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stringlengths 33
2.6k
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values | llama8b_solve_rate
float64 0.06
0.59
|
|---|---|---|---|
We draw 6 circles of equal radius on the surface of a unit sphere such that the circles do not intersect. What is the maximum possible radius of these circles?
|
\frac{\sqrt{2}}{2}
|
olympiads
| 0.15625
|
If three times a two-digit number minus 4 is a multiple of 10 and four times the number minus 15 is greater than 60 and less than 100, then what is the two-digit number?
|
28
|
olympiads
| 0.296875
|
Let \( f(x) \) be a convex function defined on an open interval \( (a, b) \), and let \( x_{1}, x_{2}, x_{3}, x_{4} \in (a, b) \) with \( x_{1} < x_{2} < x_{3} < x_{4} \). Then,
$$
\begin{array}{c}
\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}} \leqslant f^{\prime}-\left(x_{2}\right) \leqslant f^{\prime}+\left(x_{2}\right) \leqslant \\
f^{\prime}-\left(x_{3}\right) \leqslant f^{\prime}+\left(x_{3}\right) \leqslant \frac{f\left(x_{4}\right)-f\left(x_{3}\right)}{x_{4}-x_{3}}
\end{array}
$$
|
\frac{f(x_2) - f(x_1)}{x_2 - x_1} \leq f^{\prime-}(x_2) \leq f^{\prime+}(x_2) \leq f^{\prime-}(x_3) \leq f^{\prime+}(x_3) \leq \frac{f(x_4) - f(x_3)}{x_4 - x_3}
|
olympiads
| 0.0625
|
Several people need to pay 800 francs for legal costs. However, three of them do not have money, so the others need to add 60 francs each to their share. How many participants are there in total for paying the legal fees?
|
8
|
olympiads
| 0.09375
|
Determine the base of the numeral system in which the number 12551 is represented as 30407.
|
8
|
olympiads
| 0.5
|
For which positive integer values of \( k \) does the number 1 appear among the elements of the sequence \(\left(a_{n}\right)\), where \( a_{1}=k \) and \( a_{n+1}=\frac{a_{n}}{2} \) if \( a_{n} \) is even, and \( a_{n+1}=a_{n}+5 \) if \( a_{n} \) is odd?
|
1 appears in the sequence if and only if 5 \nmid k.
|
olympiads
| 0.0625
|
Fedya was 7 kopecks short of buying a portion of ice cream, and Masha was only 1 kopeck short. However, even when they combined all their money, it was still not enough to buy one portion of ice cream. How much did one portion of ice cream cost?
|
7
|
olympiads
| 0.171875
|
A *pucelana* sequence is an increasing sequence of $16$ consecutive odd numbers whose sum is a perfect cube. How many pucelana sequences are there with $3$ -digit numbers only?
|
2
|
aops_forum
| 0.15625
|
$1989$ equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.
|
4
|
aops_forum
| 0.3125
|
Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37} b(a\plus{}d)\equiv b\pmod {37} c(a\plus{}d)\equiv c\pmod{37} bc\plus{}d^2\equiv d\pmod{37} ad\minus{}bc\equiv 1\pmod{37}\end{cases}\]
|
1
|
aops_forum
| 0.1875
|
Alexey, Boris, Veniamin, and Grigory are suspected of a bank robbery. The police have determined the following:
- If Grigory is innocent, then Boris is guilty, and Alexey is innocent.
- If Veniamin is guilty, then Alexey and Boris are innocent.
- If Grigory is guilty, then Boris is also guilty.
- If Boris is guilty, then at least one of Alexey and Veniamin is guilty.
Identify those who participated in the robbery.
|
Alexey, Boris, and Grigory participated in the robbery.
|
olympiads
| 0.0625
|
A number \( a \) is randomly chosen from \( 1, 2, 3, \cdots, 10 \), and a number \( b \) is randomly chosen from \( -1, -2, -3, \cdots, -10 \). What is the probability that \( a^{2} + b \) is divisible by 3?
|
\frac{37}{100}
|
olympiads
| 0.0625
|
Find the non-negative integer solutions for the system of equations:
\[
\left\{
\begin{array}{l}
x + y = z + t \\
z + t = xy
\end{array}
\right.
\]
|
(0,0,0,0), (2,2,2,2), (1,5,2,3), (5,1,2,3), (1,5,3,2), (5,1,3,2),(2,3,1,5),(2,3,5,1),(3,2,1,5),(3,2,5,1)
|
olympiads
| 0.09375
|
Let $f(x)=\int_0^ x \frac{1}{1+t^2}dt.$ For $-1\leq x<1$ , find $\cos \left\{2f\left(\sqrt{\frac{1+x}{1-x}}\right)\right\}.$ *2011 Ritsumeikan University entrance exam/Science and Technology*
|
-x
|
aops_forum
| 0.0625
|
A traveler in the land of knights and liars hires a local resident who claims to be a knight as his guide. Later, they meet another resident. The traveler sends the guide to ask the new resident whether he is a knight or a liar. The guide returns and says that the new resident claims to be a knight. Was the guide a knight or a liar?
|
The guide is a knight.
|
olympiads
| 0.375
|
Given the vectors $\overrightarrow{O A}=\{3,1\}$ and $\overrightarrow{O B}=\{-1,2\}$, where the vector $\overrightarrow{O C}$ is perpendicular to $\overrightarrow{O B}$ and $\overrightarrow{B C}$ is parallel to $\overrightarrow{O A}$, and given that $\overrightarrow{O D}+\overrightarrow{O A}=\overrightarrow{O C}$, find $\overrightarrow{O D}$.
|
\overrightarrow{OD} = \{11, 6\}
|
olympiads
| 0.234375
|
Given the sequence \(a_{n}\) (where \(n\) is a natural number) defined as follows:
\[
a_{0} = 2 \quad \text{and} \quad a_{n} = a_{n-1} - \frac{n}{(n+1)!}, \quad \text{if} \quad n > 0
\]
Express \(a_{n}\) as a function of \(n\).
|
\frac{(n+1)! + 1}{(n+1)!}
|
olympiads
| 0.203125
|
Find all even natural numbers \( n \) for which the number of divisors (including 1 and \( n \) itself) is equal to \( \frac{n}{2} \). (For example, the number 12 has 6 divisors: \(1, 2, 3, 4, 6, 12\)).
|
\{8, 12\}
|
olympiads
| 0.0625
|
Find the ratio of the volumes obtained by the successive rotation of a parallelogram around each of its two adjacent sides.
|
\frac{a}{b}
|
olympiads
| 0.078125
|
On a plane, all points with integer coordinates $(x, y)$ such that $x^{2} + y^{2} \leq 1010$ are marked. Two players play a game (taking turns). For the first turn, the first player places a piece on any marked point and erases it. Then, on each subsequent turn, a player moves the piece to some other marked point and erases it. The lengths of the moves must always increase, and it is forbidden to move the piece to a point symmetric with respect to the center. The player who cannot make a move loses. Who among the players can guarantee a win, no matter how the opponent plays?
|
First player wins
|
olympiads
| 0.15625
|
Suppose \( A=\{1,2, \ldots, 20\} \). Call \( B \) a visionary set of \( A \) if \( B \subseteq A \), \( B \) contains at least one even integer, and \(|B| \in B\), where \(|B| \) is the cardinality of set \( B \). How many visionary sets does \( A \) have?
|
2^{20} - 512
|
olympiads
| 0.1875
|
Calculate the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty} \frac{n \sqrt[4]{11 n}+\sqrt{25 n^{4}-81}}{(n-7 \sqrt{n}) \sqrt{n^{2}-n+1}}
\]
|
5
|
olympiads
| 0.078125
|
In triangle $ABC$, the angle bisector $AM$ and the median $BN$ intersect at point $O$. It turns out that the areas of triangles $ABM$ and $MNC$ are equal. Find $\angle MON$.
|
90^ extcirc
|
olympiads
| 0.140625
|
If $h(x)$ is an even function and $g(x)$ is an odd function, and they satisfy $h(x) + g(x) \leqslant \frac{1}{x-1}$, what are $h(x)$ and $g(x)$?
|
h(x) = \frac{1}{x^2 - 1}, \quad g(x) = \frac{x}{x^2 - 1} \text{ where } x \neq \pm 1
|
olympiads
| 0.09375
|
Let $A$ and $B$ be points on a circle $\mathcal{C}$ with center $O$ such that $\angle AOB = \dfrac {\pi}2$ . Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ are internally tangent to $\mathcal{C}$ at $A$ and $B$ respectively and are also externally tangent to one another. The circle $\mathcal{C}_3$ lies in the interior of $\angle AOB$ and it is tangent externally to $\mathcal{C}_1$ , $\mathcal{C}_2$ at $P$ and $R$ and internally tangent to $\mathcal{C}$ at $S$ . Evaluate the value of $\angle PSR$ .
|
45^ extcirc
|
aops_forum
| 0.09375
|
A garden fence, similar to the one shown in the picture, had in each section (between two vertical posts) the same number of columns, and each vertical post (except for the two end posts) divided one of the columns in half. When we absentmindedly counted all the columns from end to end, counting two halves as one whole column, we found that there were a total of 1223 columns. We also noticed that the number of sections was 5 more than twice the number of whole columns in each section.
How many columns were there in each section?
|
23
|
olympiads
| 0.265625
|
Fill the numbers 1 to 9 into a grid. Each cell must contain one integer, and different cells must contain different numbers. Additionally, the sum of the numbers in the surrounding cells (cells that share a common edge with a given cell) must be an integer multiple of the number in that cell. Given that two cells already contain the numbers 4 and 5, determine: what is the maximum number that can be placed in the cell labeled \( x \)?
|
9
|
olympiads
| 0.109375
|
Find the volume of a tetrahedron whose vertices are located at points with coordinates \(\left(F_{n}, F_{n+1}, F_{n+2}\right), \quad\left(F_{n+3}, F_{n+4}, F_{n+5}\right), \quad\left(F_{n+6}, F_{n+7}, F_{n+8}\right),\) and \(\left(F_{n+9}, F_{n+10}, F_{n+11}\right)\), where \(F_{i}\) is the \(i\)-th term of the Fibonacci sequence: \(1, 1, 2, 3, 5, 8 \ldots\).
|
0
|
olympiads
| 0.140625
|
Cut a given square into two parts of equal area using a broken line (polyline) so that each segment of the broken line is parallel to a side or a diagonal of the square. Additionally, the sum of the lengths of segments parallel to the sides should equal the side length, and the sum of the lengths of segments parallel to the diagonals should equal the diagonal length. What is the minimum number of segments that such a broken line can have?
|
4
|
olympiads
| 0.375
|
If \( A \) is a positive integer such that \( \frac{1}{1 \times 3} + \frac{1}{3 \times 5} + \cdots + \frac{1}{(A+1)(A+3)} = \frac{12}{25} \), find the value of \( A \).
|
22
|
olympiads
| 0.453125
|
Let \( c \) be a prime number. If \( 11c + 1 \) is the square of a positive integer, find the value of \( c \).
|
13
|
olympiads
| 0.3125
|
Find all pairs of positive integers $ a,b$ such that \begin{align*} b^2 + b+ 1 & \equiv 0 \pmod a a^2+a+1 &\equiv 0 \pmod b . \end{align*}
|
(1, 1)
|
aops_forum
| 0.21875
|
Eddy draws $6$ cards from a standard $52$ -card deck. What is the probability that four of the cards that he draws have the same value?
|
\frac{3}{4165}
|
aops_forum
| 0.171875
|
Each of the lateral sides \(AB\) and \(CD\) of trapezoid \(ABCD\) is divided into five equal parts. Let \(M\) and \(N\) be the second division points on the lateral sides, counting from vertices \(B\) and \(C\) respectively. Find the length of \(MN\), given that the bases \(AD = a\) and \(BC = b\).
|
\frac{1}{5}(2a + 3b)
|
olympiads
| 0.109375
|
Determine the sum of all real roots of the following equation \( |x+3| - |x-1| = x+1 \).
|
-3
|
olympiads
| 0.265625
|
Farmer James has some strange animals. His hens have 2 heads and 8 legs, his peacocks have 3 heads and 9 legs, and his zombie hens have 6 heads and 12 legs. Farmer James counts 800 heads and 2018 legs on his farm. What is the number of animals that Farmer James has on his farm?
|
203
|
olympiads
| 0.09375
|
In the cube \( ABCD-A'B'C'D' \) shown in the figure, the dihedral angle \( A'-BD-C' \) is equal to \( \arccos \frac{1}{3} \) (expressed using the inverse trigonometric function).
|
arccos \frac{1}{3}
|
olympiads
| 0.328125
|
There are $n$ students standing one behind the other in a circle, with heights $h_{1}<h_{2}<\ldots<h_{n}$. At any moment, if a student of height $h_{k}$ is directly behind a student of height less than or equal to $h_{k-2}$, they can choose to switch places with that student. Determine the maximum number of exchanges that can occur among all initial configurations and possible exchange choices.
|
\binom{n}{3}
|
olympiads
| 0.109375
|
Does there exist a value of \(\alpha\) such that every term in the infinite sequence \(\cos \alpha, \cos 2 \alpha, \ldots, \cos \left(2^{n} \alpha\right), \ldots\) takes negative values?
|
There exists such \alpha
|
olympiads
| 0.0625
|
Find the sum $\sin x + \sin y + \sin z$ given that $\sin x = \tan y$, $\sin y = \tan z$, and $\sin z = \tan x$.
|
0
|
olympiads
| 0.234375
|
I wrote a two-digit natural number on a card. The sum of its digits is divisible by three. If I subtract 27 from the written number, I get another two-digit natural number formed by the same digits in reverse order. What numbers could I have written on the card?
|
63 \text{ and } 96
|
olympiads
| 0.1875
|
For the give functions in $\mathbb{N}$ :**(a)** Euler's $\phi$ function ( $\phi(n)$ - the number of natural numbers smaller than $n$ and coprime with $n$ );**(b)** the $\sigma$ function such that the $\sigma(n)$ is the sum of natural divisors of $n$ .
solve the equation $\phi(\sigma(2^x))=2^x$ .
|
1
|
aops_forum
| 0.53125
|
A chessboard with 9 horizontal and 9 vertical lines forms an $8 \times 8$ grid, resulting in $r$ rectangles, among which $s$ are squares. The value of $\frac{s}{r}$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are positive integers, and $\frac{m}{n}$ is a simplified fraction. Find the value of $m+n$.
|
125
|
olympiads
| 0.421875
|
What is the largest integer \( k \) whose square \( k^2 \) is a factor of \( 10! \)?
|
720
|
olympiads
| 0.109375
|
The edge of the cube \(E F G H E_1 F_1 G_1 H_1\) is equal to 2. Points \(A\) and \(B\) are taken on the edges \(E H\) and \(H H_1\) such that \(\frac{E A}{A H} = 2\) and \(\frac{B H}{B H_1} = \frac{1}{2}\). A plane is drawn through the points \(A\), \(B\), and \(G_1\). Find the distance from point \(E\) to this plane.
|
2 \sqrt{\frac{2}{11}}
|
olympiads
| 0.078125
|
Integrate the equation
$$
x \, dy = (x + y) \, dx
$$
and find the particular solution that satisfies the initial condition $y = 2$ when $x = -1$.
|
y = x \ln |x| - 2x
|
olympiads
| 0.078125
|
Determine all composite positive integers \( n \) for which it is possible to arrange all divisors of \( n \) that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.
|
n = pq \text{ where } p \text{ and } q \text{ are distinct primes.
|
olympiads
| 0.0625
|
To fold a paper airplane, Austin starts with a square paper $F OLD$ with side length $2$ . First, he folds corners $L$ and $D$ to the square’s center. Then, he folds corner $F$ to corner $O$ . What is the longest distance between two corners of the resulting figure?
|
\sqrt{5}
|
aops_forum
| 0.109375
|
Solve the inequality \(2 \cdot 5^{2x} \cdot \sin 2x - 3^x \geq 5^{2x} - 2 \cdot 3^x \cdot \sin 2x\).
|
\frac{\pi}{12} + k\pi \leq x \leq \frac{5\pi}{12} + k\pi, \quad k \in \mathbf{Z}
|
olympiads
| 0.078125
|
The height of a right-angled triangle, dropped to the hypotenuse, divides this triangle into two triangles. The distance between the centers of the inscribed circles of these triangles is 1. Find the radius of the inscribed circle of the original triangle.
|
\frac{1}{\sqrt{2}}
|
olympiads
| 0.078125
|
In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation $ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0$ . Find the minimum of square areas.
|
4
|
aops_forum
| 0.078125
|
Given that \( a > b > c \) and \( \frac{1}{a-b} + \frac{1}{b-c} \geq \frac{n}{a-c} \) always holds, what is the maximum value of \( n \)?
|
4
|
olympiads
| 0.140625
|
Find \(\lim _{x \rightarrow 0}\left(1+\operatorname{tg}^{2} x\right)^{2 \operatorname{ctg}^{2} x}\).
|
e^2
|
olympiads
| 0.296875
|
For each positive integer \( n \geq 1 \), we define the recursive relation given by
\[ a_{n+1} = \frac{1}{1 + a_{n}}. \]
Suppose that \( a_{1} = a_{2012} \). Find the sum of the squares of all possible values of \( a_{1} \).
|
3
|
olympiads
| 0.171875
|
Three people want to travel from city $A$ to city $B$, which is located 45 kilometers away from city $A$. They have two bicycles. The speed of a cyclist is 15 km/h, and the speed of a pedestrian is 5 km/h. What is the minimum time it will take for them to reach city $B$, given that the bicycles cannot be left unattended on the road?
|
3 \text{ hours}
|
olympiads
| 0.21875
|
Let $x$ and $y$ be real numbers such that
\[
2 < \frac{x - y}{x + y} < 5.
\]
If $\frac{x}{y}$ is an integer, what is its value?
|
-2
|
aops_forum
| 0.46875
|
On an island lives an odd number of people, each being either a knight, who always tells the truth, or a liar, who always lies. One day, all the knights declared: "I am friends with only 1 liar," while all the liars said: "I am not friends with any knights." Are there more knights or liars on the island?
|
There are more knights than liars on the island.
|
olympiads
| 0.203125
|
Let $a_1\in (0,1)$ and $(a_n)_{n\ge 1}$ a sequence of real numbers defined by $a_{n+1}=a_n(1-a_n^2),\ (\forall)n\ge 1$ . Evaluate $\lim_{n\to \infty} a_n\sqrt{n}$ .
|
\frac{\sqrt{2}}{2}
|
aops_forum
| 0.09375
|
Compute the volumes of the solids formed by the rotation of the figures bounded by the graphs of the functions. The axis of rotation is $O y$.
$$
y=\sqrt{x-1}, y=0, y=1, x=0.5
$$
|
\frac{97\pi}{60}
|
olympiads
| 0.09375
|
Let $\overline{S T}$ be a chord of a circle $\omega$ which is not a diameter, and let $A$ be a fixed point on $\overline{S T}$. For which point $X$ on minor arc $\widehat{S T}$ is the length $A X$ minimized?
|
X
|
olympiads
| 0.078125
|
Petya and Vasya are playing the following game. Petya thinks of a natural number \( x \) with a digit sum of 2012. On each turn, Vasya chooses any natural number \( a \) and finds out the digit sum of the number \( |x-a| \) from Petya. What is the minimum number of turns Vasya needs to determine \( x \) with certainty?
|
2012 moves
|
olympiads
| 0.078125
|
Consider an \( n \times n \) chessboard. A rook needs to move from the bottom-left corner to the top-right corner. The rook can only move upwards and to the right, without landing on or below the main diagonal. (The rook is on the main diagonal only at the start and end points.) How many such paths are there for the rook?
|
C_{n-1}
|
olympiads
| 0.203125
|
Find the zeros of the function \( f(z) = 1 + \cos z \) and determine their orders.
|
z_n = (2n+1)\pi \quad \text{with order 1} \quad \text{for} \, n = 0, \pm 1, \pm 2, \ldots
|
olympiads
| 0.0625
|
Given the function \( f(x) \):
\[ f(x) = \begin{cases}
\ln x & \text{if } x > 1, \\
\frac{1}{2} x + \frac{1}{2} & \text{if } x \leq 1
\end{cases} \]
If \( m < n \) and \( f(m) = f(n) \), what is the minimum value of \( n - m \)?
|
3 - 2\ln 2
|
olympiads
| 0.109375
|
A root of unity is a complex number that is a solution to \( z^{n}=1 \) for some positive integer \( n \). Determine the number of roots of unity that are also roots of \( z^{2}+a z+b=0 \) for some integers \( a \) and \( b \).
|
8
|
olympiads
| 0.359375
|
How many Friday the 13th can occur in a non-leap year?
|
3
|
olympiads
| 0.234375
|
Let us call a natural number \( n \) "squarable" if the numbers from 1 to \( n \) can be arranged in such a way that each member of the sequence, when added to its position number, gives a perfect square. For example, the number 5 is squarable because the numbers can be arranged as 32154, where \( 3+1, 2+2, 1+3, 5+4 \) are all perfect squares. Determine which of the numbers 7, 9, 11, 15 are squarable.
|
9 \text{ and } 15
|
olympiads
| 0.234375
|
Let $[ {x} ]$ be the greatest integer less than or equal to $x$ , and let $\{x\}=x-[x]$ .
Solve the equation: $[x] \cdot \{x\} = 2005x$
|
x = 0 \text{ and } x = \frac{-1}{2006}
|
aops_forum
| 0.15625
|
Find the smallest positive integer n such that n has exactly 144 positive divisors including 10 consecutive integers.
|
110880
|
olympiads
| 0.09375
|
Calculate the line integral of the first kind $\int_{L} \sqrt{x^{3} y} d l$, where $L$ is the arc of the cubic parabola $y = x^{3}$ connecting the points $O(0,0)$ and $A(1,1)$.
|
\frac{1}{54} \left( 10\sqrt{10} - 1 \right)
|
olympiads
| 0.46875
|
Plot the lines $2x + 3y = t$ and $5x - 7y = t$ in the Cartesian coordinate system, where $t$ is a real number. What will be the locus of the intersection points of these lines for all possible values of $t$?
|
y = 0.3x
|
olympiads
| 0.390625
|
How many natural numbers \(n\) are there such that
\[ 100 < \sqrt{n} < 101 ? \]
|
200
|
olympiads
| 0.078125
|
A crazy physicist has discovered a new particle called an omon. He has a machine, which takes two omons of mass $a$ and $b$ and entangles them; this process destroys the omon with mass $a$ , preserves the one with mass $b$ , and creates a new omon whose mass is $\frac 12 (a+b)$ . The physicist can then repeat the process with the two resulting omons, choosing which omon to destroy at every step. The physicist initially has two omons whose masses are distinct positive integers less than $1000$ . What is the maximum possible number of times he can use his machine without producing an omon whose mass is not an integer?
|
9
|
aops_forum
| 0.203125
|
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.
|
1 - \frac{\sqrt{2}}{2}
|
olympiads
| 0.109375
|
$ABCDEF$ is a regular hexagon. Let $R$ be the overlap between $\vartriangle ACE$ and $\vartriangle BDF$ . What is the area of $R$ divided by the area of $ABCDEF$ ?
|
\frac{1}{3}
|
aops_forum
| 0.078125
|
Into how many parts does the plane divided by \( n \) lines in general position, meaning that no two are parallel and no three lines pass through a single point?
|
1 + \frac{1}{2} n(n+1)
|
olympiads
| 0.59375
|
Find the functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that for all \( n \), the following equation holds:
\[ f(n) + f(f(n)) + f(f(f(n))) = 3n \]
|
f(n) = n
|
olympiads
| 0.5
|
In a class, each student has either 5 or 6 friends (friendship is mutual), and any two friends have a different number of friends. What is the minimum number of students, greater than 0, that can be in the class?
|
11
|
olympiads
| 0.109375
|
A merchant has inherently inaccurate scales. To the first customer, he sells one pound of goods using one scale pan. To the second customer, he measures one pound of the same goods using the other scale pan, believing that this compensates for the inaccuracy in measuring. The question is, did the merchant actually gain an advantage or incur a loss?
|
The merchant is at a loss.
|
olympiads
| 0.140625
|
In trapezoid $PQRS$, it is known that $\angle PQR=90^{\circ}$, $\angle QRS<90^{\circ}$, diagonal $SQ$ is 24 and is the angle bisector of $\angle S$, and the distance from vertex $R$ to line $QS$ is 16. Find the area of the trapezoid PQRS.
|
\frac{8256}{25}
|
olympiads
| 0.34375
|
Find the angle $\beta$ where $\frac{\pi}{2}<\beta<\pi$, given that $\operatorname{tg}(\alpha+\beta)=\frac{9}{19}$ and $\operatorname{tg} \alpha=-4$.
|
\beta = \pi - \operatorname{arctg} \, 5
|
olympiads
| 0.484375
|
The lengths of the sides of a triangle are successive terms of a geometric progression. Let \( A \) and \( C \) be the smallest and largest interior angles of the triangle respectively. If the shortest side has length \( 16 \mathrm{~cm} \) and
$$
\frac{\sin A - 2 \sin B + 3 \sin C}{\sin C - 2 \sin B + 3 \sin A} = \frac{19}{9},
$$
find the perimeter of the triangle in centimetres.
|
76
|
olympiads
| 0.078125
|
In $\triangle ABC$, $AB + AC = 7$, and the area of the triangle is 4. What is the minimum value of $\sin \angle A$?
|
\frac{32}{49}
|
olympiads
| 0.296875
|
Calculate the limit of the function:
$\lim_{x \rightarrow 0} \sqrt[x^{2}]{2-\cos x}$
|
\sqrt{e}
|
olympiads
| 0.203125
|
A positive integer is written on each of the six faces of a cube. For each vertex of the cube, we compute the product of the numbers on the three adjacent faces. The sum of these products is 1001. What is the sum of the six numbers on the faces?
|
31
|
olympiads
| 0.171875
|
Find all prime numbers \( p \) for which there exist natural numbers \( x \) and \( y \) such that \( p^x = y^3 + 1 \).
|
p = 2 \\text{ and } p = 3
|
olympiads
| 0.21875
|
The symbol $[x]$ represents the greatest integer less than or equal to the real number $x$. Find the solution to the equation $\left[3 x - 4 \frac{5}{6}\right] - 2 x - 1 = 0$.
|
6\frac{1}{2}
|
olympiads
| 0.171875
|
A cup is filled with a saline solution of $15\%$ concentration. There are three iron balls of different sizes: large, medium, and small, with their volumes in the ratio 10:5:3. Initially, the small ball is submerged in the cup of saline causing $10\%$ of the saline solution to overflow. Then the small ball is removed. Next, the medium ball is submerged in the cup and then removed. This is followed by submerging and removing the large ball. Finally, pure water is added to the cup until it is full again. What is the concentration of the saline solution in the cup at this point?
|
10\%
|
olympiads
| 0.140625
|
Let \( a \) be an integer. If the inequality \( |x+1| < a - 1.5 \) has no integral solution, find the greatest value of \( a \).
|
1
|
olympiads
| 0.203125
|
Find the solution to the system of differential equations given by:
\[
\frac{d x}{d t} = y - \frac{x}{2} - \frac{x y^3}{2}
\]
\[
\frac{d y}{d t} = -y - 2x + x^2 y^2
\]
|
The equilibrium point (x \equiv 0, y \equiv 0) is asymptotically stable.
|
olympiads
| 0.078125
|
We have four containers. The first three contain water, while the fourth is empty. The second container holds twice as much water as the first, and the third holds twice as much water as the second. We transfer half of the water from the first container, one-third of the water from the second container, and one-quarter of the water from the third container into the fourth container. Now, there are 26 liters of water in the fourth container. How much water is there in total in all the containers?
|
84 ext{ liters}
|
olympiads
| 0.109375
|
There are three identical water pools: Pool A, Pool B, and Pool C. A Type-A fountain fills Pool A, a Type-B fountain fills Pool B, and both Type-A and Type-B fountains together fill Pool C. It is known that filling Pool B takes 4 hours longer than filling Pool C, and filling Pool A takes 5 hours longer than filling Pool B. How many hours are needed to fill two-thirds of Pool C?
|
4
|
olympiads
| 0.328125
|
There are 2000 cities in Graphland; some of them are connected by roads.
For every city the number of roads going from it is counted. It is known that there are exactly two equal numbers among all the numbers obtained. What can be these numbers?
|
n or n-1
|
aops_forum
| 0.109375
|
While waiting for customers, a watermelon seller weighed 20 watermelons (each weighing 1 kg, 2 kg, 3 kg, ..., 20 kg) by balancing a watermelon on one side of the scale with one or two weights (possibly identical) on the other side of the scale. The seller recorded the weights he used on a piece of paper. What is the minimum number of different numbers that could appear in his notes if the weight of each weight is an integer number of kilograms?
|
6
|
olympiads
| 0.0625
|
Let $a_1,a_2,a_3,a_4$ be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let $i,j,k\in\{1,2,3,4\}$ with $i \neq j$ , $j\neq k$ , and $k\neq i $ . Determine the maximum number of triples $(i,j,k)$ for which $$ ({\rm gcd}(a_i,a_j))^2|a_k. $$
|
8
|
aops_forum
| 0.09375
|
An isosceles triangle with a base \( a \) and a base angle \( \alpha \) is inscribed in a circle. Additionally, a second circle is constructed, which is tangent to both of the triangle's legs and the first circle. Find the radius of the second circle.
|
\frac{a}{2 \sin \alpha (1 + \cos \alpha)}
|
olympiads
| 0.0625
|
China Mathematical Olympiad 2018 Q6
Given the positive integer $n ,k$ $(n>k)$ and $ a_1,a_2,\cdots ,a_n\in (k-1,k)$ ,if positive number $x_1,x_2,\cdots ,x_n$ satisfying:For any set $\mathbb{I} \subseteq \{1,2,\cdots,n\}$ , $|\mathbb{I} |=k$ ,have $\sum_{i\in \mathbb{I} }x_i\le \sum_{i\in \mathbb{I} }a_i$ , find the maximum value of $x_1x_2\cdots x_n.$
|
a_1 a_2 \cdots a_n
|
aops_forum
| 0.0625
|
Draw the triangle $ABC$ and an arbitrary line $PQ$. If the distances from the vertices $A$, $B$, and $C$ and the midpoints of the sides to the line $PQ$ are $a$, $b$, $c$, respectively, and $a_{1}$, $b_{1}$, $c_{1}$, respectively, then
$$
a + b + c = a_{1} + b_{1} + c_{1}
$$
|
a + b + c = a_1 + b_1 + c_1
|
olympiads
| 0.578125
|
How many cells does the diagonal cross in a grid rectangle with dimensions \(199 \times 991\)?
|
1189
|
olympiads
| 0.390625
|
Find the smallest four-digit number that gives a remainder of 5 when divided by 6.
|
1001
|
olympiads
| 0.5
|
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