Let $p$ be a prime and $m \geq 2$ be an integer. Prove that the equation \[ \frac{ x^p + y^p } 2 = \left( \frac{ x+y } 2 \right)^m \] has a positive integer solution $(x, y) \neq (1, 1)$ if and only if $m = p$. [i]Romania[/i]

In a game similar to three card monte, the dealer places three cards on the table: the queen of spades and two red cards. The cards are placed in a row, and the queen starts in the center; the card configuration is thus RQR. The dealer proceeds to move. With each move, the dealer randomly switches the center card with one of the two edge cards (so the configuration after the first move is either RRQ or QRR). What is the probability that, after 2004 moves, the center card is the queen?

$P$ is any point of the tetrahedron $ABCD$ except $D$. Show that at least one of the three distances $DA$, $DB$, $DC$ exceeds at least one of the distances $PA$, $PB$ and $PC$.

(a) Prove that (p^2)-1 is divisible by 24 if p is a prime number greater than 3. (b) Prove that (p^2)-(q^2) is divisible by 24 if p and q are prime numbers greater than 3.

For every integer $n\geqslant2$ prove the inequality $$ \frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{2^{n-2}}{n!}\leqslant\frac{3}{2}, $$ where $k!=1\cdot2\cdot\ldots\cdot k$.

Let $n\geq 1$ be odd integer. There are $n$ arrows are arranged from left to right, such that each arrow points either to the left or to the right. Prove that there exists an arrow that is pointed to by exactly as many arrows as it is pointing to. Note: For example, for $n=5$ and the arrangement $\rightarrow\rightarrow\leftarrow\leftarrow\rightarrow$, the successive arrows (from the left) point respectively to $4$, $3$, $2$, $3$, $0$ arrows.

Let $S=\{1,2,3,...,12\}$. How many subsets of $S$, excluding the empty set, have an even sum but not an even product? [i]Proposed by Gabriel Wu[/i]

Suppse that $X=\{1,2, \ldots, 2^{1996}-1\}$, prove that there exist a subset $A$ that satisfies these conditions: a) $1\in A$ and $2^{1996}-1\in A$; b) Every element of $A$ except $1$ is equal to the sum of two (possibly equal) elements from $A$; c) The maximum number of elements of $A$ is $2012$. [i]Romania[/i]

A set of points of the plane is called [i] obtuse-angled[/i] if every three of it's points are not collinear and every triangle with vertices inside the set has one angle $ >91^o$. Is it correct that every finite [i] obtuse-angled[/i] set can be extended to an infinite [i]obtuse-angled[/i] set? (UK)

Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers $x,y$ taken from two different subsets, the number $x^2-xy+y^2$ belongs to the third subset.

Find the largest real constant $\lambda$ such that \[\frac{\lambda abc}{a+b+c}\leq (a+b)^2+(a+b+4c)^2\] For all positive real numbers $a,b,c.$

The area of the triangle $ABC$ shown in the figure is $1$ unit. Points $D$ and $E$ lie on sides $AC$ and $BC$ respectively, and also are its ''one third'' points closer to $C$. Let $F$ be that $AE$ and $G$ are the midpoints of segment $BD$. What is the area of the marked quadrilateral $ABGF$? [img]https://cdn.artofproblemsolving.com/attachments/4/e/305673f429c86bbc58a8d40272dd6c9a8f0ab2.png[/img]

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

Let $m$ and $n$ be positive integers such that $2^n+3^m$ is divisible by $5$. Prove that $2^m+3^n$ is divisible by $5$.

Let $k$ be a positive integer. Prove that one can partition the set $\{ 0,1,2,3, \cdots ,2^{k+1}-1 \}$ into two disdinct subsets $\{ x_1,x_2, \cdots, x_{2k} \}$ and $\{ y_1, y_2, \cdots, y_{2k} \}$ such that $\sum_{i=1}^{2^k} x_i^m =\sum_{i=1}^{2^k} y_i^m$ for all $m \in \{ 1,2, \cdots, k \}$.

Let $ m_1,m_2,m_3,\dots,m_n$ be a rearrangement of the numbers $ 1,2,\dots,n$. Suppose that $ n$ is odd. Prove that the product \[ \left(m_1\minus{}1\right)\left(m_2\minus{}2\right)\cdots \left(m_n\minus{}n\right)\] is an even integer.

Let $ABC$ be an acute angled triangle. Points $E$ and $F$ are chosen on the sides $AC$ and $AB$, respectively, such that \[BC^2=BA\times BF+CE\times CA.\] Prove that for all such $E$ and $F$, circumcircle of the triangle $AEF$ passes through a fixed point different from $A$.

Prove that for all $n\in\mathbb{N}$ we have $\sum_{k=0}^n\dbinom {n}{k}^2=\dbinom {2n}{n}$.

In $\vartriangle ABC, D$ and $E$ are two points on segment $BC$ such that $BD = CE$ and $\angle BAD = \angle CAE$. Prove that $\vartriangle ABC$ is isosceles

$2022$ teams participated in an underwater polo tournament, each two teams played exactly once against each other. Team receives $2, 1, 0$ points for win, draw, and loss correspondingly. It turned out that all teams got distinct numbers of points. In the final standings the teams were ordered by the total number of points. A few days later, organizers realized that the results in the final standings were wrong due to technical issues: in fact, each match that ended with a draw according to them in fact had a winner, and each match with a winner in fact ended with a draw. It turned out that all teams still had distinct number of points! They corrected the standings, and ordered them by the total number of points. Could the correct order turn out to be the reversed initial order? [i](Proposed by Fedir Yudin)[/i]

A ray emanating from the vertex $A$ of the triangle $ABC$ intersects the side $BC$ at $X$ and the circumcircle of triangle $ABC$ at $Y$. Prove that $\frac{1}{AX}+\frac{1}{XY}\geq \frac{4}{BC}$.

In how many ways can the following figure be tiled with $2 \times 1$ dominos? [asy] defaultpen(linewidth(.8)); size(5.5cm); int i; for(i = 1; i<6; i = i+1) { draw((.5 + i,6-i)--(.5 + i,i-6)--(-(.5 + i),i-6)--(-(.5 + i),6-i)--cycle);} draw((.5,5)--(.5,-5)^^(-.5,5)--(-.5,-5)^^(5.5,0)--(-5.5,0)); [/asy] [i]Proposed by Lewis Chen[/i]

Find all prime numbers $p,q$, for which $p^{q+1}+q^{p+1}$ is a perfect square. [i]Proposed by P. Boyvalenkov[/i]

The sum of the base-$ 10$ logarithms of the divisors of $ 10^n$ is $ 792$. What is $ n$? $ \textbf{(A)}\ 11\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 13\qquad \textbf{(D)}\ 14\qquad \textbf{(E)}\ 15$

For a positive integer $n$, let $S(n) $be the set of complex numbers $z = x+iy$ ($x,y \in R$) with $ |z| = 1$ satisfying $(x+iy)^n+(x-iy)^n = 2x^n$ . (a) Determine $S(n)$ for $n = 2,3,4$. (b) Find an upper bound (depending on $n$) of the number of elements of $S(n)$ for $n > 5$.