problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
In the equation \( x^{3} + ax^{2} + bx + 6 = 0 \), determine \( a \) and \( b \) such that one root of the equation is 2 and another root is 3. What is the value of the third root? | -1 | 0.875 |
Let \( n \) be the smallest positive integer with exactly 2015 positive factors. What is the sum of the (not necessarily distinct) prime factors of \( n \)? For example, the sum of the prime factors of 72 is \( 2 + 2 + 2 + 3 + 3 = 14 \). | 116 | 0.875 |
Given a positive integer \( n \). What fraction of the non-empty subsets of \( \{1, 2, \ldots, 2n\} \) has an odd smallest element? | \frac{2}{3} | 0.5 |
In the square \(ABCD\), \(E\) and \(F\) are trisection points of the diagonal \(BD\). The line extending \(AE\) intersects \(BC\) at \(G\), and the line extending \(GF\) intersects \(AD\) at \(H\). Given that the area of \(\triangle DHF\) is 12, find the area of the square \(ABCD\). | 288 | 0.75 |
Find all natural numbers \( x \) that satisfy the following conditions: the product of the digits of \( x \) is equal to \( 44x - 86868 \), and the sum of the digits is a perfect cube. | 1989 | 0.125 |
While waiting for customers, a watermelon seller sequentially weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg), balancing each watermelon on one side of the scale with one or two weights on the other side (possibly identical weights). The seller recorded on a piece of paper the weights of the scales he used. What is the smallest number of different weights that could appear in his records if the mass of each weight is an integer kilogram? | 6 | 0.125 |
For an industrial internship, 30 students are provided with 15 spots in Moscow, 8 spots in Tula, and 7 spots in Voronezh. What is the probability that two specific students will be assigned to the same city for their internship? | \frac{154}{435} | 0.5 |
In 60 chandeliers (each with 4 shades), the shades need to be replaced. Each electrician takes 5 minutes to replace one shade. A total of 48 electricians will be working. No more than one shade can be replaced in a chandelier at the same time. What is the minimum time required to replace all the shades in all the chandeliers? | 25 | 0.625 |
Given a triangle $ABC$ and a point $O$ inside it. Construct a segment with a midpoint at point $O$, whose endpoints lie on the boundary of triangle $ABC$. What is the maximum number of solutions this problem can have? | 3 | 0.625 |
Three boys \( B_{1}, B_{2}, B_{3} \) and three girls \( G_{1}, G_{2}, G_{3} \) are to be seated in a row according to the following rules:
1) A boy will not sit next to another boy and a girl will not sit next to another girl,
2) Boy \( B_{1} \) must sit next to girl \( G_{1} \).
If \( s \) is the number of different such seating arrangements, find the value of \( s \). | 40 | 0.25 |
Given the ellipse \( C \) passing through the point \( M(1,2) \) with foci at \((0, \pm \sqrt{6})\) and the origin \( O \) as the center, a line \( l \) parallel to \( OM \) intersects the ellipse \( C \) at points \( A \) and \( B \). Find the maximum area of \( \triangle OAB \). | 2 | 0.875 |
Find all integers \( n \) such that \( 2^n + 3 \) is a perfect square. Similarly, find all integers \( n \) such that \( 2^n + 1 \) is a perfect square. | 3 | 0.625 |
In how many ways can 4 purple balls and 4 green balls be placed into a 4x4 grid such that every row and column contains one purple ball and one green ball? Only one ball may be placed in each box, and rotations and reflections of a single configuration are considered different. | 216 | 0.875 |
Given the function \(f(x)=\sin ^{4} \frac{k x}{10}+\cos ^{4} \frac{k x}{10}\), where \(k\) is a positive integer, if for any real number \(a\), it holds that \(\{f(x) \mid a<x<a+1\}=\{f(x) \mid x \in \mathbb{R}\}\), find the minimum value of \(k\). | 16 | 0.625 |
The lengths of two parallel sides of a rectangle are 1 cm. Additionally, it is known that the rectangle can be divided by two perpendicular lines into four smaller rectangles, three of which have an area of at least \(1 \, \mathrm{cm}^2\), and the fourth has an area of at least \(2 \, \mathrm{cm}^2\). What is the minimum possible length of the other two sides of the rectangle? | 3+2\sqrt{2} | 0.125 |
A circle with radius 3 passes through vertex $B$, the midpoints of sides $AB$ and $BC$, and touches side $AC$ of triangle $ABC$. Given that angle $BAC$ is acute and $\sin \angle BAC = \frac{1}{3}$, find the area of triangle $ABC$. | 16 \sqrt{2} | 0.25 |
The angle formed by the bisector of angle \( ABC \) with its sides is 6 times smaller than the angle adjacent to angle \( ABC \). Find angle \( ABC \). | 45^\circ | 0.875 |
Does there exist an infinite sequence of natural numbers such that for any natural number \( k \), the sum of any \( k \) consecutive terms of this sequence is divisible by \( k+1 \)? | \text{No} | 0.375 |
Find all natural numbers \( n \) such that \( n^{4} + 4^{n} \) is prime. | 1 | 0.5 |
Zhenya had 9 cards with numbers from 1 to 9. He lost the card with the number 7. Is it possible to arrange the remaining 8 cards in a row so that any two adjacent cards form a number divisible by 7? | \text{No} | 0.125 |
There is a basket of apples. If Group A divides the apples, each person gets 3 apples and there are 10 apples left. If Group B divides them, each person gets 4 apples and there are 11 apples left. If Group C divides them, each person gets 5 apples and there are 12 apples left. What is the minimum number of apples in the basket? | 67 | 0.625 |
Let $ABC$ be a triangle with circumcircle $\Gamma$. A line intersects $[AB], [BC], [CA]$ at $X, D, Y$ respectively. The circumcircle of $ADX$ intersects $\Gamma$ at $Z$. Let $V = \Gamma \cap (ZD)$ and $W = \Gamma \cap (YZ)$. Show that $AB = VW$. | AB = VW | 0.875 |
Let $S$ be a set composed of positive integers with the property that for any $x \in S$, removing $x$ from $S$ results in the remaining numbers having an arithmetic mean that is a positive integer. Additionally, $1 \in S$ and $2016$ is the largest element in $S$. Find the maximum value of $|S|$. | 32 | 0.375 |
There are 5555 children, numbered 1 to 5555, sitting around a circle in order. Each child has an integer in hand: the child numbered 1 has the integer 1, the child numbered 12 has the integer 21, the child numbered 123 has the integer 321, and the child numbered 1234 has the integer 4321. It is known that the sum of the integers held by any 2005 consecutive children is equal to 2005. What is the integer held by the child numbered 5555? | -4659 | 0.125 |
Given that $f(x)$ is a function defined on $\mathbf{R}$, with $f(1)=1$ and for any $x \in \mathbf{R}$, it holds that $f(x+5) \geq f(x)+5$ and $f(x+1) \leq f(x)+1$. If $g(x)=f(x)+1-x$, find the value of $g(2002)$. | 1 | 0.625 |
The integers \( n \) and \( m \) satisfy the inequalities \( 3n - m < 5 \), \( n + m > 26 \), and \( 3m - 2n < 46 \). What values can \( 2n + m \) take? List all possible options. | 36 | 0.875 |
Let \( AB \) and \( CD \) be perpendicular segments intersecting at point \( P \). Suppose that \( AP = 2 \), \( BP = 3 \), and \( CP = 1 \). If all the points \( A \), \( B \), \( C \), and \( D \) lie on a circle, find the length of \( DP \). | 6 | 0.625 |
On the fields $A$, $B$, and $C$ in the bottom left corner of a chessboard, there are white rooks. You can make moves according to the usual chess rules. However, after any move, each rook must be protected by another rook. Is it possible, in several moves, to rearrange the rooks so that each lands on the square marked with the same letter in the top right corner? | \text{No} | 0.625 |
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$A(0, 2, -4), B(8, 2, 2), C(6, 2, 4)$ | 0.96 | 0.625 |
A cuckoo clock rings "cuckoo" every hour, with the number of rings corresponding to the hour shown by the hour hand (e.g., at 7:00, it rings 7 times). One morning, Maxim approached the clock at 9:05 and started moving the minute hand until 7 hours had passed. How many times did the clock ring "cuckoo" during this period? | 43 | 0.875 |
Determine the smallest positive integer \( n \) such that \( n \) is divisible by 20, \( n^2 \) is a perfect cube, and \( n^3 \) is a perfect square. | 1000000 | 0.625 |
In the village where Glafira lives, there is a small pond that is filled by springs at the bottom. Glafira discovered that a herd of 17 cows completely drank this pond in 3 days. After some time, the springs refilled the pond, and then 2 cows drank it in 30 days. How many days will it take for one cow to drink this pond? | 75 | 0.875 |
27 identical dice were glued together to form a $3 \times 3 \times 3$ cube in such a way that any two adjacent small dice have the same number of dots on the touching faces. How many dots are there on the surface of the large cube? | 189 | 0.375 |
Given the function \( f(x) = a x - \frac{3}{2} x^2 \) has a maximum value of no more than \( \frac{1}{6} \), and when \( x \in \left[ \frac{1}{4}, \frac{1}{2} \right] \), \( f(x) \geqslant \frac{1}{8} \), find the value of \( a \). | 1 | 0.75 |
Given \( x > y > 0 \) and \( xy = 1 \), find the minimum value of \( \frac{3x^3 + 125y^3}{x-y} \). | 25 | 0.875 |
In the finals of a beauty contest among giraffes, there were two finalists: the Tall one and the Spotted one. There are 135 voters divided into 5 districts, each district is divided into 9 precincts, and each precinct has 3 voters. Voters in each precinct choose the winner by majority vote; in a district, the giraffe that wins in the majority of precincts is the winner; finally, the giraffe that wins in the majority of districts is declared the winner of the final. The Tall giraffe won. What is the minimum number of voters who could have voted for the Tall giraffe? | 30 | 0.375 |
A merchant had 10 barrels of sugar, which he arranged into a pyramid as shown in the illustration. Each barrel, except one, was numbered. The merchant accidentally arranged the barrels such that the sum of the numbers along each row equaled 16. Could you rearrange the barrels such that the sum of the numbers along each row equals the smallest possible number? Note that the central barrel (which happened to be number 7 in the illustration) does not count in the sum. | 13 | 0.125 |
Find all positive integers \( n \) for which \( x^n + (x+2)^n + (2-x)^n = 0 \) has an integral solution. | 1 | 0.75 |
Let \( a_{1}, a_{2}, a_{3}, a_{4} \) be any permutation of \{1, 2, 3, 4\\}. Let \( f \) be a mapping from \{1, 2, 3, 4\} to \{1, 2, 3, 4\} such that \( f(i) \neq i \) for all \( i \). Consider the table \(\left[ \begin{array}{cccc} a_{1} & a_{2} & a_{3} & a_{4} \\ f(a_{1}) & f(a_{2}) & f(a_{3}) & f(a_{4}) \end{array} \right] \). Two tables \( M \) and \( N \) are said to be different if they differ in at least one corresponding entry. Determine the number of different tables satisfying these conditions. | 216 | 0.75 |
The lateral edge of a regular triangular pyramid is \(\sqrt{5}\), and the height of the pyramid is 1. Find the dihedral angle at the base. | 45^\circ | 0.5 |
The mathematician Fibonacci, while studying the problem of rabbit reproduction, discovered a sequence of numbers: \(1, 1, 2, 3, 5, 8, 13, \cdots\). The characteristic of this sequence is that the first two numbers are both 1, and starting from the third number, each number is equal to the sum of the two preceding numbers. A sequence formed in this way is called the Fibonacci sequence, denoted as \(\{a_n\}\). Calculate the following expression using the Fibonacci sequence:
$$
\begin{array}{l}
\left(a_{1} a_{3}+a_{2} a_{4}+a_{3} a_{5}+\cdots+a_{2019} a_{2021}\right)- \\
\left(a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+\cdots+a_{2020}^{2}\right) \\
=
\end{array}
$$ | 1 | 0.5 |
Let \( S \) be a subset of \(\{1, 2, 3, \ldots, 12\}\) such that it is impossible to partition \( S \) into \( k \) disjoint subsets, each of whose elements sum to the same value, for any integer \( k \geq 2 \). Find the maximum possible sum of the elements of \( S \). | 77 | 0.125 |
Let \( A = \{1, 2, \ldots, 2002\} \) and \( M = \{1001, 2003, 3005\} \). For any non-empty subset \( B \) of \( A \), \( B \) is called an \( M \)-free set if the sum of any two elements in \( B \) does not belong to \( M \). If \( A = A_1 \cup A_2 \), \( A_1 \cap A_2 = \varnothing \), and both \( A_1 \) and \( A_2 \) are \( M \)-free sets, then the ordered pair \(\left(A_{1}, A_{2}\right)\) is called an \( M \)-partition of \( A \). Find the number of all \( M \)-partitions of \( A \). | 2^{501} | 0.125 |
Alison folds a square piece of paper in half along the dashed line. After opening the paper out again, she then folds one of the corners onto the dashed line. What is the value of $\alpha$?
A) 45
B) 60
C) 65
D) 70
E) 75 | 75 | 0.625 |
Determine all quadruples $(p, q, r, n)$ of prime numbers $p, q, r$ and positive integers $n$ for which
$$
p^{2}=q^{2}+r^{n}
$$
is satisfied. | (5, 3, 2, 4) | 0.625 |
The set of all positive integers can be divided into two disjoint subsets, $\{f(1), f(2), \cdots, f(n), \cdots\}$ and $\{g(1), g(2), \cdots, g(n), \cdots\}$, where $f(1)<f(2)<\cdots<f(n)<\cdots$ and $g(1)<g(2)<\cdots<g(n)<\cdots$ satisfy the condition $g(n)=f[f(n)]+1$ for $n \geq 1$. Find $f(240)$. | 388 | 0.375 |
Two circles intersect at points $A$ and $B$. Through these points, two chords are drawn in arbitrary directions, intersecting the first circle at points $C$ and $D$, and the second circle at points $C^{\prime}$ and $D^{\prime}$. Show that the chords $CD$ and $C^{\prime}D^{\prime}$ are parallel. | CD \parallel C'D' | 0.75 |
In a winter camp, Vanya and Grisha share a room. Each night they draw lots to determine who will turn off the light before bed. The switch is near the door, so the loser has to walk back to bed in complete darkness, bumping into chairs.
Usually, Vanya and Grisha draw lots without any particular method, but this time Grisha has come up with a special way:
- "Let's flip a coin. If heads come up on an even-numbered throw, we stop flipping the coin: I win. If tails come up on an odd-numbered throw, you win."
a) What is the probability that Grisha wins?
b) Find the expected number of coin flips until the outcome is decided. | 2 | 0.25 |
Compute the definite integral:
$$
\int_{1 / 24}^{1 / 3} \frac{5 \sqrt{x+1}}{(x+1)^{2} \sqrt{x}} \, dx
$$ | 3 | 0.75 |
How many natural numbers less than 1000 are multiples of 4 and do not contain the digits 1, 3, 4, 5, 7, 9 in their representation? | 31 | 0.375 |
There are fewer than 30 students in a class. The probability that a randomly selected girl is an honor student is $\frac{3}{13}$, and the probability that a randomly selected boy is an honor student is $\frac{4}{11}$. How many honor students are there in the class? | 7 | 0.875 |
A dragon has 40 piles of gold coins, with the number of coins in any two piles differing. After the dragon plundered a neighboring city and brought back more gold, the number of coins in each pile increased by either 2, 3, or 4 times. What is the minimum number of different piles of coins that could result? | 14 | 0.75 |
On Ming's way to the swimming pool, there are 200 trees. On his round trip, Ming marked some trees with red ribbons. On his way to the swimming pool, he marked the 1st tree, the 6th tree, the 11th tree, and so on, marking every 4th tree. On his way back, he marked the 1st tree he encountered, the 9th tree, the 17th tree, and so on, marking every 7th tree. How many trees are unmarked when he returns home? | 140 | 0.375 |
Given the sequence \(\left\{a_{n}\right\}\) that satisfies:
\[
\begin{array}{l}
a_{1}=\frac{9}{4}, \\
2 a_{n+1} a_{n}-7 a_{n+1}-3 a_{n}+12=0 \text{ for } n \geq 1.
\end{array}
\]
(1) Let \(c_{n}=a_{n}-2\). Find the general term formula for the sequence \(\left\{c_{n}\right\}\).
(2) Let \([x]\) denote the greatest integer less than or equal to the real number \(x\), and define \(b_{n}=\frac{n^2}{n+1} a_{n}\). Find the largest positive integer \(n\) such that \(\sum_{k=1}^{n}\left[b_{k}\right] \leq 2019\). | 45 | 0.5 |
Let \(\mathbb{N}\) denote the set of positive integers. Let \(\varphi: \mathbb{N} \rightarrow \mathbb{N}\) be a bijective function and assume that there exists a finite limit
$$
\lim _{n \rightarrow \infty} \frac{\varphi(n)}{n} = L
$$
What are the possible values of \(L\)? | 1 | 0.75 |
Find the maximum volume of a tetrahedron inscribed in a cylinder with a base radius \( R \) and height \( h \). | \frac{2}{3} R^2 h | 0.5 |
The base of a hexagonal prism is a regular hexagon, and the side edges are perpendicular to the base. It is known that all the vertices of the hexagonal prism lie on the same spherical surface, and the volume of the hexagonal prism is \(\frac{9}{8}\). The perimeter of the base is 3. Calculate the volume of the sphere. | \frac{4}{3} \pi | 0.125 |
A taxi driver wasn't very polite, and an upset Mr. Wilkins asked for his number.
- "Do you want to know my number?" said the driver. "Well, okay. If you divide it by 2, 3, 4, 5, or 6, you'll get a remainder of 1, but it divides evenly by 11. I'll also say that, among all drivers who could say the same about their number, mine is the smallest."
What number did the driver have? | 121 | 0.75 |
Note that if you flip a sheet with numbers written on it, the digits $0, 1, 8$ remain unchanged, $6$ and $9$ swap places, and the rest lose their meaning. How many nine-digit numbers exist that remain unchanged when the sheet is flipped? | 1500 | 0.375 |
For the real numbers \(a, b, c, d\) the following equations apply:
\[ a = \sqrt{45 - \sqrt{21 - a}}, \]
\[ b = \sqrt{45 + \sqrt{21 - b}}, \]
\[ c = \sqrt{45 - \sqrt{21 + c}}, \]
\[ d = \sqrt{45 + \sqrt{21 + d}}. \]
Show that \(abcd = 2004\). | 2004 | 0.75 |
Tetrahedron \(ABCD\) has side lengths \(AB = 6\), \(BD = 6\sqrt{2}\), \(BC = 10\), \(AC = 8\), \(CD = 10\), and \(AD = 6\). The distance from vertex \(A\) to face \(BCD\) can be written as \(\frac{a \sqrt{b}}{c}\), where \(a, b, c\) are positive integers, \(b\) is square-free, and \(\operatorname{gcd}(a, c) = 1\). Find \(100a + 10b + c\). | 2851 | 0.375 |
A motorboat departed from dock A downstream. Simultaneously, a boat departed from dock B upstream towards the motorboat. After some time, they met. At the moment of their meeting, a second boat departed from B and after some time met the motorboat. The distance between the points of the first and second meetings is $\frac{56}{225}$ of the distance AB. The flow velocities, the motorboat, and the boats' speeds are constant (the boats' speeds are the same), and the ratio of the motorboat's speed to the boat's speed is $2:3$. What fraction of the distance AB had the motorboat traveled by the time it met the second boat?
Answer: $\frac{161}{225}$ or $\frac{176}{225}$. | \frac{176}{225} | 0.5 |
Given the acute angles \( \alpha \) and \( \beta \) satisfy
\[ \sin \beta = m \cos (\alpha + \beta) \cdot \sin \alpha \quad (m > 0, \alpha + \beta \neq \frac{\pi}{2}) \]
let \( x = \tan \alpha \) and \( y = \tan \beta \).
(1) Find the expression of \( y = f(x) \);
(2) Under the condition in (1), determine the maximum value of the function \( y \) when \( \alpha \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right) \). | \frac{m}{m + 2} | 0.125 |
How many three-digit numbers are there in which any two adjacent digits differ by 3? | 20 | 0.375 |
Given \( x, y, z \in \mathbf{R}^{+} \) and \( x + y + z = 1 \), find the minimum value of \( \frac{1}{x} + \frac{4}{y} + \frac{9}{z} \). | 36 | 0.875 |
In how many ways can you choose $k$ items from $n$ items? | \frac{n!}{k!(n-k)!} | 0.125 |
In a trapezoid, the longer base is 5, and one of the non-parallel sides is 3. It is known that one diagonal is perpendicular to this non-parallel side, while the other diagonal bisects the angle between this non-parallel side and the base. Find the area of the trapezoid. | 9.6 | 0.125 |
Calculate: \( 39 \frac{18}{19} \times 18 \frac{19}{20} = 757 \frac{1}{380} \) | 757 \frac{1}{380} | 0.625 |
Given point \( P \), circle \( k \), and the secant line \( AB \) passing through \( P \) such that \( PA = AB = 1 \). The tangents from \( P \) to circle \( k \) touch the circle at points \( C \) and \( D \). The intersection of \( AB \) and \( CD \) is point \( M \). What is the distance \( PM \)? | \frac{4}{3} | 0.125 |
Triangle $ABC$ is equilateral. Point $M$ is marked on side $AC$, and point $N$ is marked on side $BC$, with $MC = BN = 2AM$. Segments $MB$ and $AN$ intersect at point $Q$. Find the angle $CQB$. | 90^\circ | 0.875 |
How many sequences \( a_{1}, a_{2}, \ldots, a_{8} \) of zeroes and ones have \( a_{1}a_{2} + a_{2}a_{3} + \cdots + a_{7}a_{8} = 5 \)? | 9 | 0.25 |
Given that \(x, y, z\) are all positive real numbers and satisfy \(x + y + z = 1\), find the minimum value of the function \(f(x, y, z) = \frac{3x^2 - x}{1 + x^2} + \frac{3y^2 - y}{1 + y^2} + \frac{3z^2 - z}{1 + z^2}\), and provide a proof. | 0 | 0.75 |
In a grove, there are four types of trees: birches, spruces, pines, and aspens. There are 100 trees in total. It is known that among any 85 trees, there are trees of all four types. What is the smallest number of any trees in this grove that must include trees of at least three types? | 69 | 0.625 |
Calculate: \(2011 - (9 \times 11 \times 11 + 9 \times 9 \times 11 - 9 \times 11) =\) | 130 | 0.625 |
There are several (more than one) consecutive natural numbers written on a board, the sum of which equals 2016. What can be the smallest of these numbers? | 1 | 0.5 |
On a rectangular sheet of paper, a picture in the shape of a "cross" is drawn from two rectangles $ABCD$ and $EFGH$, with sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$. | 52.5 | 0.25 |
At a physical education class, 27 seventh graders attended, some of whom brought one ball each. Occasionally during the class, a seventh grader would give their ball to another seventh grader who did not have a ball.
At the end of the class, \( N \) seventh graders said: "I received balls less often than I gave them away!" Find the maximum possible value of \( N \) given that nobody lied. | 13 | 0.625 |
Ilya takes a triplet of numbers and transforms it following the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and the smallest numbers in the triplet after the 1989th application of this rule, if the initial triplet of numbers was \(\{70, 61, 20\}\)? If the question allows for multiple solutions, list them all as a set. | 50 | 0.625 |
Vera bought 6 notebooks less than Misha and Vasya together, and Vasya bought 10 notebooks less than Vera and Misha together. How many notebooks did Misha buy? | 8 | 0.75 |
Two parallel chords of a circle have lengths 24 and 32 respectively, and the distance between them is 14. What is the length of another parallel chord midway between the two chords? | 2\sqrt{249} | 0.25 |
Pasha, Masha, Tolya, and Olya ate 88 candies, with each of them eating at least one candy. Masha and Tolya together ate 57 candies, but Pasha ate the most candies. How many candies did Olya eat? | O = 1 | 0.75 |
Inside the triangle \( ABC \), a point \( M \) is taken such that the following property is known: if the sum of the squares of all sides of the triangle is added to three times the sum of all squares of the distances from point \( M \) to the vertices of the triangle, the resulting value does not exceed \( 24 \cdot x \). Find the side of the triangle \( y \), given that the area of the triangle \( ABC \) is at least \( \sqrt{3} \cdot x \). If necessary, round the found value to two decimal places. | y = 2 \sqrt{x} | 0.625 |
What is 20% of \(3 \frac{3}{4}\)? | \frac{3}{4} | 0.5 |
Oleg has four cards, each with a natural number on each side (a total of 8 numbers). He considers all possible sets of four numbers where the first number is written on the first card, the second number on the second card, the third number on the third card, and the fourth number on the fourth card. For each set of four numbers, he writes the product of the numbers in his notebook. What is the sum of the eight numbers on the cards if the sum of the sixteen numbers in Oleg’s notebook is $330? | 21 | 0.5 |
Masha was given a box with differently colored beads (each bead having a unique color, with a total of $\mathrm{n}$ beads in the box). Masha chose seven beads for her dress and decided to try out all possible combinations of these beads on the dress (thus, Masha considers sewing one, two, three, four, five, six, or seven beads, with the order of the beads not mattering to her). Then she counted the total number of combinations and was very surprised that the number turned out to be odd.
1) What number did Masha get?
2) Is it true that when selecting from a set with an even number of beads, Masha could get an even number of combinations?
3) Is it true that if the order of the beads sewn onto the dress mattered to Masha, she could have ended up with either an even or odd number of combinations? | \text{Yes} | 0.125 |
A triangle was cut into two triangles. Find the maximum value of $N$ such that among the 6 angles of these two triangles, exactly $N$ are the same. | 4 | 0.25 |
Let the function \( f(x) = \sin^4 \left( \frac{kx}{10} \right) + \cos^4 \left( \frac{kx}{10} \right) \), where \( k \) is a positive integer. If for any real number \( a \), the set \(\{ f(x) \mid a < x < a+1 \} = \{ f(x) \mid x \in \mathbf{R} \}\), then find the minimum value of \( k \). | 16 | 0.375 |
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? | 15 | 0.625 |
Two siblings sold their flock of sheep. Each sheep was sold for as many florins as the number of sheep originally in the flock. They divided the revenue by giving out 10 florins at a time. First, the elder brother took 10 florins, then the younger brother, then the elder again, and so on. In the end, the younger brother received less than 10 florins, so the elder brother gave him his knife, making their earnings equal. How much is the knife worth in florins? | 2 | 0.625 |
Inside the triangle $ABC$, a point $D$ is chosen such that $\angle BAD = 60^\circ$ and $\angle ABC = \angle BCD = 30^\circ$. It is given that $AB = 15$ and $CD = 8$. Find the length of segment $AD$. If necessary, round the answer to 0.01 or write the answer as a common fraction. | 3.5 | 0.375 |
Given a rectangular playing field of size \( 13 \times 2 \) and an unlimited number of dominoes of sizes \( 2 \times 1 \) and \( 3 \times 1 \), the playing field is to be covered completely without gaps and without overlapping, and no domino should extend beyond the playing field. Additionally, all dominoes must be oriented the same way, i.e., their long sides must be parallel to each other.
How many such coverings are possible? | 257 | 0.375 |
The set $M$ consists of $n$ numbers, where $n$ is odd and $n > 1$. It is such that if any element is replaced by the sum of the remaining $n-1$ elements of $M$, the sum of all $n$ elements does not change. Find the product of all $n$ elements in the set $M$. | 0 | 0.875 |
If we select $a_{1}, a_{2}, a_{3}$ from the numbers $1, 2, \cdots, 14$ in ascending order such that both $a_{2} - a_{1} \geqslant 3$ and $a_{3} - a_{2} \geqslant 3$ are satisfied, how many different ways are there to select these numbers? | 120 | 0.75 |
The village council of the secret pipeline is gathering around a round table, where each arriving member can sit in any available seat. How many different seating arrangements are possible if 7 participants join the council? (Two arrangements are considered identical if the same people are sitting to the left and right of each participant, and empty seats are not considered.) | 720 | 0.875 |
Two players, A and B, take turns removing stones from a pile of \( n \) stones. Player A starts first and can take any number of stones, but must take at least one and not all of them. Following this, each player must take a number of stones that is a divisor of the number of stones the other player took in the previous turn. The player who takes the last stone wins. What is the smallest value of \( n > 1992 \) for which player B has a winning strategy? | 2048 | 0.25 |
Consider a regular hexagon \(ABCDEF\). A frog starts at vertex \(A\). Each time it jumps, it can move to one of the two adjacent vertices. If the frog reaches vertex \(D\) within 5 jumps, it stops jumping. If it does not reach vertex \(D\) within 5 jumps, it stops after making 5 jumps. How many different ways can the frog jump from the start until it stops? | 26 | 0.375 |
In a lathe workshop, parts are turned from steel blanks, one part from one blank. The shavings left after processing three blanks can be remelted to get exactly one blank. How many parts can be made from nine blanks? What about from fourteen blanks? How many blanks are needed to get 40 parts? | 27 | 0.25 |
There are 20 chairs in a room, colored either blue or red. Each chair is occupied by either a knight or a liar. Knights always tell the truth and liars always lie. Initially, each occupant claimed that they were sitting on a blue chair. Later, they switched seats somehow, after which half of them claimed to be sitting on blue chairs and the other half claimed to be sitting on red chairs. How many knights are now sitting on red chairs? | 5 | 0.875 |
How many solutions does the equation
$$
\left\lfloor\frac{x}{20}\right\rfloor = \left\lfloor\frac{x}{17}\right\rfloor
$$
have over the set of positive integers?
Here, $\lfloor a\rfloor$ denotes the greatest integer less than or equal to \(a\). | 56 | 0.375 |
Let \( ABC \) be a right triangle with the hypotenuse \( BC \) measuring \( 4 \) cm. The tangent at \( A \) to the circumcircle of \( ABC \) meets the line \( BC \) at point \( D \). Suppose \( BA = BD \). Let \( S \) be the area of triangle \( ACD \), expressed in square centimeters. Calculate \( S^2 \). | 27 | 0.5 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.