problem stringlengths 10 5.15k | answer dict |
|---|---|
A team of fishermen planned to catch 1800 centners of fish within a certain timeframe. During one-third of this period, there was a storm, causing them to fall short of their daily target by 20 centners each day. However, on the remaining days, the team managed to catch 20 centners more than the daily norm and completed the planned goal one day ahead of schedule. How many centners of fish were they planning to catch daily? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can two rooks be arranged on a chessboard such that one cannot capture the other? (A rook can capture another if it is on the same row or column of the chessboard). | {
"answer": "3136",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Take a clay sphere of radius 13, and drill a circular hole of radius 5 through its center. Take the remaining "bead" and mold it into a new sphere. What is this sphere's radius? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Teams A, B, and C need to complete two projects, $A$ and $B$. The workload of project $B$ is $\frac{1}{4}$ more than the workload of project $A$. If teams A, B, and C work alone, they can finish project $A$ in 20 days, 24 days, and 30 days respectively. To complete these two projects simultaneously, team A is assigned to project $A$, and teams B and C work together on project $B$. After a few days, team C joins team A to complete project $A$. How many days did team C and team B work together? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangular pyramid \( S-ABC \) has a base that is an equilateral triangle with a side length of 4. It is given that \( AS = BS = \sqrt{19} \) and \( CS = 3 \). Find the surface area of the circumscribed sphere of the triangular pyramid \( S-ABC \). | {
"answer": "\\frac{268\\pi}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three shepherds met on a large road, each driving their respective herds. Jack says to Jim:
- If I give you 6 pigs for one horse, your herd will have twice as many heads as mine.
And Dan remarks to Jack:
- If I give you 14 sheep for one horse, your herd will have three times as many heads as mine.
Jim, in turn, says to Dan:
- And if I give you 4 cows for one horse, your herd will become 6 times larger than mine.
The deals did not take place, but can you still say how many heads of livestock were in the three herds? | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a, b, c, d, e \) be positive integers whose sum is 2018. Let \( M = \max (a+b, b+c, c+d, d+e) \). Find the smallest possible value of \( M \). | {
"answer": "673",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Positive real numbers \( x, y, z \) satisfy
\[
\left\{
\begin{array}{l}
\frac{2}{5} \leqslant z \leqslant \min \{x, y\}, \\
xz \geqslant \frac{4}{15}, \\
yz \geqslant \frac{1}{5}.
\end{array}
\right.
\]
Find the maximum value of \( \frac{1}{x} + \frac{2}{y} + \frac{3}{z} \). | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We traveled by train from Anglchester to Klinkerton. But an hour after the train started, a locomotive malfunction was discovered. We had to continue the journey at a speed that was $\frac{3}{5}$ of the original speed. As a result, we arrived in Klinkerton with a delay of 2 hours, and the driver said that if the breakdown had occurred 50 miles further, the train would have arrived 40 minutes earlier.
What is the distance from Anglchester to Klinkerton? | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the final round of a giraffe beauty contest, two giraffes named Tall and Spotted have made it to this stage. There are 105 voters divided into 5 districts, each district divided into 7 sections, with each section having 3 voters. Voters select the winner in their section by majority vote; in a district, the giraffe winning the majority of sections wins the district; finally, the giraffe winning the majority of districts is declared the contest winner. The giraffe named Tall won. What is the minimum number of voters who could have voted for him? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On side \(BC\) and on the extension of side \(AB\) through vertex \(B\) of triangle \(ABC\), points \(M\) and \(K\) are located, respectively, such that \(BM: MC = 4: 5\) and \(BK: AB = 1: 5\). Line \(KM\) intersects side \(AC\) at point \(N\). Find the ratio \(CN: AN\). | {
"answer": "5/24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum
$$
\frac{1}{1 \times 2 \times 3}+\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\cdots+\frac{1}{100 \times 101 \times 102}
$$
can be expressed as $\frac{a}{b}$, a fraction in its simplest form. Find $a+b$. | {
"answer": "12877",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Refer to the diagram, $P$ is any point inside the square $O A B C$ and $b$ is the minimum value of $P O + P A + P B + P C$. Find $b$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The price of an item is an integer number of yuan. With 100 yuan, you can buy up to 3 items. Person A and Person B each have a certain number of 100-yuan bills. The amount of money Person A has can buy at most 7 items, and the amount of money Person B has can buy at most 14 items. Together, they can buy 1 more item than the sum of what each can buy individually. What is the price of each item in yuan? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Entrepreneurs Vasiliy Petrovich and Petr Gennadievich opened a clothing factory "ViP." Vasiliy Petrovich invested 200 thousand rubles, while Petr Gennadievich invested 350 thousand rubles. The factory was successful, and after a year, Anastasia Alekseevna approached them with an offer to buy part of the shares. They agreed, and after the deal, each owned a third of the company's shares. Anastasia Alekseevna paid 1,100,000 rubles for her share. Determine who of the entrepreneurs is entitled to a larger portion of this money. In the answer, write the amount he will receive. | {
"answer": "1000000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lengths of the diagonals of a rhombus and the length of its side form a geometric progression. Find the sine of the angle between the side of the rhombus and its longer diagonal, given that it is greater than \( \frac{1}{2} \). | {
"answer": "\\sqrt{\\frac{\\sqrt{17}-1}{8}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many parallelograms with sides 1 and 2, and angles \(60^{\circ}\) and \(120^{\circ}\), can be placed inside a regular hexagon with side length 3? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( x + y + z = xy + yz + zx \), find the minimum value of \( \frac{x}{x^2 + 1} + \frac{y}{y^2 + 1} + \frac{z}{z^2 + 1} \). | {
"answer": "-1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A smaller square was cut out from a larger square in such a way that one side of the smaller square lies on a side of the original square. The perimeter of the resulting octagon is $40\%$ greater than the perimeter of the original square. By what percentage is the area of the octagon less than the area of the original square? | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside an angle of $60^{\circ}$, there is a point located at distances $\sqrt{7}$ and $2 \sqrt{7}$ from the sides of the angle. Find the distance of this point from the vertex of the angle. | {
"answer": "\\frac{14 \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a $4 \times 4$ grid, place candies according to the following requirements: (1) Each cell must contain candies; (2) In adjacent cells, the left cell has 1 fewer candy than the right cell and the upper cell has 2 fewer candies than the lower cell; (3) The bottom-right cell contains 20 candies. How many candies are there in total on the grid? (Adjacent cells are those that share a common edge) | {
"answer": "248",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \(a, b, c, d\) belong to the interval \([-7.5, 7.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line passing through the intersection point of the medians of triangle \(ABC\) intersects the sides \(BA\) and \(BC\) at points \(A^{\prime}\) and \(C_1\) respectively. Given that:
\(BA^{\prime} < BA = 3\), \(BC = 2\), and \(BA^{\prime} \cdot BC_1 = 3\). Find \(BA^{\prime}\). | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a clock, there are two hands: the hour hand and the minute hand. At a random moment in time, the clock stops. Find the probability that the angle between the hands on the stopped clock is acute. | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number, when raised to the sixth power, has digits which, when arranged in ascending order, are:
$$
0,2,3,4,4,7,8,8,9
$$
What is this number? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each edge of a regular tetrahedron is divided into three equal parts. Through each point of division, two planes are drawn, each parallel to one of the two faces of the tetrahedron that do not pass through this point. Into how many parts do the constructed planes divide the tetrahedron? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of solutions to the equation $\sin |x| = |\cos x|$ in the closed interval $[-10\pi, 10\pi]$ is __. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a tribal council meeting, 60 people spoke in turn. Each of them said only one phrase. The first three speakers all said the same thing: "I always tell the truth!" The next 57 speakers also said the same phrase: "Among the previous three speakers, exactly two of them told the truth." What is the maximum number of speakers who could have been telling the truth? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A number is the product of four prime numbers. What is this number if the sum of the squares of the four prime numbers is 476? | {
"answer": "1989",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petrov writes down odd numbers: \(1, 3, 5, \ldots, 2013\), and Vasechkin writes down even numbers: \(2, 4, \ldots, 2012\). Each of them calculates the sum of all the digits of all their numbers and tells it to the star student Masha. Masha subtracts Vasechkin's result from Petrov's result. What is the outcome? | {
"answer": "1007",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A heavy concrete platform anchored to the seabed in the North Sea supported an oil rig that stood 40 m above the calm water surface. During a severe storm, the rig toppled over. The catastrophe was captured from a nearby platform, and it was observed that the top of the rig disappeared into the depths 84 m from the point where the rig originally stood.
What is the depth at this location? (Neglect the height of the waves.) | {
"answer": "68.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square is constructed on one side of a regular octagon, outward. In the octagon, two diagonals intersect at point \( B \) (see the drawing). Find the measure of angle \( ABC \).
(A polygon is called regular if all its sides are equal and all its angles are equal.) | {
"answer": "22.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A, B, C \) be points on the same plane with \( \angle ACB = 120^\circ \). There is a sequence of circles \( \omega_0, \omega_1, \omega_2, \ldots \) on the same plane (with corresponding radii \( r_0, r_1, r_2, \ldots \) where \( r_0 > r_1 > r_2 > \cdots \)) such that each circle is tangent to both segments \( CA \) and \( CB \). Furthermore, \( \omega_i \) is tangent to \( \omega_{i-1} \) for all \( i \geq 1 \). If \( r_0 = 3 \), find the value of \( r_0 + r_1 + r_2 + \cdots \). | {
"answer": "\\frac{3}{2} + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two coal mines, Mine A and Mine B. Each gram of coal from Mine A releases 4 calories of heat when burned, and each gram of coal from Mine B releases 6 calories of heat when burned. The price per ton of coal at the production site is 20 yuan for Mine A and 24 yuan for Mine B. It is known that the transportation cost for a ton of coal from Mine A to city $N$ is 8 yuan. What should the transportation cost per ton of coal from Mine B to city $N$ be for it to be more economical than transporting coal from Mine A? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), side \(AC = 42\). The angle bisector \(CL\) is divided by the point of intersection of the angle bisectors of the triangle in the ratio \(2:1\) from the vertex. Find the length of side \(AB\) if the radius of the circle inscribed in triangle \(ABC\) is 14. | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \(a, b, c, d\) belong to the interval \([-8.5, 8.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | {
"answer": "306",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cube with its eight vertices labeled with the numbers $1, 2, 3, \cdots, 8$ in any order, define the number on each edge as $|i-j|$, where $i$ and $j$ are the labels of the edge’s endpoints. Let $S$ be the sum of the numbers on all the edges. Find the minimum value of $S$. | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given 1 coin of 0.1 yuan, 1 coin of 0.2 yuan, 1 coin of 0.5 yuan, 4 coins of 1 yuan, and 2 coins of 5 yuan, how many different amounts of money can be paid using any combination of these coins? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that 5 students each specialize in one subject (Chinese, Mathematics, Physics, Chemistry, History) and there are 5 test papers (one for each subject: Chinese, Mathematics, Physics, Chemistry, History), a teacher randomly distributes one test paper to each student. Calculate the probability that at least 4 students receive a test paper not corresponding to their specialized subject. | {
"answer": "89/120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the maximum possible value of the expression
$$
27abc + a\sqrt{a^2 + 2bc} + b\sqrt{b^2 + 2ca} + c\sqrt{c^2 + 2ab}
$$
where \(a, b, c\) are positive real numbers such that \(a + b + c = \frac{1}{\sqrt{3}}\). | {
"answer": "\\frac{2}{3 \\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of the expression \((\sqrt{36-4 \sqrt{5}} \sin x-\sqrt{2(1+\cos 2 x)}-2) \cdot (3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y)\). If the answer is not an integer, round it to the nearest whole number. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Add 3 digits after 325 to make a six-digit number such that it is divisible by 3, 4, and 5, and make this number as small as possible. What is the new six-digit number? | {
"answer": "325020",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence where each term is either 1 or 2, begins with the term 1, and between the $k$-th term 1 and the $(k+1)$-th term 1 there are $2^{k-1}$ terms of 2 (i.e., $1,2,1,2,2,1,2,2,2,2,1,2,2,2,2,2,2,2,2,1, \cdots$), what is the sum of the first 1998 terms in this sequence? | {
"answer": "3985",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya thought of a four-digit number and wrote down the product of each pair of its adjacent digits on the board. After that, he erased one product, and the numbers 20 and 21 remained on the board. What is the smallest number Vasya could have in mind? | {
"answer": "3745",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral
$$
\int_{0}^{\pi / 2} \frac{\sin x}{2+\sin x} \, dx
$$ | {
"answer": "\\frac{\\pi}{2} - \\frac{2 \\pi}{3 \\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two groups have an equal number of students. Each student studies at least one language: English or French. It is known that 5 people in the first group and 5 in the second group study both languages. The number of students studying French in the first group is three times less than in the second group. The number of students studying English in the second group is four times less than in the first group. What is the minimum possible number of students in one group? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a target on the wall consisting of five zones: a central circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the number of points for hitting each zone is inversely proportional to the probability of hitting that zone and that hitting the bullseye is worth 315 points. How many points is hitting the blue (second to last) zone worth? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the inequality $\cos \alpha_{1} \cos \alpha_{2} \cdots \cos \alpha_{n} + \sin \alpha_{1} \sin \alpha_{2} \cdots \sin \alpha_{n} \leqslant M$ always holds, then what is the minimum value of $M$? | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest root of the equation $|\sin (2 \pi x) - \cos (\pi x)| = ||\sin (2 \pi x)| - |\cos (\pi x)|$, which belongs to the interval $\left(\frac{1}{4}, 2\right)$. | {
"answer": "1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle with sides of lengths 3, 4, and 5. Three circles are constructed with radii of 1, centered at each vertex of the triangle. Find the total area of the portions of the circles that lie within the triangle. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fluffball and Shaggy the squirrels ate a basket of berries and a pack of seeds containing between 50 and 65 seeds, starting and finishing at the same time. Initially, Fluffball ate berries while Shaggy ate seeds. Later, they swapped tasks. Shaggy ate berries six times faster than Fluffball, and seeds three times faster. How many seeds did Shaggy eat if Shaggy ate twice as many berries as Fluffball? | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two divisions of an oil extraction company compete with each other for the amount of oil extracted. Depending on the availability of free tanks, each of them can extract the amount of tons defined by the following payoff matrix:
$$
\text { Extraction }=\left(\begin{array}{cc}
12 & 22 \\
32 & 2
\end{array}\right)
$$
Determine the optimal extraction strategies for both divisions and the value of the game. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 10 boys, each with a unique weight and height. For any two boys, $\mathbf{A}$ and $\mathbf{B}$, if $\mathbf{A}$ is heavier than $\mathbf{B}$, or if $\mathbf{A}$ is taller than $\mathbf{B}$, then $\mathbf{A}$ is not considered worse than $\mathbf{B}$. A boy who is not worse than the other 9 boys is called an "outstanding boy". Determine the maximum number of "outstanding boys" there can be among these 10 boys. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The principal of a certain school decided to take a photo of the graduating class of 2008. He arranged the students in parallel rows, all with the same number of students, but this arrangement was too wide for the field of view of his camera. To solve this problem, the principal decided to take one student from each row and place them in a new row. This arrangement displeased the principal because the new row had four students fewer than the other rows. He then decided to take one more student from each of the original rows and place them in the newly created row, and noticed that now all the rows had the same number of students, and finally took his photo. How many students appeared in the photo? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right triangle \( ABC \) with leg lengths equal to 1, a point \( P \) is chosen on one of the sides of the triangle. Find the maximum value of \( PA \cdot PB \cdot PC \). | {
"answer": "\\frac{\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
289. A remarkable number. Find a number such that its fractional part, its integer part, and the number itself form a geometric progression. | {
"answer": "\\frac{1+\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The set \( A = \left\{ x \left\lvert\, x = \left\lfloor \frac{5k}{6} \right\rfloor \right., k \in \mathbb{Z}, 100 \leqslant k \leqslant 999 \right\} \), where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \( x \). Determine the number of elements in set \( A \). | {
"answer": "750",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a natural number $N$, if at least eight out of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called an "Eight Immortals Number." What is the smallest "Eight Immortals Number" greater than $2000$? | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At the refreshment point, four athletes drank all the stock of Coke-Loco lemonade. If only athlete Bystrov had drunk half as much, one-tenth of the lemonade would have been left. If, additionally, athlete Shustrov had also drunk half as much, one-eighth of the lemonade would have been left. If, additionally to both of them, athlete Vostrov had also drunk half as much, one-third of the lemonade would have been left. What portion of the lemonade would be left if only athlete Pereskochizaborov had drunk half as much? | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Out of 60 right-angled triangles with legs of 2 and 3, a rectangle was formed. What can be the maximum perimeter of this rectangle? | {
"answer": "184",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice and Bob play on a $20 \times 20$ grid. Initially, all the cells are empty. Alice starts and the two players take turns placing stones on unoccupied cells. On her turn, Alice places a red stone on an empty cell that is not at a distance of $\sqrt{5}$ from any other cell containing a red stone. On his turn, Bob places a blue stone on an unoccupied cell. The game ends when a player can no longer place a stone. Determine the largest $K$ such that Alice can ensure to place at least $K$ red stones regardless of how Bob places his stones. | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many total strikes do the clocks make in a day if they strike once every half an hour and strike $1,2,3 \ldots 12$ times each hour? | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside a $60^{\circ}$ angle, there is a point situated at distances of $\sqrt{7}$ cm and $2\sqrt{7}$ cm from the sides of the angle. Find the distance from this point to the vertex of the angle. | {
"answer": "\\frac{14\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vitya collects toy cars from the "Retro" series. The problem is that the total number of different models in the series is unknown — it's a big commercial secret. However, it is known that different cars are produced in equal quantities, so it can be assumed that all models are evenly and randomly distributed across different online stores.
On different websites, Vitya found several offers, but upon closer inspection, it turned out that among the offered cars, there were only 12 different ones. Vitya is almost convinced that there are only 12 cars in total and that further search is pointless. But who knows?
How many more offers from other stores does Vitya need to review to be convinced that there are indeed 12 models in the series? Vitya considers himself convinced of something if the probability of this event is greater than 0.99. | {
"answer": "58",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 18 identical cars in a train. In some cars, exactly half of the seats are free, in others, exactly one-third of the seats are free, and in the remaining cars, all seats are occupied. At the same time, exactly one-ninth of all seats in the whole train are free. How many cars have all seats occupied? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest positive integer \( n \) such that \( n^{3} + 4n^{2} - 15n - 18 \) is the cube of an integer. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many times in a day is the angle between the hour and minute hands exactly $19^{\circ}$? | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Matěj had written six different natural numbers in a row in his notebook. The second number was double the first, the third was double the second, and similarly, each subsequent number was double the previous one. Matěj copied all these numbers into the following table in random order, one number in each cell.
The sum of the two numbers in the first column of the table was 136, and the sum of the numbers in the second column was double that, or 272. Determine the sum of the numbers in the third column of the table. | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Farmer Tim starts walking from the origin following the path \((t, \sin t)\) where \(t\) is the time in minutes. Five minutes later, Alex enters the forest and follows the path \((m, \cos t)\) where \(m\) is the time since Alex started walking. What is the greatest distance between Alex and Farmer Tim while they are walking these paths? | {
"answer": "\\sqrt{29}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four princesses each guessed a two-digit number, and Ivan guessed a four-digit number. After they wrote their numbers in a row in some order, they got the sequence 132040530321. Find Ivan's number. | {
"answer": "5303",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the angles \(A\) and \(B\) are \(45^{\circ}\) and \(30^{\circ}\) respectively, and \(CM\) is the median. The circles inscribed in triangles \(ACM\) and \(BCM\) touch segment \(CM\) at points \(D\) and \(E\). Find the area of triangle \(ABC\) if the length of segment \(DE\) is \(4(\sqrt{2} - 1)\). | {
"answer": "16(\\sqrt{3}+1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain number with a sum of digits equal to 2021 was divided by 7 and resulted in a number composed exclusively of the digit 7. How many digits 7 can this number contain? If there are multiple answers, provide their sum. | {
"answer": "503",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A four-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, if according to standards, a stove should not be located closer than 1 meter from the main gas pipe. | {
"answer": "1/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the board, six numbers are written in a row. It is known that each number, starting from the third, is equal to the product of the two preceding numbers, and the fifth number is equal to 108. Find the product of all six numbers in this row. | {
"answer": "136048896",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine how many different ways there are to assign the elements of the set \( M = \{1, 2, 3, 4, 5\} \) into three ordered sets \( A, B,\) and \( C \) such that the following conditions are satisfied: each element must belong to at least one of the sets, the intersection of all three sets is empty, and the intersection of any two sets is not empty. (i.e., \( A \cup B \cup C = M, A \cap B \cap C = \varnothing \), and \( A \cap B \neq \varnothing, B \cap C \neq \varnothing, C \cap A \neq \varnothing \)) | {
"answer": "1230",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \cos ^{8}\left(\frac{x}{2}\right) dx
$$ | {
"answer": "\\frac{35\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( PROBLEMZ \) be a regular octagon inscribed in a circle of unit radius. Diagonals \( MR \) and \( OZ \) meet at \( I \). Compute \( LI \). | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A chunk fell out of a dictionary. The first page of the chunk has the number 213, and the number of the last page is written using the same digits in a different order. How many pages are in the chunk that fell out? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular 2017-gon, all diagonals are drawn. Peter randomly selects some $\mathrm{N}$ diagonals. What is the smallest $N$ such that there are guaranteed to be two diagonals of the same length among the selected ones? | {
"answer": "1008",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two advanced Level 3 students participated in a university chess tournament. Each participant plays against all others exactly once. A win is worth 1 point, a draw is worth 0.5 points, and a loss is worth 0 points. The total scores of the two Level 3 students sum up to 6.5 points. All university students scored the same amount of points. How many university students participated in the competition? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a ball, there were 29 boys and 15 girls. Some boys danced with some girls (no more than once with each partner). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have mentioned? | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 10 cards, each card is written with two different numbers from $1, 2, 3, 4, 5$, and the numbers on any two cards are not completely the same. These 10 cards are to be placed into five boxes labeled $1, 2, 3, 4, 5$, with the rule that a card with numbers $i, j$ can only be placed in box $i$ or box $j$. A placement is deemed "good" if the number of cards in box 1 is greater than the number of cards in any other box. Find the total number of "good" placements. | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many 4-digit positive multiples of 4 can be formed from the digits 0, 1, 2, 3, 4, 5, 6 such that each digit appears without repetition? | {
"answer": "176",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an isosceles triangle \(XYZ\) with \(XY = YZ\) and an angle at the vertex equal to \(96^{\circ}\). Point \(O\) is located inside triangle \(XYZ\) such that \(\angle OZX = 30^{\circ}\) and \(\angle OXZ = 18^{\circ}\). Find the measure of angle \(\angle YOX\). | {
"answer": "78",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cat is going up a stairwell with ten stairs. The cat can jump either two or three stairs at each step, or walk the last step if necessary. How many different ways can the cat go from the bottom to the top? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The natural numbers from 1951 to 1982 are arranged in a certain order one after another. A computer reads two consecutive numbers from left to right (i.e., the 1st and 2nd, the 2nd and 3rd, etc.) until the last two numbers. If the larger number is on the left, the computer swaps their positions. Then the computer reads in the same manner from right to left, following the same rules to change the positions of the two numbers. After reading, it is found that the number at the 100th position has not changed its position in either direction. Find this number. | {
"answer": "1982",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of positive integer solutions \((x, y, z, w)\) to the equation \(x + y + z + w = 25\) that satisfy \(x < y\). | {
"answer": "946",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a horizontal plane, touching each other, there lie 4 spheres with a radius of \( 24-12\sqrt{2} \), and their centers form a square. On top above the center of the square, a fifth sphere of the same radius is placed over the four spheres. Find the distance from its highest point to the plane. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
58 balls of two colors - red and blue - are arranged in a circle. It is known that the number of consecutive triplets of balls with a majority of red balls is equal to the number of triplets with a majority of blue balls. What is the minimum possible number of red balls? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Does there exist a three-digit number whose cube ends in three sevens? | {
"answer": "753",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the sides $AB$ and $AC$ of triangle $ABC$ lie points $K$ and $L$, respectively, such that $AK: KB = 4:7$ and $AL: LC = 3:2$. The line $KL$ intersects the extension of side $BC$ at point $M$. Find the ratio $CM: BC$. | {
"answer": "8 : 13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a $6 \times 6$ square in which all cells are white. In one move, you are allowed to change the color of both cells in any domino (rectangle consisting of two cells) to the opposite color. What is the minimum number of moves required to obtain a square with a checkerboard pattern? Do not forget to explain why fewer moves would not be sufficient. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a peculiar computer with a button. If the current number on the screen is a multiple of 3, pressing the button will divide it by 3. If the current number is not a multiple of 3, pressing the button will multiply it by 6. Xiaoming pressed the button 6 times without looking at the screen, and the final number displayed on the computer was 12. What is the smallest possible initial number on the computer? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bullet was fired perpendicular to a moving express train with a speed of \( c = 60 \frac{\text{km}}{\text{hr}} \). The bullet pierced a windowpane on both sides of the car. How are the two holes positioned relative to each other if the bullet's speed was \( c' = 40 \frac{\text{m}}{\text{sec}} \) and the width of the car was \( a = 4 \text{m} \)? | {
"answer": "1.667",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A part of a book has fallen out. The number of the first fallen page is 387, and the number of the last page consists of the same digits but in a different order. How many sheets fell out of the book? | {
"answer": "176",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the numbers \(1, 2, 3, \ldots, 1000\). Find the largest number \(m\) with the following property: no matter which \(m\) of these numbers are removed, among the remaining \(1000-m\) numbers, there will be two such that one of them divides the other. | {
"answer": "499",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Winnie-the-Pooh eats 3 cans of condensed milk and a jar of honey in 25 minutes, while Piglet eats them in 55 minutes. Pooh eats one can of condensed milk and 3 jars of honey in 35 minutes, while Piglet eats them in 1 hour 25 minutes. How long will it take for them to eat 6 cans of condensed milk together? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a cube with edge length 1, two cross-sections in the form of regular hexagons are made. Find the length of the segment at which these cross-sections intersect. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain type of ray, when passing through a glass plate, attenuates to $\text{a}\%$ of its original intensity for every $1 \mathrm{~mm}$ of thickness. It was found that stacking 10 pieces of $1 \mathrm{~mm}$ thick glass plates results in the same ray intensity as passing through a single $11 \mathrm{~mm}$ thick glass plate. This indicates that the gaps between the plates also cause attenuation. How many pieces of $1 \mathrm{~mm}$ thick glass plates need to be stacked together to ensure the ray intensity is not greater than that passing through a single $20 \mathrm{~mm}$ thick glass plate? (Note: Assume the attenuation effect of each gap between plates is the same.) | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An integer has exactly 4 prime factors, and the sum of the squares of these factors is 476. Find this integer. | {
"answer": "1989",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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