Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$, $$f(f(x-y)-yf(x))=xf(y).$$

A magician and his assistant perform in front of an audience of many people. On the stage there is an $8$×$8$ board, the magician blindfolds himself, and then the assistant goes inviting people from the public to write down the numbers $1, 2, 3, 4, . . . , 64$ in the boxes they want (one number per box) until completing the $64$ numbers. After the assistant covers two adyacent boxes, at her choice. Finally, the magician removes his blindfold and has to $“guess”$ what number is in each square that the assistant. Explain how they put together this trick. $Clarification:$ Two squares are adjacent if they have a common side

For which values of the real parameter $r$ the equation $r^2 x^2+2rx+4=28r^2$ has two distinct integer roots?

Find all triplets $(a,b,p)$ where $a,b$ are positive integers and $p$ is a prime number satisfying: $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$

Let $H$ be the orthocenter of acute-angled triangle $ABC$ and $M$ be the midpoint of $AH$. Line $BH$ cuts $AC$ at $D$. Consider point $E$ such that $BC$ is the perpendicular bisector of $DE$. Segments $CM$ and $AE$ intersect at $F$. Show that $BF$ is perpendicular to $CM$. [i]Proposed by Germán Puga[/i]

Let $x\leq y\leq z$ be real numbers such that $x+y+z=12$, $x^2+y^2+z^2=54$. Prove that: a) $x\leq 3$ and $z\geq 5$ b) $xy$, $yz$, $zx\in [9,25]$

The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism. A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box? $\textbf{(A) }\frac{24}{5}\qquad\textbf{(B) }\frac{42}{5}\qquad\textbf{(C) }\frac{81}{5}\qquad\textbf{(D) }30\qquad\textbf{(E) }48$

Let $ABC$ be a triangle. A point $P$ is chosen inside $\triangle ABC$ such that $\angle BPC+\angle BAC=180^{\circ}$. The lines $AP,BP,CP$ intersect $BC,CA,AB$ at $P_A,P_B,P_C$ respectively. Let $X_A$ be the second intersection of the circumcircles of $\triangle ABC$ and $\triangle AP_BP_C$ . Similarly define $X_B,X_C$. Let $B'$ be the intersection of lines $AX_A,CX_C$, and let $C'$ be the intersection of lines $AX_A,BX_B$. Prove that lines $BB'$ and $CC'$ intersect on the circumcircle of $\triangle AP_BP_C$.

In trapezoid $ ABCD$ with $ \overline{BC}\parallel\overline{AD}$, let $ BC\equal{}1000$ and $ AD\equal{}2008$. Let $ \angle A\equal{}37^\circ$, $ \angle D\equal{}53^\circ$, and $ m$ and $ n$ be the midpoints of $ \overline{BC}$ and $ \overline{AD}$, respectively. Find the length $ MN$.

Let $N$ be the set of natural numbers, and let $f: N \to N$ be a function satisfying $f(x) + f(x + 2) < 2 f(x + 1)$ for any $x \in N$. Prove that there exists a straight line in the $xy$-plane which contains infinitely many points with coordinates $(n,f(n))$.

A unit square is removed from the corner of an $n\times n$ grid where $n \geq 2$. Prove that the remainder can be covered by copies of the "L-shapes" consisting of $3$ or $5$ unit square, as depicted in the figure below. Every square must be covered once and the L-shapes must not go over the bounds of the grid. [asy] import geometry; draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle); draw((-3.5,1)--(-2.5,1)--(-2.5,0)); draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle); draw((1.5,0)--(1.5,1)); draw((2.5,0)--(2.5,1)); draw((0.5,1)--(1.5,1)); draw((0.5,2)--(1.5,2)); [/asy][i]Estonian Olympiad, 2009[/i]

Find all functions $f : Z^+ \to Z^+$ satisfying $f (1) = 2, f (2) \ne 4$, and max $\{f (m) + f (n), m + n\} |$ min $\{2m + 2n, f (m + n) + 1\}$ for all $m, n \in Z^+$.

The sequence $\left(a_n \right)$ is defined by $a_1=1, \ a_2=2$ and $$a_{n+2} = 2a_{n+1}-pa_n, \ \forall n \ge 1,$$ for some prime $p.$ Find all $p$ for which there exists $m$ such that $a_m=-3.$

Solve in positive integers: \[x^{y^2+1}+y^{x^2+1}=2^z\]

A positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn he must replace some even integer $n$ on the board with $n/2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends. After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of $a$ and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.

Find all functions $f: Q \to R$ satisfying $f (x + y) = f (x)f (y) - f(xy) + 1$ for all $x,y \in Q$

Let $n$ be a positive integer. A square $ABCD$ is divided into $n^2$ identical small squares by drawing $(n-1)$ equally spaced lines parallel to the side $AB$ and another $(n- 1)$ equally spaced lines parallel to $BC$, thus giving rise to $(n+1)^2$ intersection points. The points $A, C$ are coloured red and the points $B, D$ are coloured blue. The rest of the intersection points are coloured either red or blue. Prove that the number of small squares having exactly $3$ vertices of the same colour is even.

Let $F$ be an increasing real function defined for all $x, 0 \le x \le 1$, satisfying the conditions (i) $F (\frac{x}{3}) = \frac{F(x)}{2}$. (ii) $F(1- x) = 1 - F(x)$. Determine $F(\frac{173}{1993})$ and $F(\frac{1}{13})$ .

A chess knight has injured his leg and is limping. He alternates between a normal move and a short move where he moves to any diagonally neighbouring cell. The limping knight moves on a $5 \times 6$ cell chessboard starting with a normal move. What is the largest number of moves he can make if he is starting from a cell of his own choice and is not allowed to visit any cell (including the initial cell) more than once?

Pentagon $ABCDE$ is inscribed in circle $\Omega$. Line $AD$ meets $CE$ at $F$, and $P (\neq E, F)$ is a point on segment $EF$. The circumcircle of triangle $AFP$ meets $\Omega$ at $Q(\neq A)$ and $AC$ at $R(\neq A)$. Line $AD$ meets $BQ$ at $S$, and the circumcircle of triangle $DES$ meets line $BQ, BD$ at $T(\neq S), U(\neq D)$, respectively. Prove that if $F, P, T, S$ are concyclic, then $P, T, R, U$ are concyclic.

Let $A$, $B$, $C$ and $D$ be points on the same circumference with $\angle BCD=90^\circ$. Let $P$ and $Q$ be the projections of $A$ onto $BD$ and $CD$, respectively. Prove that $PQ$ cuts the segment $AC$ into equal parts.

A group of children is sitting around a round table . At first, each child has an even number of candies. Each turn, each child gives half of his candies to the child sitting at his right. If, after a turn, a child has an odd number of candies, the teacher gives him\her an extra candy. Show that after a finite number of rounds all children will have the same number of candies.

Let be given a triangle $ABC$. Points $P$ on side $AC$ and $Y$ on the production of $CB$ beyond $B$ are chosen so that $Y$ subtends equal angles with $AP$ and $PC$. Similarly, $Q$ on side $BC$ and $X$ on the production of $AC$ beyond $C$ are such that $X$ subtends equal angles with $BQ$ and $QC$. Lines $YP$ and $XB$ meet at $R$, $XQ$ and $YA$ meet at $S$, and $XB$ and $YA$ meet at $D$. Prove that $PQRS$ is a parallelogram if and only if $ACBD$ is a cyclic quadrilateral.

Let $\triangle ABC$ be an acute-angled triangle with orthocentre $H$ and circumcentre $O$. Let $R$ be the radius of the circumcircle. \begin{align*} \text{Let }\mathit{A'}\text{ be the point on }\mathit{AO}\text{ (extended if necessary) for which }\mathit{HA'}\perp\mathit{AO}. \\ \text{Let }\mathit{B'}\text{ be the point on }\mathit{BO}\text{ (extended if necessary) for which }\mathit{HB'}\perp\mathit{BO}. \\ \text{Let }\mathit{C'}\text{ be the point on }\mathit{CO}\text{ (extended if necessary) for which }\mathit{HC'}\perp\mathit{CO}.\end{align*} Prove that $HA'+HB'+HC'<2R$ (Note: The orthocentre of a triangle is the intersection of the three altitudes of the triangle. The circumcircle of a triangle is the circle passing through the triangle’s three vertices. The circummcentre is the centre of the circumcircle.)

A merida path of order $2n$ is a lattice path in the first quadrant of $xy$- plane joining $(0,0)$ to $(2n,0)$ using three kinds of steps $U=(1,1)$, $D= (1,-1)$ and $L= (2,0)$, i.e. $U$ joins $x,y)$ to $(x+1,y+1)$ etc... An ascent in a merida path is a maximal string of consecutive steps of the form $U$. If $S(n,k)$ denotes the number of merdia paths of order $2n$ with exactly $k$ ascents, compute $S(n,1)$ and $S(n,n-1)$.