Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

Let $ ABC$ be a triangle with $ AB \equal{} AC$ . The angle bisectors of $ \angle C AB$ and $ \angle AB C$ meet the sides $ B C$ and $ C A$ at $ D$ and $ E$ , respectively. Let $ K$ be the incentre of triangle $ ADC$. Suppose that $ \angle B E K \equal{} 45^\circ$ . Find all possible values of $ \angle C AB$ . [i]Jan Vonk, Belgium, Peter Vandendriessche, Belgium and Hojoo Lee, Korea [/i]

Characterize the set of positive integers $n$ such that, for all integers $a$, the sequence $a$, $a^2$, $a^3$, $\cdots$ is periodic modulo $n$.

There are two circles of perimeter $1979$. Let $1979$ points be marked on the first circle, and several arcs with the total length of $1$ on the second. Show that it is possible to place the second circle onto the first in such a way that none of the marked points is covered by a marked arc.

Which natural numbers can be expressed as the difference of squares of two integers?

In multiplying two positive integers $a$ and $b$, Ron reversed the digits of the two-digit number $a$. His errorneous product was $161$. What is the correct value of the product of $a$ and $b$? $ \textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224 $

Given two integers $m,n$ which are greater than $1$. $r,s$ are two given positive real numbers such that $r<s$. For all $a_{ij}\ge 0$ which are not all zeroes,find the maximal value of the expression \[f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^{\frac{s}{r}})^{\frac{1}{s}}}.\]

Does there exist a set $S$ of $100$ points in a plane such that the center of mass of any $10$ points in $S$ is also a point in $S$?

Show that if the integers $a_1$; $\dots$ $a_m$ are nonzero and for each $k =0; 1; \dots ;n$ ($n < m - 1$), $a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$; then the sequence $a_1, \dots, a_m$ contains at least $n+1$ pairs of consecutive terms having opposite signs. [i]O. Musin[/i]

Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.

On Binary Island, residents communicate using special paper. Each piece of paper is a $1 \times n$ row of initially uncolored squares. To send a message, each square on the paper must either be colored either red or green. Unfortunately the paper on the island has become damaged, and each sheet of paper has $10$ random consecutive squares each of which is randomly colored red or green. Malmer and Weven would like to develop a scheme that allows them to send messages of length $2016$ between one another. They would like to be able to send any message of length $2016$, and they want their scheme to work with perfect accuracy. What is the smallest value of $n$ for which they can develop such a strategy? [i]Note that when sending a message, one can see which 10 squares are colored and what colors they are. One also knows on which square the message begins, and on which square the message ends.[/i]

The cells of an $n \times n$ table are filled with the numbers $1,2,\dots,n$ for the first row, $n+1,n+2,\dots,2n$ for the second, and so on until $n^2-n,n^2-n+1,\dots,n^2$ for the $n$-th row. Peter picks $n$ numbers from this table such that no two of them lie on the same row or column. Peter then calculates the sum $S$ of the numbers he has chosen. Prove that Peter always gets the same number for $S$, no matter how he chooses his $n$ numbers.

A group of $4046$ friends will play a videogame tournament. For that, $2023$ of them will go to one room which the computers are labeled with $a_1,a_2,\dots,a_{2023}$ and the other $2023$ friends go to another room which the computers are labeled with $b_1,b_2,\dots,b_{2023}$. The player of computer $a_i$ [b]always[/b] challenges the players of computer $b_i,b_{i+2},b_{i+3},b_{i+4}$(the player doesn't challenge $b_{i+1}$). After the first round, inside both rooms, the players may switch the computers. After the reordering, all the players realize that they are challenging the same players of the first round. Prove that if [b]one[/b] player has [b]not[/b] switched his computer, then all the players have not switched their computers.

Two players play a game on four piles of pebbles labeled with the numbers $1,2,3,4$ respectively. The players take turns in an alternating fashion. On his or her turn, a player selects integers $m$ and $n$ with $1\leq m<n\leq 4$, removes $m$ pebbles from pile $n$, and places one pebble in each of the piles $n-1,n-2,\dots,n-m$. A player loses the game if he or she cannot make a legal move. For each starting position, determine the player with a winning strategy.

Solve for real $x,y$: $x+y=2$ $x^5+y^5=82$

The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$. Determine $f (2014)$. $N_0=\{0,1,2,...\}$

The excircle of a triangle $ABC$ corresponding to $A$ touches the side $BC$ at $K$ and the rays $AB$ and $AC$ at $P$ and $Q$, respectively. The lines $OB$ and $OC$ intersect $PQ$ at $M$ and $N$, respectively. Prove that $$\frac{QN}{AB}=\frac{NM}{BC}=\frac{MP}{CA}.$$

Let $ABCD$ be a trapezium in which $AB$ is parallel to $CD$. The circles on $AD$ and $BC$ as diameters intersect at two distinct points $P$ and $Q$. Prove that the lines $PQ,AC,BD$ are concurrent.

For Kelvin the Frog's birthday, Alex the Kat gives him one brick weighing $x$ pounds, two bricks weighing $y$ pounds, and three bricks weighing $z$ pounds, where $x,y,z$ are positive integers of Kelvin the Frog's choice. Kelvin the Frog has a balance scale. By placing some combination of bricks on the scale (possibly on both sides), he wants to be able to balance any item of weight $1,2,\ldots,N$ pounds. What is the largest $N$ for which Kelvin the Frog can succeed?

I have two cents and Bill has $n$ cents. Bill wants to buy some pencils, which come in two different packages. One package of pencils costs 6 cents for 7 pencils, and the other package of pencils costs a [i]dime for a dozen[/i] pencils (i.e. 10 cents for 12 pencils). Bill notes that he can spend [b]all[/b] $n$ of his cents on some combination of pencil packages to get $P$ pencils. However, if I [i]give my two cents[/i] to Bill, he then notes that he can instead spend [b]all[/b] $n+2$ of his cents on some combination of pencil packages to get fewer than $P$ pencils. What is the smallest value of $n$ for which this is possible? Note: Both times Bill must spend [b]all[/b] of his cents on pencil packages, i.e. have zero cents after either purchase.

Suppose $a$ real number so that there is a non-constant polynomial $P (x)$ such that $\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}$ for each real number $x$, with $x+\pi \ne 0$ and $P(x+\pi)\ne 0$. Show that $a$ is a natural number.

In the adjoining figure, $AB$ is a diameter of the circle, $CD$ is a chord parallel to $AB$, and $AC$ intersects $BD$ at $E$, with $\angle AED = \alpha$. The ratio of the area of $\triangle CDE$ to that of $\triangle ABE$ is [asy] size(200); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair A=(-1,0), B=(1,0), E=(0,-.4), C=(.6,-.8), D=(-.6,-.8), E=(0,-.8/(1.6)); draw(unitcircle); draw(A--B--D--C--A); draw(Arc(E,.2,155,205)); label("$A$",A,W); label("$B$",B,C); label("$C$",C,C); label("$D$",D,W); label("$\alpha$",E-(.2,0),W); label("$E$",E,N);[/asy] $ \textbf{(A)}\ \cos\ \alpha\qquad\textbf{(B)}\ \sin\ \alpha\qquad\textbf{(C)}\ \cos^2\alpha\qquad\textbf{(D)}\ \sin^2\alpha\qquad\textbf{(E)}\ 1 - \sin\ \alpha $

For any real number $p\geq1$ consider the set of all real numbers $x$ with \[p<x<\left(2+\sqrt{p+\frac{1}{4}}\right)^2.\] Prove that from any such set one can select four mutually distinct natural numbers $a, b, c, d$ with $ab=cd.$

What is the greatest number of positive integers lesser than or equal to 2016 we can choose such that it doesn't have two of them differing by 1,2, or 6?

Let $p, k$ and $x$ be positive integers such that $p \geq k$ and $x < \left[ \frac{p(p-k+1)}{2(k-1)} \right]$, where $[q]$ is the largest integer no larger than $q$. Prove that when $x$ balls are put into $p$ boxes arbitrarily, there exist $k$ boxes with the same number of balls.