Find all pairs $(x, y)$ of positive rational numbers such that $x^{y}=y^{x}$.

Let $n > 1$ be an integer and let $a_1, a_2,... , a_n$ be $n$ different integers. Show that the polynomial $f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1$ is not divisible by any polynomial with integer coefficients and of degree greater than zero but less than $n$ and such that the highest power of $x$ has coefficient $1$.

Sequence $x_1 , x_2 , ..., $ with $x_1=20$ ; $x_2=12$ for all $n\geq 1$ such that $x_{n+2}=x_n+x_{n+1}+2\sqrt{x_{n}*x_{n+1}+121} $then prove that $x_{2013}$ is an integer number.

Let $A=\{1,2,\ldots,1997\}$ be a set. Find the samllest integer $k>1$ such that in each subset $M{}$ of $A{}$, which cointain $k{}$ elements, there is a multiple of the smallest element from $M{}$, different from itself.

In $\triangle ABC\ T$ is a point lies on the internal angle bisector of $B$. Let $\omega$ be circle with diameter $BT$. $\omega$ intersects with $BA$ and $BC$ at $P$ and $Q$,respectively. A circle passes through $A$ and tangent to $\omega$ at $P$ intersects with $AC$ again at $X$ . A circle passes through $B$ and tangent to $\omega$ at $Q$ intersects with $AC$ again at $Y$ . Prove that $TX=TY$

Let $ n,k$ be given positive integers satisfying $ k\le 2n \minus{} 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m \equal{} f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.

In the vertexes of a regular $ 31$-gon there are written the numbers from $ 1$ to $ 31$, ordered increasingly, clockwise oriented. We are allowed to perform an operation which consists in taking any three vertexes, namely the ones who have written $ a$,$ b$, and $ c$ and change them into $ c$, $ a\minus{}\frac{1}{10}$ and $ b\plus{}\frac{1}{10}$ respectively ( $ a$ becomes $ c$, $ b$ becomes $ a\minus{}\frac{1}{10}$ and $ c$ turns into $ b\plus{}\frac{1}{10}$ Prove that after applying several operations we can reach the state in which the numbers in the vertexes are the numbers from $ 1$ to $ 31$, ordered increasingly,anti-clockwise oriented.

Several circles are drawn on the plane and all points of their meeting or touching are marked. May be that each circle contains exactly four marked points and exactly four marked points lie on each circle?

A $3 \times 7$ is covered without overlap by $3$ shapes of tiles: $2 \times 2$, $1 \times 4$, and $1 \times 1$, shown below. What is the minimum possible number of $1 \times 1$ tiles used? [center][img width=70]https://wiki-images.artofproblemsolving.com//e/ee/2024-AMC8-q7.png[/img][/center] $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

A parabola $y = ax^{2} + bx + c$ has vertex $(4,2)$. If $(2,0)$ is on the parabola, then $abc$ equals $ \textbf{(A)}\ -12\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 12$

Considers a quadrilateral $[ABCD]$ that has an inscribed circle and a circumscribed circle. The sides $[AD]$ and $[BC]$ are tangent to the circle inscribed at points $E$ and $F$, respectively. Prove that $AE \cdot F C = BF \cdot ED$. [img]https://1.bp.blogspot.com/-6o1fFTdZ69E/X4XMo98ndAI/AAAAAAAAMno/7FXiJnWzJgcfSn-qSRoEAFyE8VgxmeBjwCLcBGAsYHQ/s0/2005%2BPortugal%2Bp5.png[/img]

A positive integer $N$ is [i]apt[/i] if for each integer $0 < k < 1009$, there exists exactly one divisor of $N$ with a remainder of $k$ when divided by $1009$. For a prime $p$, suppose there exists an [i]apt[/i] positive integer $N$ where $\tfrac Np$ is an integer but $\tfrac N{p^2}$ is not. Find the number of possible remainders when $p$ is divided by $1009$. [i]Proposed by Evan Chang[/i]

In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Let $M$ be the midpoint of side $AB$, $G$ be the centroid of $\triangle ABC$, and $E$ be the foot of the altitude from $A$ to $BC$. Compute the area of quadrilateral $GAME$. [i]Proposed by Evin Liang[/i] [hide=Solution][i]Solution[/i]. $\boxed{23}$ Use coordinates with $A = (0,12)$, $B = (5,0)$, and $C = (-9,0)$. Then $M = \left(\dfrac{5}{2},6\right)$ and $E = (0,0)$. By shoelace, the area of $GAME$ is $\boxed{23}$.[/hide]

On a line $\ell$ there exists $3$ points $A, B$, and $C$ where $B$ is located between $A$ and $C$. Let $\Gamma_1, \Gamma_2, \Gamma_3$ be circles with $AC, AB$, and $BC$ as diameter respectively; $BD$ is a segment, perpendicular to $\ell$ with $D$ on $\Gamma_1$. Circles $\Gamma_4, \Gamma_5, \Gamma_6$ and $\Gamma_7$ satisfies the following conditions: $\bullet$ $\Gamma_4$ touches $\Gamma_1, \Gamma_2$, and$ BD$. $\bullet$ $\Gamma_5$ touches $\Gamma_1, \Gamma_3$, and $BD$. $\bullet$ $\Gamma_6$ touches $\Gamma_1$ internally, and touches $\Gamma_2$ and $\Gamma_3$ externally. $\bullet$ $\Gamma_7$ passes through $B$ and the tangent points of $\Gamma_2$ with $\Gamma_6$, and $\Gamma_3$ with $\Gamma_6$. Show that the circles $\Gamma_4, \Gamma_5$, and $\Gamma_7$ are congruent.

$a$, $b$ and $c$ are positive integers with $\textrm{gcd}(a,b,c)=1$. Is it true that there exist a positive integer $n$ such that $a^k+b^k+c^k$ is not divisible by $2^n$ for all $k$?

Find the sum of all positive integers $n$ such that $\tau(n)^2=2n$, where $\tau(n)$ is the number of positive integers dividing $n$. [i]Proposed by Michael Kural[/i]

Let $ f(x)\equal{}\left(1\plus{}\frac{1}{x}\right)^{x}\ (x>0)$. Find $ \lim_{n\to\infty}\left\{f\left(\frac{1}{n}\right)f\left(\frac{2}{n}\right)f\left(\frac{3}{n}\right)\cdots\cdots f\left(\frac{n}{n}\right)\right\}^{\frac{1}{n}}$.

Consider a $\triangle {ABC}$, with $AC \perp BC$. Consider a point $D$ on $AB$ such that $CD=k$, and the radius of the inscribe circles on $\triangle {ADC}$ and $\triangle {CDB}$ are equals. Prove that the area of $\triangle {ABC}$ is equal to $k^2$.

A prime $p$ has decimal digits $p_{n}p_{n-1} \cdots p_0$ with $p_{n}>1$. Show that the polynomial $p_{n}x^{n} + p_{n-1}x^{n-1}+\cdots+ p_{1}x + p_0$ cannot be represented as a product of two nonconstant polynomials with integer coefficients

On a $24$ hour clock, there are two times after $01:00$ for which the time expressed in the form $hh:mm$ and in minutes are both perfect squares. One of these times is $01:21$, since $121$ and $60+21 = 81$ are both perfect squares. Find the other time, expressed in the form $hh:mm$.

Suppose $\alpha, \beta$ are two positive rational numbers. Assume for some positive integers $m,n$, it is known that $\alpha^{\frac 1n}+\beta^{\frac 1m}$ is a rational number. Prove that each of $\alpha^{\frac 1n}$ and $\beta^{\frac 1m}$ is a rational number.

An equilateral $12$-gon has side length $10$ and interior angle measures that alternate between $90^\circ$, $90^\circ$, and $270^\circ$. Compute the area of this $12$-gon. [i]Proposed by Connor Gordon[/i]

Find $\lim_{n\to\infty} \int_0^{\pi} x|\sin 2nx| dx\ (n=1,\ 2,\ \cdots)$. [i]1992 Japan Women's University entrance exam/Physics, Mathematics[/i]

Let $A$ be a positive real number. What are the possible values of $\displaystyle\sum_{j=0}^{\infty} x_j^2, $ given that $x_0, x_1, \cdots$ are positive numbers for which $\displaystyle\sum_{j=0}^{\infty} x_j = A$?

Along a circle, 10 iron weights have been placed. Between every two weights there is a brass ball. The mass of each ball is equal to the difference of the masses of its neighboring weights. Prove that it is possible to divide the balls among two pans so as to make the balance in equilibrium. Proposed by V. Proizvolov