In a remote island, a language in which every word can be written using only the letters $a$, $b$, $c$, $d$, $e$, $f$, $g$ is spoken. Let's say two words are [i]synonymous[/i] if we can transform one into the other according to the following rules: i) Change a letter by another two in the following way: \[a \rightarrow bc,\ b \rightarrow cd,\ c \rightarrow de,\ d \rightarrow ef,\ e \rightarrow fg,\ f\rightarrow ga,\ g\rightarrow ab\] ii) If a letter is between other two equal letters, these can be removed. For example, $dfd \rightarrow f$. Show that all words in this language are synonymous.

Let $D$ and $E$ be points in the interiors of sides $AB$ and $AC$, respectively, of a triangle $ABC$, such that $DB = BC = CE$. Let the lines $CD$ and $BE$ meet at $F$. Prove that the incentre $I$ of triangle $ABC$, the orthocentre $H$ of triangle $DEF$ and the midpoint $M$ of the arc $BAC$ of the circumcircle of triangle $ABC$ are collinear.

Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation \[ f(x^2+y^2+2f(xy)) = (f(x+y))^2. \] for all $x,y \in \mathbb{R}$.

Let $ABC$ be a triangle and $\omega$ its circumcircle. Point $D$ lies on the arc $BC$ (not containing $A$) of $\omega$ and is different from $B, C$ and the midpoint of arc $BC$ . The tangent line to $\omega$ at $D$ intersects lines $BC, CA,AB$ at $A', B',C'$ respectively. Lines $BB'$ and $CC'$ intersect at $E$. Line $AA' $ intersects again circle $\omega$ at $F$. Prove that the three points $D,E,F$ are colinear. Malik Talbi

We say that a strictly increasing positive real sequence $a_1,a_2,\cdots $ is an [i]elf sequence[/i] if for any $c>0$ we can find an $N$ such that $a_n<cn$ for $n=N,N+1,\cdots$. Furthermore, we say that $a_n$ is a [i]hat[/i] if $a_{n-i}+a_{n+i}<2a_n$ for $\displaystyle 1\le i\le n-1$. Is it true that every elf sequence has infinitely many hats?

Two students $A$ and $B$ play a game on a $20 \text{ x } 20$ chessboard. It is known that two squares are said to be [i]adjacent[/i] if the two squares have a common side. At the beginning, there is a chess piece in a certain square of the chessboard. Given that $A$ will be the first one to move the chess piece, $A$ and $B$ will alternately move this chess piece to an adjacent square. Also, the common side of any pair of adjacent squares can only be passed once. If the opponent cannot move anymore, then he will be declared the winner (to clarify since the wording wasn’t that good, you lose if you can’t move). Who among $A$ and $B$ has a winning strategy? Justify your claim.

If $A=\{1,2,...,4s-1,4s\}$ and $S \subseteq A$ such that $\mid S \mid =2s+2$, prove that in $S$ we can find three distinct numbers $x$, $y$ and $z$ such that $x+y=2z$

In acute scalene triangle $ABC$ the external angle bisector of $\angle BAC$ meet $BC$ at point $X$.Lines $l_b$ and $l_c$ which tangents of $B$ and $C$ with respect to $(ABC)$.The line pass through $X$ intersects $l_b$ and $l_c$ at points $Y$ and $Z$ respectively. Suppose $(AYB)\cap(AZC)=N$ and $l_b\cap l_c=D$. Show that $ND$ is angle bisector of $\angle YNZ$. Proposed by [i]Alireza Haghi[/i]

The sides of an equilateral triangle are divided into n equal parts by $n-1$ points on each side. Through these points one draws parallel lines to the sides of the triangle. Thus, the initial triangle is divides into $n^2$ equal equilateral triangles. In every vertex of such a triangle there is a beetle. The beetles start crawling simultaneously, with equal speed, along the sides of the small triangles. When they reach a vertex, the beetles change the direction of their movement by $60^{\circ}$ or by $120^{\circ}$ . a) Prove that, if $n \geq 7$, the beetles can move indefinitely on the sides of the small triangles without two beetles ever meeting in a vertex of a small triangle. b) Determine all the values of $n \geq 1$ for which the beetles can move along the sides of the small triangles without meeting in their vertices.

Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$. [i]Proposed by Japan[/i]

Let $ABC$ be a triangle with $AB = 7, BC = 5,$ and $CA = 6$. Let $D$ be a variable point on segment $BC$, and let the perpendicular bisector of $AD$ meet segments $AC, AB$ at $E, F,$ respectively. It is given that there is a point $P$ inside $\triangle ABC$ such that $\frac{AP}{PC} = \frac{AE}{EC}$ and $\frac{AP}{PB} = \frac{AF}{FB}$. The length of the path traced by $P$ as $D$ varies along segment $BC$ can be expressed as $\sqrt{\frac{m}{n}}\sin^{-1}\left(\sqrt \frac 17\right)$, where $m$ and $n$ are relatively prime positive integers, and angles are measured in radians. Compute $100m + n$. [i]Proposed by Edward Wan[/i]

Let $I$ be the incenter of an acute-angled triangle $ABC$. Let $P$, $Q$, $R$ be points on sides $AB$, $BC$, $CA$ respectively, such that $AP=AR$, $BP=BQ$ and $\angle PIQ = \angle BAC$. Prove that $QR \perp AC$.

The diagonals of a quadrilateral $ABCD$ are perpendicular: $AC\perp BD$. Four squares, $ABEF,BCGH,CDIJ,DAKL$, are erected externally on its sides. The intersection points of the pairs of straight lines $CL,DF; DF,AH; AH,BJ; BJ,CL$ are denoted by $P_1,Q_1,R_1, S_1$, respectively, and the intersection points of the pairs of straight lines $AI,BK; BK,CE;$ $ CE,DG; DG,AI$ are denoted by $P_2,Q_2,R_2, S_2$, respectively. Prove that $P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2.$

$ n$ people decide to play a game. There are $ n\minus{}1$ ropes and each of its two ends are in hand of one of the players, in such a way that ropes and players form a tree. (Each person can hold more than rope end.) At each step a player gives one of the rope ends he is holding to another player. The goal is to make a path of length $ n\minus{}1$ at the end. But the game regulations change before game starts. Everybody has to give one of his rope ends only two one of his neighbors. Let $ a$ and $ b$ be minimum steps for reaching to goal in these two games. Prove that $ a\equal{}b$ if and only if by removing all players with one rope end (leaves of the tree) the remaining people are on a path. (the remaining graph is a path.) [img]http://i37.tinypic.com/2l9h1tv.png[/img]

Points $C_1$ and $C_2$ lie on side $AB$ of triangle $ABC$, where the point $C_1$ belongs to the segment $AC_2$ and $\angle ACC_1= \angle BCC_2$. On segments $CC_1$ and $CC_2$ points $A'$ and $B'$ are taken such that $\angle CAA'= \angle CBB' = \angle C_1CC_2$. Prove that the center of the circle $(CA'B')$ lies on the perpendicular bisector of the segment $AB$.

In the $xOy$ system, consider the points $A_n(n,n^3)$ with $n\in \mathbb{N}^*$ and the point $B(0,1)$. Prove that a) for any positive integers $k>j>i\ge 1$, the points $A_i,A_j,A_k$ cannot be collinear. b) for any positive integers $i_k>i_{k-1}>\ldots>i_1\ge 1$, we have \[\mu(\widehat{A_{i_1}OB})+\mu(\widehat{A_{i_2}OB})+\cdots+\mu(\widehat{A_{i_k}OB})<\frac{\pi}{2}\] [i]***[/i]

In the interior of the convex polygon $A_1A_2...A_{2n}$ there is point $M$. Prove that at least one side of the polygon has not intersection points with the lines $MA_i$, $1\le i\le 2n$. (Spain)

Let $P$ be a polynomial of degree greater than or equal to $4$ with integer coefficients. An integer $x$ is called $P$-[i]representable[/i] if there exists integer numbers $a$ and $b$ such that $x = P(a) - P(b)$. Prove that, if for all $N \geq 0$, more than half of the integers of the set $\{0,1,\dots,N\}$ are $P$-[i]representable[/i], then all the even integers are $P$-[i]representable[/i] or all the odd integers are $P$-[i]representable[/i].

Let $ABCD$ be a square with side length $6$. Let $E$ be a point on the perimeter of $ABCD$ such that the area of $\triangle{AEB}$ is $\frac{1}{6}$ the area of $ABCD$. Find the maximum possible value of $CE^2$. [i]Proposed by Anthony Yang[/i]

Let \( ABC \) be an acute-angled scalene triangle with circumcenter \( O \). Denote by \( M \), \( N \), and \( P \) the midpoints of sides \( BC \), \( CA \), and \( AB \), respectively. Let \( \omega \) be the circle passing through \( A \) and tangent to \( OM \) at \( O \). The circle \( \omega \) intersects \( AB \) and \( AC \) at points \( E \) and \( F \), respectively (where \( E \) and \( F \) are distinct from \( A \)). Let \( I \) be the midpoint of segment \( EF \), and let \( K \) be the intersection of lines \( EF \) and \( NP \). Prove that \( AO = 2IK \) and that triangle \( IMO \) is isosceles.

For every positive integer $n$, define $$f(n)=\sum_{p\mid n}{p^{k_p}},$$where the sum is taken over all positive prime divisors $p$ of $n$, and $k_p$ is the unique integer satisfying $$p^{k_p}\leqslant n<p^{k_p+1}.$$Find$$\limsup_{n\to \infty} \frac{f(n)\log \log n}{n\log n} .$$

Two robots are programmed to communicate numbers using different bases. The first robot states: "I communicate in base 10, which interestingly is a perfect square. You communicate in base 16, which is not a perfect square." The second robot states: "I find it more interesting that the sum of our bases is the factorial of an integer." The second robot is referring to the factorial of which integer?

Solve the quasi-homogeneous equation \[\ddot{x}=x^5+x^2\dot{x}\]

Let $ABCD$ be a square of side length $4$. Points $E$ and $F$ are chosen on sides $BC$ and $DA$, respectively, such that $EF = 5$. Find the sum of the minimum and maximum possible areas of trapezoid $BEDF$. [i]Proposed by Andrew Wu[/i]

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for any two reals $x$ and $y$, we have $f\left( f\left( x+y\right) \right) +xy=f\left( x+y\right) +f\left( x\right) f\left( y\right) $.