $a=3$, and $f(B)=0$. Find the value of side $b$. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p,$ $q,$ $r,$ $s$ be real numbers such that $p + q + r + s = 10$ and
\[ pq + pr + ps + qr + qs + rs = 20. \]
Find the largest possible value of $s$. | {
"answer": "\\frac{5 + \\sqrt{105}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{\begin{array}{l}{x=2+2\cos\varphi}\\{y=2\sin\varphi}\end{array}\right.$ ($\varphi$ is the parameter). Taking the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, the polar equation of curve $C_{2}$ is $\rho =4\sin \theta$. <br/>$(Ⅰ)$ Find the general equation of curve $C_{1}$ and the Cartesian equation of $C_{2}$; <br/>$(Ⅱ)$ Given that the polar equation of curve $C_{3}$ is $\theta =\alpha$, $0 \lt \alpha\ \ \lt \pi$, $\rho \in R$, point $A$ is the intersection point of curves $C_{3}$ and $C_{1}$, point $B$ is the intersection point of curves $C_{3}$ and $C_{2}$, and both $A$ and $B$ are different from the origin $O$, and $|AB|=4\sqrt{2}$, find the value of the real number $\alpha$. | {
"answer": "\\frac{3\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
You are given a number composed of three different non-zero digits, 7, 8, and a third digit which is not 7 or 8. Find the minimum value of the quotient of this number divided by the sum of its digits. | {
"answer": "11.125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $tanA=\frac{2}{3}$, find the value of $\cos A$. | {
"answer": "\\frac{3\\sqrt{13}}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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