problem stringlengths 10 5.15k | answer dict |
|---|---|
Given that points \( B \) and \( C \) are in the fourth and first quadrants respectively, and both lie on the parabola \( y^2 = 2px \) where \( p > 0 \). Let \( O \) be the origin, and \(\angle OBC = 30^\circ\) and \(\angle BOC = 60^\circ\). If \( k \) is the slope of line \( OC \), find the value of \( k^3 + 2k \). | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a sequence of convex polygons \( P_n \) as follows. \( P_0 \) is an equilateral triangle with side length 1. \( P_{n+1} \) is obtained from \( P_n \) by cutting off the corners one-third of the way along each side (for example, \( P_1 \) is a regular hexagon with side length \(\frac{1}{3}\)). Find \( \lim_{n \to \infty} \) area(\( P_n \)). | {
"answer": "\\frac{\\sqrt{3}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers $1000^{2}, 1001^{2}, 1002^{2}, \ldots$ have their last two digits discarded. How many of the first terms of the resulting sequence form an arithmetic progression? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A seven-digit number has the following properties: the hundreds digit is twice the ten millions digit, the tens digit is twice the hundred thousands digit, the units digit is twice the ten thousands digit, the thousands digit is 0, and it must be divisible by a five-digit number \( a \). What is \( a \)? | {
"answer": "10002",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Adva is a regular tetrahedron with side length \( s \), and there are three spheres associated with it. The first sphere passes through the vertices of the tetrahedron, the second intersects the midpoints of the edges, and the third is inscribed such that it touches the faces of the tetrahedron. How do the surface areas of these spheres compare to each other? | {
"answer": "9:3:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Through the midpoint $D$ of the base of an isosceles triangle, a line is drawn at an angle of $30^{\circ}$ to this base, on which the angle $\angle ACB$ intercepts a segment $EF$. It is known that $ED = 6$ and $FD = 4$. Find the height of the triangle drawn to the base. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the minimum number of convex pentagons needed to form a convex 2011-gon? | {
"answer": "670",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three young married couples were captured by cannibals. Before eating the tourists, the cannibals decided to weigh them. The total weight of all six people was not an integer, but the combined weight of all the wives was exactly 171 kg. Leon weighed the same as his wife, Victor weighed one and a half times more than his wife, and Maurice weighed twice as much as his wife. Georgette weighed 10 kg more than Simone, who weighed 5 kg less than Elizabeth. While the cannibals argued over who to eat first, five of the six young people managed to escape. The cannibals only ate Elizabeth's husband. How much did he weigh? | {
"answer": "85.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya was running down an escalator, counting the steps. Exactly halfway down, he tripped and tumbled the rest of the way (he tumbles 3 times faster than he runs). How many steps are on the escalator if Petya counted 20 steps with his feet (before falling) and 30 steps with his sides (after falling)? | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pedestrian left city $A$ at noon heading towards city $B$. A cyclist left city $A$ at a later time and caught up with the pedestrian at 1 PM, then immediately turned back. After returning to city $A$, the cyclist turned around again and met the pedestrian at city $B$ at 4 PM, at the same time as the pedestrian.
By what factor is the cyclist's speed greater than the pedestrian's speed? | {
"answer": "5/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an integer sequence \(\{a_i\}\) defined as follows:
\[ a_i = \begin{cases}
i, & \text{if } 1 \leq i \leq 5; \\
a_1 a_2 \cdots a_{i-1} - 1, & \text{if } i > 5.
\end{cases} \]
Find the value of \(\sum_{i=1}^{2019} a_i^2 - a_1 a_2 \cdots a_{2019}\). | {
"answer": "1949",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system \( xOy \), find the area of the graph formed by all points \( (x, y) \) that satisfy \( \lfloor x \rfloor \cdot \lfloor y \rfloor = 2013 \), where \( \lfloor x \rfloor \) represents the greatest integer less than or equal to the real number \( x \). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$C$ is a point on the extension of diameter $A B$, $C D$ is a tangent, and the angle $A D C$ is $110^{\circ}$. Find the angular measure of arc $B D$. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate
$$
\frac{1 \times 2 \times 4+2 \times 4 \times 8+3 \times 6 \times 12+4 \times 8 \times 16}{1 \times 3 \times 9+2 \times 6 \times 18+3 \times 9 \times 27+4 \times 12 \times 36}
$$
Only a numerical answer is expected here. The answer must be given in the form of an irreducible fraction (i.e., in the form $\frac{a}{b}$ where $a$ and $b$ are two integers with no common divisors). | {
"answer": "8/27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of the following expression:
$$
|\cdots||| x_{1}-x_{2}\left|-x_{3}\right|-x_{4}\left|-\cdots-x_{1990}\right|,
$$
where \( x_{1}, x_{2}, \cdots, x_{1990} \) are distinct natural numbers from 1 to 1990. | {
"answer": "1989",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\( BL \) is the angle bisector of triangle \( ABC \). Find its area if it is known that \( |AL| = 3 \), \( |BL| = 6\sqrt{5} \), and \( |CL| = 4 \). | {
"answer": "\\frac{21 \\sqrt{55}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(a > 0\), \(b > 0\), \(c > 1\), and \(a + b = 1\). Find the minimum value of \(\left(\frac{2a + b}{ab} - 3\right)c + \frac{\sqrt{2}}{c - 1}\). | {
"answer": "4 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(ABC\) be a triangle with \(AB=7\), \(BC=9\), and \(CA=4\). Let \(D\) be the point such that \(AB \parallel CD\) and \(CA \parallel BD\). Let \(R\) be a point within triangle \(BCD\). Lines \(\ell\) and \(m\) going through \(R\) are parallel to \(CA\) and \(AB\) respectively. Line \(\ell\) meets \(AB\) and \(BC\) at \(P\) and \(P'\) respectively, and \(m\) meets \(CA\) and \(BC\) at \(Q\) and \(Q'\) respectively. If \(S\) denotes the largest possible sum of the areas of triangles \(BPP'\), \(RP'Q'\), and \(CQQ'\), determine the value of \(S^2\). | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Engineer Sergei received a research object with a volume of approximately 200 monoliths (a container designed for 200 monoliths, which was almost completely filled). Each monolith has a specific designation (either "sand loam" or "clay loam") and genesis (either "marine" or "lake-glacial" deposits). The relative frequency (statistical probability) that a randomly chosen monolith is "sand loam" is $\frac{1}{9}$. Additionally, the relative frequency that a randomly chosen monolith is "marine clay loam" is $\frac{11}{18}$. How many monoliths with lake-glacial genesis does the object contain if none of the sand loams are marine? | {
"answer": "77",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( M = \{1, 2, 3, \cdots, 1995\} \) and \( A \subseteq M \), with the constraint that if \( x \in A \), then \( 19x \notin A \). Find the maximum value of \( |A| \). | {
"answer": "1890",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the integral \(\int_{0}^{\pi / 2} \frac{\sin^{3} x}{2 + \cos x} \, dx\). | {
"answer": "3 \\ln \\left(\\frac{2}{3}\\right) + \\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a positive number \(r\) such that the set \(T=\left\{(x, y) \mid x, y \in \mathbf{R}\right.\) and \(\left.x^{2}+(y-7)^{2} \leqslant r^{2}\right\}\) is a subset of the set \(S=\{(x, y) \mid x, y \in \mathbf{R}\right.\) and for any \(\theta \in \mathbf{R}\), \(\cos 2\theta + x \cos \theta + y \geqslant 0\},\) determine the maximum value of \(r\). | {
"answer": "4 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the acute triangle \( \triangle ABC \), the sides \( a, b, c \) are opposite to the angles \( \angle A, \angle B, \angle C \) respectively, and \( a, b, c \) form an arithmetic sequence. Also, \( \sin (A - C) = \frac{\sqrt{3}}{2} \). Find \( \sin (A + C) \). | {
"answer": "\\frac{\\sqrt{39}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the three sides of a triangle be integers \( l \), \( m \), and \( n \) with \( l > m > n \). It is known that \( \left\{\frac{3^l}{10^4}\right\} = \left\{\frac{3^m}{10^4}\right\} = \left\{\frac{3^n}{10^4}\right\} \), where \( \{x\} \) denotes the fractional part of \( x \). Determine the minimum value of the perimeter of the triangle. | {
"answer": "3003",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
2001 coins, each valued at 1, 2, or 3, are arranged in a row. The coins are placed such that:
- Between any two coins of value 1, there is at least one other coin.
- Between any two coins of value 2, there are at least two other coins.
- Between any two coins of value 3, there are at least three other coins.
What is the largest number of coins with a value of 3 that can be in the row? | {
"answer": "501",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The steamboat "Rarity" travels for three hours at a constant speed after leaving the city, then drifts with the current for an hour, then travels for three hours at the same speed, and so on. If the steamboat starts its journey in city A and goes to city B, it takes it 10 hours. If it starts in city B and goes to city A, it takes 15 hours. How long would it take to travel from city A to city B on a raft? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( p \) is a prime number and \( r \) is the remainder when \( p \) is divided by 210, if \( r \) is a composite number that can be expressed as the sum of two perfect squares, find \( r \). | {
"answer": "169",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A collector has \( N \) precious stones. If he takes away the three heaviest stones, then the total weight of the stones decreases by \( 35\% \). From the remaining stones, if he takes away the three lightest stones, the total weight further decreases by \( \frac{5}{13} \). Find \( N \). | {
"answer": "10",
"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 exactly one cell in each row and each column is colored.
There are exactly 96 ways to color five cells in a $5 \times 5$ grid without the corner cell, such that exactly one cell in each row and each column is colored.
How many ways are there to color five cells in a $5 \times 5$ grid without two corner cells, such that exactly one cell in each row and each column is colored? | {
"answer": "78",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle \( ACE \) with a point \( B \) on segment \( AC \) and a point \( D \) on segment \( CE \) such that \( BD \) is parallel to \( AE \). A point \( Y \) is chosen on segment \( AE \), and segment \( CY \) is drawn, intersecting \( BD \) at point \( X \). If \( CX = 5 \) and \( XY = 3 \), what is the ratio of the area of trapezoid \( ABDE \) to the area of triangle \( BCD \)? | {
"answer": "39/25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function \( f(x) = \max \left\{\sin x, \cos x, \frac{\sin x + \cos x}{\sqrt{2}}\right\} \) (for \( x \in \mathbb{R} \)) has a maximum value and a minimum value. Find the sum of these maximum and minimum values. | {
"answer": "1 - \\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways are there to list the numbers 1 to 10 in some order such that every number is either greater or smaller than all the numbers before it? | {
"answer": "512",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During breaks, schoolchildren played table tennis. Any two schoolchildren played no more than one game against each other. At the end of the week, it turned out that Petya played half, Kolya - a third, and Vasya - one fifth of the total number of games played during the week. What could be the total number of games played during the week if it is known that at least two games did not involve Vasya, Petya, or Kolya? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the interior angles \(A, B, C\) of triangle \(\triangle ABC\) are opposite to the sides \(a, b, c\) respectively, and that \(A - C = \frac{\pi}{2}\), and \(a, b, c\) form an arithmetic sequence, find the value of \(\cos B\). | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A = \{ x \mid 5x - a \leqslant 0 \} \) and \( B = \{ x \mid 6x - b > 0 \} \), where \( a, b \in \mathbb{N}_+ \). If \( A \cap B \cap \mathbb{N} = \{ 2, 3, 4 \} \), find the number of integer pairs \((a, b)\). | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{8} x \, dx
$$ | {
"answer": "\\frac{35\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular piece of paper with a length of 20 cm and a width of 12 cm is folded along its diagonal (refer to the diagram). What is the perimeter of the shaded region formed? | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One day, Xiao Ming took 100 yuan to go shopping. In the first store, he bought several items of product A. In the second store, he bought several items of product B. In the third store, he bought several items of product C. In the fourth store, he bought several items of product D. In the fifth store, he bought several items of product E. In the sixth store, he bought several items of product F. The prices of the six products are all different integers, and Xiao Ming spent the same amount of money in all six stores. How much money does Xiao Ming have left? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside the cube \( ABCD A_1 B_1 C_1 D_1 \) is located the center \( O \) of a sphere with a radius of 10. The sphere intersects the face \( A A_1 D_1 D \) in a circle with a radius of 1, the face \( A_1 B_1 C_1 D_1 \) in a circle with a radius of 1, and the face \( C D D_1 C_1 \) in a circle with a radius of 3. Find the length of the segment \( O D_1 \). | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All vertices of a regular tetrahedron \( A B C D \) are located on one side of the plane \( \alpha \). It turns out that the projections of the vertices of the tetrahedron onto the plane \( \alpha \) are the vertices of a certain square. Find the value of \(A B^{2}\), given that the distances from points \( A \) and \( B \) to the plane \( \alpha \) are 17 and 21, respectively. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A necklace consists of 80 beads of red, blue, and green colors. It is known that in any segment of the necklace between two blue beads, there is at least one red bead, and in any segment of the necklace between two red beads, there is at least one green bead. What is the minimum number of green beads in this necklace? (The beads in the necklace are arranged cyclically, meaning the last one is adjacent to the first one.) | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya cut an 8x8 square along the borders of the cells into parts of equal perimeter. It turned out that not all parts are equal. What is the maximum possible number of parts he could get? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\angle BAC = 90^{\circ}$ and the quadrilateral $ADEF$ is a square with side length 1, find the maximum value of $\frac{1}{AB} + \frac{1}{BC} + \frac{1}{CA}$. | {
"answer": "2 + \\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x \) be a positive integer, and write \( a = \left\lfloor \log_{10} x \right\rfloor \) and \( b = \left\lfloor \log_{10} \frac{100}{x} \right\rfloor \). Here \( \lfloor c \rfloor \) denotes the greatest integer less than or equal to \( c \). Find the largest possible value of \( 2a^2 - 3b^2 \). | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a box, there are 3 red, 4 gold, and 5 silver stars. Stars are randomly drawn one by one from the box and placed on a Christmas tree. What is the probability that a red star is placed on the top of the tree, no more red stars are on the tree, and there are exactly 3 gold stars on the tree, if a total of 6 stars are drawn from the box? | {
"answer": "5/231",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the year 2009, there is a property that rearranging the digits of the number 2009 cannot yield a smaller four-digit number (numbers do not start with zero). In what subsequent year does this property first repeat again? | {
"answer": "2022",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x, y, z \in (0, +\infty) \) and \(\frac{x^2}{1+x^2} + \frac{y^2}{1+y^2} + \frac{z^2}{1+z^2} = 2 \), find the maximum value of \(\frac{x}{1+x^2} + \frac{y}{1+y^2} + \frac{z}{1+z^2}\). | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The value of the expression \(10 - 10.5 \div [5.2 \times 14.6 - (9.2 \times 5.2 + 5.4 \times 3.7 - 4.6 \times 1.5)]\) is | {
"answer": "9.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system \( xOy \), the area of the region corresponding to the set of points \( K = \{(x, y) \mid (|x| + |3y| - 6)(|3x| + |y| - 6) \leq 0 \} \) is ________. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside triangle \(ABC\), a point \(O\) is chosen such that \(\angle ABO = \angle CAO\), \(\angle BAO = \angle BCO\), and \(\angle BOC = 90^{\circ}\). Find the ratio \(AC : OC\). | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the board, the number 27 is written. Every minute, the number is erased from the board and replaced with the product of its digits increased by 12. For example, after one minute, the number on the board will be $2 \cdot 7 + 12 = 26$. What number will be on the board after an hour? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tanya wrote a certain two-digit number on a piece of paper; to Sveta, who was sitting opposite her, the written number appeared different and was 75 less. What number did Tanya write? | {
"answer": "91",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( z \) be a complex number with a modulus of 1. Then the maximum value of \(\left|\frac{z+\mathrm{i}}{z+2}\right|\) is \(\ \ \ \ \ \ \). | {
"answer": "\\frac{2\\sqrt{5}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A car left the city for the village, and simultaneously, a cyclist left the village for the city. When the car and the cyclist met, the car immediately turned around and went back to the city. As a result, the cyclist arrived in the city 35 minutes later than the car. How many minutes did the cyclist spend on the entire trip, given that his speed is 4.5 times less than the speed of the car? | {
"answer": "55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A polygon is said to be friendly if it is regular and it also has angles that, when measured in degrees, are either integers or half-integers (i.e., have a decimal part of exactly 0.5). How many different friendly polygons are there? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci $F_{1}$ and $F_{2}$ respectively, draw a line $l$ through the right focus that intersects the ellipse at points $P$ and $Q$. Find the maximum area of the inscribed circle of triangle $F_{1} PQ$. | {
"answer": "\\frac{9 \\pi}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The school organized a picnic with several attendees. The school prepared many empty plates. Each attendee who arrives will count the empty plates and then take one plate for food (no one can take more than one plate). The first attendee will count all the empty plates, the second will count one fewer, and so on. The last attendee will find 4 empty plates remaining. It is known that the sum of the total number of plates prepared by the school and the total number of attendees is 2015. How many people attended the picnic in total? | {
"answer": "1006",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regular decagon (10-sided polygon) \(A B C D E F G H I J\) has an area of 2017 square units. Determine the area (in square units) of the rectangle \(C D H I\). | {
"answer": "806.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the point \( P \) lies in the plane of the right triangle \( \triangle ABC \) with \( \angle BAC = 90^\circ \), and \( \angle CAP \) is an acute angle. Also given are the conditions:
\[ |\overrightarrow{AP}| = 2, \quad \overrightarrow{AP} \cdot \overrightarrow{AC} = 2, \quad \overrightarrow{AP} \cdot \overrightarrow{AB} = 1. \]
Find the value of \( \tan \angle CAP \) when \( |\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AP}| \) is minimized. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane Cartesian coordinate system \(xOy\), the set of points
$$
\begin{aligned}
K= & \{(x, y) \mid(|x|+|3 y|-6) \cdot \\
& (|3 x|+|y|-6) \leqslant 0\}
\end{aligned}
$$
corresponds to an area in the plane with the measurement of ______. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of two teams, Team A and Team B, sends 7 players in a predetermined order to participate in a Go contest. The players from both teams compete sequentially starting with Player 1 from each team. The loser of each match is eliminated, and the winner continues to compete with the next player from the opposing team. This process continues until all the players of one team are eliminated, and the other team wins. How many different possible sequences of matches can occur in this contest? | {
"answer": "3432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At around 8 o'clock in the morning, two cars left the fertilizer plant one after another, heading toward Happy Village. Both cars travel at a speed of 60 kilometers per hour. At 8:32, the distance the first car had traveled from the fertilizer plant was three times the distance traveled by the second car. At 8:39, the distance the first car had traveled from the fertilizer plant was twice the distance traveled by the second car. At what exact time did the first car leave the fertilizer plant? | {
"answer": "8:11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many natural five-digit numbers have the product of their digits equal to 2000? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cubes. As is known, the whole space can be filled with equal cubes. At each vertex, eight cubes will converge. Therefore, by appropriately truncating the vertices of the cubes and joining the adjacent truncated parts into a single body, it is possible to fill the space with regular octahedra and the remaining bodies from the cubes. What will these bodies be? If we maximize the size of the octahedra, what portion of the space will they occupy? | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A typesetter scattered part of a set - a set of a five-digit number that is a perfect square, written with the digits $1, 2, 5, 5,$ and $6$. Find all such five-digit numbers. | {
"answer": "15625",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A child gave Carlson 111 candies. They ate some of them right away, 45% of the remaining candies went to Carlson for lunch, and a third of the candies left after lunch were found by Freken Bok during cleaning. How many candies did she find? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( N \) be the total number of students in the school before the New Year, among which \( M \) are boys, making up \( k \) percent of the total. This means \( M = \frac{k}{100} N \), or \( 100M = kN \).
After the New Year, the number of boys became \( M+1 \), and the total number of students became \( N+3 \). If the boys now make up \( \ell \) percent (with \( \ell < 100 \) since there are definitely still some girls in the school), then:
\[ 100(M+1) = \ell(N+3) \]
Recalling the equality \( 100M = kN \), we find that:
\[ \ell N + 3\ell = 100M + 100 = kN + 100 \]
Thus, \( 100 - 3\ell = (\ell - k)N \). If \( 3\ell < 100 \) (that is, if girls make up less than one-third), then \( N \) is a natural divisor of the positive number \( 100 - 3\ell < 100 \), and therefore \( N \) is less than 100. If \( 3\ell \) is greater than 100 (it clearly cannot be equal to 100), then \( (k - \ell)N = 3\ell - 100 \leq 3 \cdot 99 - 100 = 197 \), and the number of students from the previous year is at most 197. | {
"answer": "197",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate \( \frac{2}{1} \times \frac{2}{3} \times \frac{4}{3} \times \frac{4}{5} \times \frac{6}{5} \times \frac{6}{7} \times \frac{8}{7} \). Express the answer in decimal form, accurate to two decimal places. | {
"answer": "1.67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a sheet of graph paper, two rectangles are outlined. The first rectangle has a vertical side shorter than the horizontal side, and for the second rectangle, the opposite is true. Find the maximum possible area of their intersection if the first rectangle contains 2015 cells and the second one contains 2016 cells. | {
"answer": "1302",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside a right angle with vertex \(O\), there is a triangle \(OAB\) with a right angle at \(A\). The height of the triangle \(OAB\), dropped to the hypotenuse, is extended past point \(A\) to intersect with the side of the angle at point \(M\). The distances from points \(M\) and \(B\) to the other side of the angle are \(2\) and \(1\) respectively. Find \(OA\). | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: \( 4\left(\sin ^{3} \frac{49 \pi}{48} \cos \frac{49 \pi}{16} + \cos ^{3} \frac{49 \pi}{48} \sin \frac{49 \pi}{16}\right) \cos \frac{49 \pi}{12} \). | {
"answer": "0.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the measure of angle \( B \widehat{A} D \), given that \( D \widehat{A C}=39^{\circ} \), \( A B = A C \), and \( A D = B D \). | {
"answer": "47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let squares of one kind have a side of \(a\) units, another kind have a side of \(b\) units, and the original square have a side of \(c\) units. Then the area of the original square is given by \(c^{2}=n a^{2}+n b^{2}\).
Numbers satisfying this equation can be obtained by multiplying the equality \(5^{2}=4^{2}+3^{2}\) by \(n=k^{2}\). For \(n=9\), we get \(a=4, b=3, c=15\). | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Gabor wanted to design a maze. He took a piece of grid paper and marked out a large square on it. From then on, and in the following steps, he always followed the lines of the grid, moving from grid point to grid point. Then he drew some lines within the square, totaling 400 units in length. These lines became the walls of the maze. After completing the maze, he noticed that it was possible to reach any unit square from any other unit square in exactly one way, excluding paths that pass through any unit square more than once. What is the side length of the initially drawn large square? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a taxi, one passenger can sit in the front and three in the back. In how many ways can the four passengers be seated if one of them wants to sit by the window? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length $1$, let $P$ be a moving point on the space diagonal $B C_{1}$ and $Q$ be a moving point on the base $A B C D$. Find the minimum value of $D_{1} P + P Q$. | {
"answer": "1 + \\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest two-digit number \( N \) such that the sum of digits of \( 10^N - N \) is divisible by 170. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An odd six-digit number is called "just cool" if it consists of digits that are prime numbers, and no two identical digits are adjacent. How many "just cool" numbers exist? | {
"answer": "729",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( I \) is the incenter of \( \triangle ABC \) and \( 5 \overrightarrow{IA} = 4(\overrightarrow{BI} + \overrightarrow{CI}) \). Let \( R \) and \( r \) be the radii of the circumcircle and the incircle of \( \triangle ABC \) respectively. If \( r = 15 \), then find \( R \). | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We call a number antitriangular if it can be expressed in the form \(\frac{2}{n(n+1)}\) for some natural number \(n\). For how many numbers \(k\) (where \(1000 \leq k \leq 2000\)) can the number 1 be expressed as the sum of \(k\) antitriangular numbers (not necessarily distinct)? | {
"answer": "1001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Construct spheres that are tangent to 4 given spheres. If we accept the point (a sphere with zero radius) and the plane (a sphere with infinite radius) as special cases, how many such generalized spatial Apollonian problems exist? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integers \( n \) is \(\frac{2n^3 - 12n^2 - 2n + 12}{n^2 + 5n - 6}\) equal to an integer? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the numbers \(1000^{2}, 1001^{2}, 1002^{2}, \ldots\), the last two digits are discarded. How many of the first terms in the resulting sequence form an arithmetic progression? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Gru and the Minions plan to make money through cryptocurrency mining. They chose Ethereum as one of the most stable and promising currencies. They bought a system unit for 9499 rubles and two graphics cards for 31431 rubles each. The power consumption of the system unit is 120 W, and for each graphics card, it is 125 W. The mining speed for one graphics card is 32 million hashes per second, allowing it to earn 0.00877 Ethereum per day. 1 Ethereum equals 27790.37 rubles. How many days will it take for the team's investment to pay off, considering electricity costs of 5.38 rubles per kWh? (20 points) | {
"answer": "165",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points \( A, B, C \), and \( D \) are located on a line such that \( AB = BC = CD \). Segments \( AB \), \( BC \), and \( CD \) serve as diameters of circles. From point \( A \), a tangent line \( l \) is drawn to the circle with diameter \( CD \). Find the ratio of the chords cut on line \( l \) by the circles with diameters \( AB \) and \( BC \). | {
"answer": "\\sqrt{6}: 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(\sqrt{9-8 \sin 50^{\circ}}=a+b \csc 50^{\circ}\) where \(a, b\) are integers, find \(ab\). | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a real number \(a\), and for any \(k \in [-1, 1]\), when \(x \in (0, 6]\), the inequality \(6 \ln x + x^2 - 8x + a \leq kx\) always holds. Determine the maximum value of \(a\). | {
"answer": "6 - 6 \\ln 6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a singing contest, a Rooster, a Crow, and a Cuckoo were contestants. Each jury member voted for one of the three contestants. The Woodpecker tallied that there were 59 judges, and that the sum of votes for the Rooster and the Crow was 15, the sum of votes for the Crow and the Cuckoo was 18, and the sum of votes for the Cuckoo and the Rooster was 20. The Woodpecker does not count well, but each of the four numbers mentioned is off by no more than 13. How many judges voted for the Crow? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two players, \(A\) and \(B\), play rock-paper-scissors continuously until player \(A\) wins 2 consecutive games. Suppose each player is equally likely to use each hand sign in every game. What is the expected number of games they will play? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(0<\theta<\pi\), a complex number \(z_{1}=1-\cos \theta+i \sin \theta\) and \(z_{2}=a^{2}+a i\), where \(a \in \mathbb{R}\), it is known that \(z_{1} z_{2}\) is a pure imaginary number, and \(\bar{a}=z_{1}^{2}+z_{2}^{2}-2 z_{1} z_{2}\). Determine the value of \(\theta\) when \(\bar{a}\) is a negative real number. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Team A and Team B each have 7 players who compete in a predetermined order in a Go competition. Initially, Player 1 from each team competes. The loser is eliminated, and the winner competes next against the loser's team Player 2, and so on, until all players from one team are eliminated. The remaining team wins. How many different possible competition sequences can occur? | {
"answer": "3432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
By solving the inequality \(\sqrt{x^{2}+3 x-54}-\sqrt{x^{2}+27 x+162}<8 \sqrt{\frac{x-6}{x+9}}\), find the sum of its integer solutions within the interval \([-25, 25]\). | {
"answer": "310",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \(a\) and \(b\) are such that \(|a| \neq |b|\) and \(\frac{a+b}{a-b} + \frac{a-b}{a+b} = 6\). Find the value of the expression \(\frac{a^{3} + b^{3}}{a^{3} - b^{3}} + \frac{a^{3} - b^{3}}{a^{3} + b^{3}}\). | {
"answer": "\\frac{18}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A waiter at the restaurant U Šejdíře always adds the current date to the bill: he increases the total amount spent by as many crowns as the day of the month it is.
In September, a group of three friends dined at the restaurant twice. The first time, each person paid separately, and the waiter added the date to each bill, resulting in each person being charged 168 CZK. Four days later, they had lunch again and ordered exactly the same as before. This time, however, one person paid for all three. The waiter added the date to the bill only once and asked for 486 CZK in total. The friends were puzzled that although the prices on the menu had not changed, the lunch was cheaper this time, and they uncovered the waiter’s scam. What was the date?
(Hint: Determine what their total bill would have been if each person paid separately the second time as well.) | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base of a pyramid is a square. The height of the pyramid intersects the diagonal of the base. Find the maximum volume of such a pyramid if the perimeter of the diagonal cross-section that contains the height of the pyramid is 5. | {
"answer": "\\frac{\\sqrt{5}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$A, B, C, D$ attended a meeting, and each of them received the same positive integer. Each person made three statements about this integer, with at least one statement being true and at least one being false. Their statements are as follows:
$A:\left(A_{1}\right)$ The number is less than 12;
$\left(A_{2}\right)$ 7 cannot divide the number exactly;
$\left(A_{3}\right)$ 5 times the number is less than 70.
$B:\left(B_{1}\right)$ 12 times the number is greater than 1000;
$\left(B_{2}\right)$ 10 can divide the number exactly;
$\left(B_{3}\right)$ The number is greater than 100.
$C:\left(C_{1}\right)$ 4 can divide the number exactly;
$\left(C_{2}\right)$ 11 times the number is less than 1000;
$\left(C_{3}\right)$ 9 can divide the number exactly.
$D:\left(D_{1}\right)$ The number is less than 20;
$\left(D_{2}\right)$ The number is a prime number;
$\left(D_{3}\right)$ 7 can divide the number exactly.
What is the number? | {
"answer": "89",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Through the vertices \( A \) and \( C \) of triangle \( ABC \), lines are drawn perpendicular to the bisector of angle \( ABC \), intersecting lines \( CB \) and \( BA \) at points \( K \) and \( M \) respectively. Find \( AB \) if \( BM = 10 \) and \( KC = 2 \). | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the fraction \(\frac{5}{1+\sqrt[3]{32 \cos ^{4} 15^{\circ}-10-8 \sqrt{3}}}\). Simplify the expression under the cubic root to a simpler form, and then reduce the fraction. | {
"answer": "1 - \\sqrt[3]{4} + \\sqrt[3]{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a hyperbola \( x^{2} - y^{2} = t \) (where \( t > 0 \)), the right focus is \( F \). Any line passing through \( F \) intersects the right branch of the hyperbola at points \( M \) and \( N \). The perpendicular bisector of \( M N \) intersects the \( x \)-axis at point \( P \). When \( t \) is a positive real number (not equal to zero), find the value of \( \frac{|F P|}{|M N|} \). | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive integer \( n \) is said to be increasing if, by reversing the digits of \( n \), we get an integer larger than \( n \). For example, 2003 is increasing because, by reversing the digits of 2003, we get 3002, which is larger than 2003. How many four-digit positive integers are increasing? | {
"answer": "4005",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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