problem stringlengths 10 5.15k | answer dict |
|---|---|
Given point \( A(4,0) \) and \( B(2,2) \), while \( M \) is a moving point on the ellipse \(\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1\), the maximum value of \( |MA| + |MB| \) is ______. | {
"answer": "10 + 2\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Along the southern shore of a boundless sea stretches an archipelago of an infinite number of islands. The islands are connected by an endless chain of bridges, and each island is connected by a bridge to the shore. In the event of a strong earthquake, each bridge independently has a probability $p=0.5$ of being destroyed. What is the probability that after a strong earthquake, one can travel from the first island to the shore using the remaining intact bridges? | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the integers 1 to 2020, there are a total of 1616 integers that are not multiples of 5. These 1616 numbers need to be divided into groups (each group may have a different number of elements), such that the difference (larger number minus smaller number) between any two numbers in the same group is a prime number. What is the minimum number of groups required? | {
"answer": "404",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside the tetrahedron \( ABCD \), points \( X \) and \( Y \) are given. The distances from point \( X \) to the faces \( ABC, ABD, ACD, BCD \) are \( 14, 11, 29, 8 \) respectively. The distances from point \( Y \) to the faces \( ABC, ABD, ACD, BCD \) are \( 15, 13, 25, 11 \) respectively. Find the radius of the inscribed sphere of tetrahedron \( ABCD \). | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a four-digit number $\overline{a b c d}$ meets the condition $a + b = c + d$, it is called a "good number." For instance, 2011 is a "good number." How many "good numbers" are there? | {
"answer": "615",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate plane $xOy$, the equation of the ellipse $C$ is $\frac{x^{2}}{9}+\frac{y^{2}}{10}=1$. Let $F$ be the upper focus of $C$, $A$ be the right vertex of $C$, and $P$ be a moving point on $C$ located in the first quadrant. Find the maximum value of the area of the quadrilateral $OAPF$. | {
"answer": "\\frac{3 \\sqrt{11}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In what ratio does the point \( P \) divide the perpendicular segment dropped from vertex \( A \) of a regular tetrahedron \( ABCD \) to the face \( BCD \), given that the lines \( PB \), \( PC \), and \( PD \) are mutually perpendicular to each other? | {
"answer": "1:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The legs of a right triangle are 3 and 4. Find the area of the triangle with vertices at the points of tangency of the incircle with the sides of the triangle. | {
"answer": "6/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On New Year's Eve, Santa Claus gave the children the following task: by using all nine digits from 1 to 9 exactly once, insert either "+" or "-" between each pair of adjacent digits so that the result yields all possible two-digit prime numbers. How many such numbers can be obtained? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the distance between the midpoints of the non-parallel sides of different bases of a regular triangular prism, each of whose edges is 2. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular 2017-gon \( A_1 A_2 \cdots A_{2017} \) is inscribed in a unit circle \( O \). If two different vertices \( A_i \) and \( A_j \) are chosen randomly, what is the probability that \( \overrightarrow{O A_i} \cdot \overrightarrow{O A_j} > \frac{1}{2} \)? | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( M \) lies on the side of a regular hexagon with side length 12. Find the sum of the distances from point \( M \) to the lines containing the remaining sides of the hexagon. | {
"answer": "36\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the 3-digit decimal number \( n = \overline{abc} \) satisfies that \( a \), \( b \), and \( c \) form an arithmetic sequence, then what is the maximum possible value of a prime factor of \( n \)? | {
"answer": "317",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Person A and person B start simultaneously from points A and B, respectively, and move towards each other. When person A reaches the midpoint C of A and B, person B is still 240 meters away from point C. When person B reaches point C, person A has already moved 360 meters past point C. What is the distance between points C and D, where person A and person B meet? | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A train has 18 identical cars. In some of the cars, half of the seats are free, in others, one third of the seats are free, and in the remaining cars, all the seats are occupied. In the entire train, exactly one ninth of all seats are free. How many cars have all seats occupied? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the measure of the angle
$$
\delta=\arccos \left(\left(\sin 2903^{\circ}+\sin 2904^{\circ}+\cdots+\sin 6503^{\circ}\right)^{\cos 2880^{\circ}+\cos 2881^{\circ}+\cdots+\cos 6480^{\circ}}\right)
$$ | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle \( ABCD \), a circle \(\omega\) is constructed using side \( AB \) as its diameter. Let \( P \) be the second intersection point of segment \( AC \) with circle \(\omega\). The tangent to \(\omega\) at point \( P \) intersects segment \( BC \) at point \( K \) and passes through point \( D \). Find \( AD \), given that \( KD = 36 \). | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set \( T \) consist of integers between 1 and \( 2^{30} \) whose binary representations contain exactly two 1s. If one number is randomly selected from the set \( T \), what is the probability that it is divisible by 9? | {
"answer": "5/29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of pairs of integers \((x ; y)\) that satisfy the equation \(y^{2} - xy = 700000000\). | {
"answer": "324",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle of size $1000 \times 1979$ is divided into cells. Into how many parts will it be divided if one diagonal is drawn in it? | {
"answer": "2978",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the vertices of a convex 2020-gon, numbers are placed such that among any three consecutive vertices, there is both a vertex with the number 7 and a vertex with the number 6. On each segment connecting two vertices, the product of the numbers at these two vertices is written. Andrey calculated the sum of the numbers written on the sides of the polygon and obtained the sum \( A \), while Sasha calculated the sum of the numbers written on the diagonals connecting vertices one apart and obtained the sum \( C \). Find the largest possible value of the difference \( C - A \). | {
"answer": "1010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of a triangle is $4 \sqrt{21}$, its perimeter is 24, and the segment of the angle bisector from one of the vertices to the center of the inscribed circle is $\frac{\sqrt{30}}{3}$. Find the longest side of the triangle. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the country of Draconia, there are red, green, and blue dragons. Each dragon has three heads; each head always tells the truth or always lies. Each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each dragon's heads made the following statements:
- 1st head: "To my left is a green dragon."
- 2nd head: "To my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."
What is the maximum number of red dragons that could have been at the table? | {
"answer": "176",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle inscribed in an isosceles trapezoid divides its lateral side into segments equal to 4 and 9. Find the area of the trapezoid. | {
"answer": "156",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle, whose center lies on the line \( y = b \), intersects the parabola \( y = \frac{12}{5} x^2 \) at least at three points; one of these points is the origin, and two of the remaining points lie on the line \( y = \frac{12}{5} x + b \). Find all values of \( b \) for which this configuration is possible. | {
"answer": "169/60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the tetrahedron \( OABC \), \(\angle AOB = 45^\circ\), \(\angle AOC = \angle BOC = 30^\circ\). Find the cosine value of the dihedral angle \(\alpha\) between the planes \( AOC \) and \( BOC \). | {
"answer": "2\\sqrt{2} - 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A packet of seeds was passed around the table. The first person took 1 seed, the second took 2 seeds, the third took 3 seeds, and so on: each subsequent person took one more seed than the previous one. It is known that on the second round, a total of 100 more seeds were taken than on the first round. How many people were sitting at the table? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( m \) and \( n \) be positive integers satisfying
\[ m n^{2} + 876 = 4 m n + 217 n. \]
Find the sum of all possible values of \( m \). | {
"answer": "93",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A four-digit number $\overline{a b c d}$ (where digits can repeat and are non-zero) is called a "good number" if it satisfies the conditions $\overline{a b}<20$, $\overline{b c}<21$, and $\overline{c d}<22$. How many such "good numbers" are there? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a triangle with sides \(AB = 4\), \(BC = 2\), and \(AC = 3\), an incircle is inscribed. Find the area of triangle \(AMN\), where \(M\) and \(N\) are the points of tangency of this incircle with sides \(AB\) and \(AC\) respectively. | {
"answer": "\\frac{25 \\sqrt{15}}{64}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a kindergarten class, there are two (small) Christmas trees and five children. The teachers want to divide the children into two groups to form a ring around each tree, with at least one child in each group. The teachers distinguish the children but do not distinguish the trees: two configurations are considered identical if one can be converted into the other by swapping the trees (along with the corresponding groups) or by rotating each group around its tree. In how many ways can the children be divided into groups? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane Cartesian coordinate system \( xOy \), the circle \( \Omega \) intersects the parabola \( \Gamma: y^{2} = 4x \) at exactly one point, and the circle \( \Omega \) is tangent to the x-axis at the focus \( F \) of \( \Gamma \). Find the radius of the circle \( \Omega \). | {
"answer": "\\frac{4 \\sqrt{3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two ferries travel between two opposite banks of a river at constant speeds. Upon reaching a bank, each immediately starts moving back in the opposite direction. The ferries departed from opposite banks simultaneously, met for the first time 700 meters from one of the banks, continued onward to their respective banks, turned back, and met again 400 meters from the other bank. Determine the width of the river. | {
"answer": "1700",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \) for \( n = 1, 2, \cdots \). Find the smallest positive integer \( n \) such that \( S_{n} > 10 \). | {
"answer": "12367",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\left\{a_{n}\right\}$ is an arithmetic sequence with a nonzero common difference and $\left\{b_{n}\right\}$ is a geometric sequence where $a_{1}=3, b_{1}=1, a_{2}=b_{2}, 3a_{5}=b_{3}$, and there exist constants $\alpha$ and $\beta$ such that for every positive integer $n$, $a_{n} = \log_{\alpha} b_{n} + \beta$, find the value of $\alpha + \beta$. | {
"answer": "\\sqrt[3]{3} + 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( LOVER \) be a convex pentagon such that \( LOVE \) is a rectangle. Given that \( OV = 20 \) and \( LO = VE = RE = RL = 23 \), compute the radius of the circle passing through \( R, O \), and \( V \). | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the 20 numbers 11, 12, 13, 14, ... 30, how many numbers at a minimum must be taken to ensure that among the selected numbers, there will definitely be a pair whose sum is a multiple of ten? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Santa Claus has 36 identical gifts distributed among 8 bags. Each bag contains at least 1 gift, and the number of gifts in each of the 8 bags is unique. From these bags, select some bags such that the total number of gifts in the selected bags can be evenly divided among 8 children, with each child receiving at least one gift. Determine how many different ways the selection can be made. | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product $8 \cdot 78$ and the increasing sequence of integers ${3, 15, 24, 48, \cdots}$, which consists of numbers that are divisible by 3 and are one less than a perfect square, are given. What is the remainder when the 1994th term in this sequence is divided by 1000? | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x, y \in \mathbb{N}$, find the maximum value of $y$ such that there exists a unique value of $x$ satisfying the following inequality:
$$
\frac{9}{17}<\frac{x}{x+y}<\frac{8}{15}.
$$ | {
"answer": "112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area enclosed by the graph \( x^{2}+y^{2}=|x|+|y| \) on the \( xy \)-plane. | {
"answer": "\\pi + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a magical tree with 63 fruits. On the first day, 1 fruit will fall from the tree. Starting from the second day, the number of fruits falling each day increases by 1 compared to the previous day. However, if the number of fruits on the tree is less than the number that should fall on that day, then the sequence resets and starts falling 1 fruit again each day, following the original pattern. On which day will all the fruits be gone from the tree? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the 16 vertices of a $3 \times 3$ grid comprised of 9 smaller unit squares, what is the probability that any three chosen vertices form a right triangle? | {
"answer": "9/35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest possible number in decimal notation where all the digits are different, and the sum of its digits is 37. | {
"answer": "976543210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among all the simple fractions where both the numerator and the denominator are two-digit numbers, find the smallest fraction that is greater than $\frac{3}{5}$. Provide the numerator of this fraction in your answer. | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A \) be a set containing only positive integers, and for any elements \( x \) and \( y \) in \( A \), \(|x-y| \geq \frac{x y}{30}\). Determine at most how many elements \( A \) may contain. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a convex quadrilateral \(ABCD\), \(X\) is the midpoint of the diagonal \(AC\). It is known that \(CD \parallel BX\). Find \(AD\), given that \(BX = 3\), \(BC = 7\), and \(CD = 6\). | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the inequality \( ab + b^2 + c^2 \geq \lambda(a + b)c \) holds for all positive real numbers \( a, b, c \) that satisfy \( b + c \geq a \), then the maximum value of the real number \( \lambda \) is \(\quad\) . | {
"answer": "\\sqrt{2} - \\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In 2006, the revenues of an insurance company increased by 25% and the expenses increased by 15% compared to the previous year. The company's profit (revenue - expenses) increased by 40%. What percentage of the revenues were the expenses in 2006? | {
"answer": "55.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many points on the hyperbola \( y = \frac{2013}{x} \) are there such that the tangent line at those points intersects both coordinate axes at points with integer coordinates? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( M \) is the midpoint of the height \( D D_{1} \) of a regular tetrahedron \( ABCD \), find the dihedral angle \( A-M B-C \) in radians. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( M = \{1, 2, \cdots, 10\} \), and \( A_1, A_2, \cdots, A_n \) be distinct non-empty subsets of \( M \). If \(i \neq j\), then \( A_i \cap A_j \) can have at most two elements. Find the maximum value of \( n \). | {
"answer": "175",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A ball invites 2018 couples, each assigned to areas numbered $1, 2, \cdots, 2018$. The organizer specifies that at the $i$-th minute of the ball, the couple in area $s_i$ (if any) moves to area $r_i$, and the couple originally in area $r_i$ (if any) exits the ball. The relationship is given by:
$$
s_i \equiv i \pmod{2018}, \quad r_i \equiv 2i \pmod{2018},
$$
with $1 \leq s_i, r_i \leq 2018$. According to this rule, how many couples will still be dancing after $2018^2$ minutes? (Note: If $s_i = r_i$, the couple in area $s_i$ remains in the same area and does not exit the ball). | {
"answer": "1009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The altitude \(AH\) and the angle bisector \(CL\) of triangle \(ABC\) intersect at point \(O\). Find the angle \(BAC\) if it is known that the difference between the angle \(COH\) and half of the angle \(ABC\) is \(46^\circ\). | {
"answer": "92",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Olga Ivanovna, the homeroom teacher of class 5B, is organizing a "Mathematical Ballet". She wants to arrange the boys and girls so that exactly 2 boys are at a distance of 5 meters from each girl. What is the maximum number of girls that can participate in the ballet if it is known that 5 boys are participating? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The task is given a finite increasing sequence \( a_{1}, a_{2}, \ldots, a_{n} \) (\(n \geq 3\)) of natural numbers, and for all \( k \leq n-2 \), the equality \( a_{k+2}=3 a_{k+1}-2 a_{k}-1 \) holds. The sequence must necessarily contain the term \( a_{k}=2021 \). Determine the maximum number of three-digit numbers divisible by 25 that this sequence can contain. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Teacher Shi distributed cards with the numbers 1, 2, 3, and 4 written on them to four people: Jia, Yi, Bing, and Ding. Then the following conversation occurred:
Jia said to Yi: "The number on your card is 4."
Yi said to Bing: "The number on your card is 3."
Bing said to Ding: "The number on your card is 2."
Ding said to Jia: "The number on your card is 1."
Teacher Shi found that statements between people with cards of the same parity (odd or even) are true, and statements between people with cards of different parity are false. Additionally, the sum of the numbers on Jia's and Ding's cards is less than the sum of the numbers on Yi's and Bing's cards.
What is the four-digit number formed by the numbers on the cards of Jia, Yi, Bing, and Ding, in that order? | {
"answer": "2341",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that for any positive integer \( n \), \( 9^{2n} - 8^{2n} - 17 \) is always divisible by \( m \), find the largest positive integer \( m \). | {
"answer": "2448",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 300 children in the "Young Photographer" club. In a session, they divided into 100 groups of 3 people each, and in every group, each member took a photograph of the other two members in their group. No one took any additional photographs. In total, there were 100 photographs of "boy+boy" and 56 photographs of "girl+girl." How many "mixed" groups were there, that is, groups containing both boys and girls? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One day, a group of young people came to the Platonic Academy located in the outskirts of Athens. The academy's gate was closed, and above the gate a sign read: "No one ignorant of geometry may enter!" Next to the sign was a diagram with four small rectangles of areas $20, 40, 48, \text{and } 42$ forming a larger rectangle. Find the shaded area to gain entry. Euclid confidently walked up to the gatekeeper, gave a number, and the gatekeeper nodded and opened the academy gate.
What was the answer given by Euclid? | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles of radius \( r \) touch each other. Additionally, each of them is externally tangent to a third circle of radius \( R \) at points \( A \) and \( B \) respectively.
Find the radius \( r \), given that \( AB = 12 \) and \( R = 8 \). | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by no more than one bridge. It is known that no more than 5 bridges lead from each island and that among any 7 islands, there are always two that are connected by a bridge. What is the maximum possible value of $N$? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a ten-digit number. From left to right:
- Its first digit indicates how many zeros are in the number.
- Its second digit indicates how many ones are in the number.
- Its third digit indicates how many twos are in the number.
- $\cdots \cdots$
- Its tenth digit indicates how many nines are in the number.
Find this ten-digit number. | {
"answer": "6210001000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The triangle $\triangle ABC$ is equilateral, and the point $P$ is such that $PA = 3 \, \text{cm}$, $PB = 4 \, \text{cm}$, and $PC = 5 \, \text{cm}$. Calculate the length of the sides of the triangle $\triangle ABC$. | {
"answer": "\\sqrt{25 + 12 \\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A \(10 \times 1\) rectangular pavement is to be covered by tiles which are either green or yellow, each of width 1 and of varying integer lengths from 1 to 10. Suppose you have an unlimited supply of tiles for each color and for each of the varying lengths. How many distinct tilings of the rectangle are there, if at least one green and one yellow tile should be used, and adjacent tiles should have different colors? | {
"answer": "1022",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya and Vasya calculated that if they walk at a speed of 4 km per hour to the neighboring village, which is 4 kilometers away, they will be 10 minutes late for the football match held there for the district championship. How should they proceed to arrive at the match on time and achieve the greatest time gain, having at their disposal a bicycle that can only be ridden by one person but goes three times faster than walking? How many minutes before the start of the match will they arrive? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( P \) be a regular polygon with 2006 sides. A diagonal of \( P \) is called good if its endpoints divide the perimeter of \( P \) into two parts, each having an odd number of sides of \( P \). The sides of \( P \) are also called good. Suppose \( P \) has been subdivided into triangles by 2003 diagonals that do not intersect at an interior point of \( P \). Find the maximum number of isosceles triangles that can appear in such a subdivision, where two sides of the isosceles triangles are good. | {
"answer": "1003",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line \( l \) passes through the focus \( F \) of the parabola \( y^2 = 4x \) and intersects the parabola at points \( A \) and \( B \). Point \( M \) is given as \( (4,0) \). Extending \( AM \) and \( BM \) intersects the parabola again at points \( C \) and \( D \), respectively. Find the value of \(\frac{S_{\triangle CDM}}{S_{\triangle ABM}}\). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right]=1994\). | {
"answer": "312",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any positive integer \( k \), let \( f_{1}(k) \) be the square of the sum of the digits of \( k \) when written in decimal notation. For \( n > 1 \), let \( f_{n}(k) = f_{1}\left(f_{n-1}(k)\right) \). What is \( f_{1992}\left(2^{1991}\right) \)? | {
"answer": "256",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An isosceles right triangle has a leg length of 36 units. Starting from the right angle vertex, an infinite series of equilateral triangles is drawn consecutively on one of the legs. Each equilateral triangle is inscribed such that their third vertices always lie on the hypotenuse, and the opposite sides of these vertices fill the leg. Determine the sum of the areas of these equilateral triangles. | {
"answer": "324",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers $1,2, \ldots, 2016$ are grouped into pairs in such a way that the product of the numbers in each pair does not exceed a certain natural number $N$. What is the smallest possible value of $N$ for which this is possible? | {
"answer": "1017072",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each positive integer \( n \), define \( A_{n} = \frac{20^{n} + 11^{n}}{n!} \), where \( n! = 1 \times 2 \times \cdots \times n \). Find the value of \( n \) that maximizes \( A_{n} \). | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Dad says he is exactly 35 years old, not counting weekends. How old is he really? | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(X_{0}\) be the interior of a triangle with side lengths 3, 4, and 5. For all positive integers \(n\), define \(X_{n}\) to be the set of points within 1 unit of some point in \(X_{n-1}\). The area of the region outside \(X_{20}\) but inside \(X_{21}\) can be written as \(a\pi + b\), for integers \(a\) and \(b\). Compute \(100a + b\). | {
"answer": "4112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pyramid with a triangular base has edges of unit length, and the angles between its edges are \(60^{\circ}, 90^{\circ},\) and \(120^{\circ}\). What is the volume of the pyramid? | {
"answer": "\\frac{\\sqrt{2}}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Several island inhabitants gather in a hut, with some belonging to the Ah tribe and the rest to the Uh tribe. Ah tribe members always tell the truth, while Uh tribe members always lie. One inhabitant said, "There are no more than 16 of us in the hut," and then added, "All of us are from the Uh tribe." Another said, "There are no more than 17 of us in the hut," and then noted, "Some of us are from the Ah tribe." A third person said, "There are five of us in the hut," and looking around, added, "There are at least three Uh tribe members among us." How many Ah tribe members are in the hut? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let point \( P \) lie on the face \( ABC \) of a tetrahedron \( ABCD \) with edge length 2. The distances from \( P \) to the planes \( DAB \), \( DBC \), and \( DCA \) form an arithmetic sequence. Find the distance from \( P \) to the plane \( DBC \). | {
"answer": "\\frac{2\\sqrt{6}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given fixed points \( A(3,0) \), \( B(0,4) \), and point \( P \) on the incircle of triangle \( \triangle AOB \) (where \( O \) is the origin), find the maximum value of \( |PA|^2 + |PB|^2 + |PO|^2 \). | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of the expression
$$
\frac{a}{x} + \frac{a+b}{x+y} + \frac{a+b+c}{x+y+z}
$$
where \( a, b, c \in [2,3] \), and the triplet of numbers \( x, y, z \) is some permutation of the triplet \( a, b, c \). | {
"answer": "15/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( T \) be the set of all positive divisors of \( 60^{100} \). \( S \) is a subset of \( T \) such that no number in \( S \) is a multiple of another number in \( S \). Find the maximum value of \( |S| \). | {
"answer": "10201",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( M \) lies on the side of a regular hexagon with a side length of 10. Find the sum of the distances from point \( M \) to the lines containing the other sides of the hexagon. | {
"answer": "30\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers \(a\) and \(b\) that satisfy \(0 \leqslant a, b \leqslant 8\) and \(b^2 = 16 + a^2\), find the sum of the maximum and minimum values of \(b - a\). | {
"answer": "12 - 4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We start with 5000 forints in our pocket to buy gifts, visiting three stores. In each store, we find a gift that we like and purchase it if we have enough money. The prices in each store are independently 1000, 1500, or 2000 forints, each with a probability of $\frac{1}{3}$. What is the probability that we are able to purchase gifts from all three stores and still have money left? | {
"answer": "17/27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Divide each natural number by the sum of the squares of its digits (for single-digit numbers, divide by the square of the number). Is there a smallest quotient among the obtained quotients, and if so, which one is it? | {
"answer": "1/9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose \(A, B\) are the foci of a hyperbola and \(C\) is a point on the hyperbola. Given that the three sides of \(\triangle ABC\) form an arithmetic sequence, and \(\angle ACB = 120^\circ\), determine the eccentricity of the hyperbola. | {
"answer": "7/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the pair of positive integers \((x, y)\) such that \(\frac{x^{2}+y^{2}}{11}\) is an integer and \(\frac{x^{2}+y^{2}}{11} \leqslant 1991\), find the number of such pairs \((x, y)\) (where \((a, b)\) and \((b, a)\) are considered different pairs if \(a \neq b\)). | {
"answer": "131",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( M_{n} = \left\{ 0 . \overline{a_{1} a_{2} \cdots a_{n}} \mid a_{i} \ \text{is either 0 or 1 for} \ i=1,2, \cdots, n-1, \ a_{n}=1 \right\} \). \( T_{n} \) is the number of elements in \( M_{n} \) and \( S_{n} \) is the sum of all elements in \( M_{n} \). Find \( \lim_{n \rightarrow \infty} \frac{S_{n}}{T_{n}} \). | {
"answer": "1/18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When measuring a part, random errors occur that follow a normal distribution with a parameter $\sigma=10$ mm. Find the probability that the measurement is made with an error not exceeding $15$ mm. | {
"answer": "0.8664",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We write on the board the equation
$$
(x-1)(x-2) \cdots(x-2016)=(x-1)(x-2) \cdots(x-2016),
$$
where there are 2016 linear factors on each side. What is the smallest positive value of $k$ such that we can omit exactly $k$ of these 4032 linear factors in such a way that there is at least one linear factor on each side, and the resulting equation has no real roots? | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a kindergarten's junior group, there are two identical small Christmas trees and five children. The teachers want to divide the children into two circles around each tree, with at least one child in each circle. The teachers distinguish the children but do not distinguish the trees: two such divisions into circles are considered the same if one can be obtained from the other by swapping the trees (along with their respective circles) and rotating each circle around its tree. In how many ways can the children be divided into the circles? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( M = \{1, 2, \cdots, 17\} \). If there exist four distinct numbers \( a, b, c, d \in M \) such that \( a + b \equiv c + d \pmod{17} \), then \( \{a, b\} \) and \( \{c, d\} \) are called a balanced pair of the set \( M \). Find the number of balanced pairs in the set \( M \). | {
"answer": "476",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a simple pendulum with a period of $T=1$ second in summer. In winter, the length of the pendulum shortens by 0.01 centimeters. How many seconds faster is this pendulum in winter over a 24-hour period compared to summer? (Round to the nearest second).
Note: The formula for the period of a simple pendulum is $T=2 \pi \sqrt{\frac{l}{g}}$, where the period $T$ is in seconds, the length $l$ is in centimeters, and the acceleration due to gravity $g$ is $980$ cm/s². | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a highway, there are checkpoints D, A, C, and B arranged in sequence. A motorcyclist and a cyclist started simultaneously from A and B heading towards C and D, respectively. After meeting at point E, they exchanged vehicles and each continued to their destinations. As a result, the first person spent 6 hours traveling from A to C, and the second person spent 12 hours traveling from B to D. Determine the distance of path AB, given that the speed of anyone riding a motorcycle is 60 km/h, and the speed on a bicycle is 25 km/h. Additionally, the average speed of the first person on the path AC equals the average speed of the second person on the path BD. | {
"answer": "340",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The country Omega grows and consumes only vegetables and fruits. It is known that in 2014, 1200 tons of vegetables and 750 tons of fruits were grown in Omega. In 2015, 900 tons of vegetables and 900 tons of fruits were grown. During the year, the price of one ton of vegetables increased from 90,000 to 100,000 rubles, and the price of one ton of fruits decreased from 75,000 to 70,000 rubles. By what percentage (%) did the real GDP of this country change in 2015, if the base year in Omega is 2014? Round your answer to two decimal places. If the real GDP of the country decreased, put a minus sign in the answer, and if it increased, put a plus sign. | {
"answer": "-9.59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two identical cylindrical vessels are connected by a small tube with a valve at the bottom. Initially, the valve is closed, and water is poured into the first vessel while oil is poured into the second vessel, such that the liquid levels are equal and are $h=40$ cm. At what level will the water be in the first vessel if the valve is opened? The density of water is 1000 kg/m³, and 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 triangle with sides 6 cm, 10 cm, and 12 cm, an inscribed circle is tangent to the two longer sides. Find the perimeter of the resulting triangle formed by the tangent line and the two longer sides. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At 30 palm trees on different parts of an uninhabited island, a sign is attached.
- On 15 of them it says: "Exactly under 15 signs a treasure is buried."
- On 8 of them it says: "Exactly under 8 signs a treasure is buried."
- On 4 of them it says: "Exactly under 4 signs a treasure is buried."
- On 3 of them it says: "Exactly under 3 signs a treasure is buried."
It is known that only those signs under which there is no treasure are truthful.
Under the smallest number of signs can a treasure be buried? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two skiers started from the same point one after another with an interval of 9 minutes. The second skier caught up with the first one 9 km from the starting point. After reaching the “27 km” mark, the second skier turned back and met the first skier at a distance of 2 km from the turning point. Find the speed of the second skier. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In parallelogram \(ABCD\), \(AB = 1\), \(BC = 4\), and \(\angle ABC = 60^\circ\). Suppose that \(AC\) is extended from \(A\) to a point \(E\) beyond \(C\) so that triangle \(ADE\) has the same area as the parallelogram. Find the length of \(DE\). | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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