Let $ a$, $ b$ and $ k$ be three positive integers. We define two sequences $ \left( a_{n}\right)$ and $ \left( b_{n}\right)$ by the starting values $ a_{1}\equal{}a$ and $ b_{1}\equal{}b$ and the recurrent equations $ a_{n\plus{}1}\equal{}ka_{n}\plus{}b_{n}$ and $ b_{n\plus{}1}\equal{}kb_{n}\plus{}a_{n}$ for each positive integer $ n$. Prove that if $ a_{1}\perp b_{1}$, $ a_{2}\perp b_{2}$ and $ a_{3}\perp b_{3}$ hold, then $ a_{n}\perp b_{n}$ holds for every positive integer $ n$. Here, the abbreviation $ x\perp y$ stands for "the numbers $ x$ and $ y$ are coprime".

[color=darkred]Let $A,B\in\mathcal{M}_4(\mathbb{R})$ such that $AB=BA$ and $\det (A^2+AB+B^2)=0$ . Prove that: \[\det (A+B)+3\det (A-B)=6\det (A)+6\det (B)\ .\][/color]

Prove that for all positive real numbers $a,b,c$ we have \[\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}\ge\sqrt{a^2+ac+c^2} \]

Given a number $n\in\mathbb{Z}^+$ and let $S$ denotes the set $\{0,1,2,...,2n+1\}$. Consider the function $f:\mathbb{Z}\times S\to [0,1]$ satisfying two following conditions simultaneously: i) $f(x,0)=f(x,2n+1)=0\forall x\in\mathbb{Z}$; ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x\in\mathbb{Z}$ and $y\in\{1,2,3,...,2n\}$. Let $F$ be the set of such functions. For each $f\in F$, let $v(f)$ be the set of values of $f$. a) Proof that $|F|=\infty$. b) Proof that for each $f\in F$ then $|v(f)|<\infty$. c) Find the maximum value of $|v(f)|$ for $f\in F$.

For some positive numbers $a$, $b$, $c$ and $d$, we know that $$ \frac{1}{a^3 + 1}+ \frac{1}{b^3 + 1}+ \frac{1}{c^3 + 1} + \frac{1}{d^3 + 1} = 2. $$ Prove that $$ \frac{1 - a}{a^2 - a + 1} + \frac{1-b}{b^2 - b + 1} + \frac{1-c}{c^2 - c + 1} +\frac{1-d}{d^2 - d + 1} \geq 0. $$

The square of $ 5\minus{}\sqrt{y^2\minus{}25}$ is: $ \textbf{(A)}\ y^2\minus{}5\sqrt{y^2\minus{}25} \qquad \textbf{(B)}\ \minus{}y^2 \qquad \textbf{(C)}\ y^2 \\ \textbf{(D)}\ (5\minus{}y)^2 \qquad \textbf{(E)}\ y^2\minus{}10\sqrt{y^2\minus{}25}$

A line $MN$ is given in the plane. Consider circles $k_1$, $k_2$ tangent to the line at points $M$, $N$, respectively, while touching each other externally. Let $X$ be the midpoint of the segment $PQ$, where $P$, $Q$ are in this order tangent points of the second common external tangent of the circles $k_1$, $k_2$. Find the locus of the points $X$ for all pairs of circles of the specified properties.

Triangles $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. If $AB = 4$, what is the perimeter of figure $ABCDEFG$? [asy] pair A,B,C,D,EE,F,G; A = (4,0); B = (0,0); C = (2,2*sqrt(3)); D = (3,sqrt(3)); EE = (5,sqrt(3)); F = (5.5,sqrt(3)/2); G = (4.5,sqrt(3)/2); draw(A--B--C--cycle); draw(D--EE--A); draw(EE--F--G); label("$A$",A,S); label("$B$",B,SW); label("$C$",C,N); label("$D$",D,NE); label("$E$",EE,NE); label("$F$",F,SE); label("$G$",G,SE); [/asy] $\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 21$

Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$.Points $X,Y$ lie on $AB,AC$,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$.Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$.

Let $ABCD$ be a square and let $P$ be a point on side $AB$. The point $Q$ lies outside the square such that $\angle ABQ = \angle ADP$ and $\angle AQB = 90^{\circ}$. The point $R$ lies on the side $BC$ such that $\angle BAR = \angle ADQ$. Prove that the lines $AR, CQ$ and $DP$ pass through a common point.

Let $a,b,c$ are positive real numbers and $\alpha$ be any real number. Denote $f(\alpha)=abc(a^{\alpha}+b^{\alpha}+c^{\alpha}), g(\alpha)=a^{2+\alpha}(b+c-a)+b^{2+\alpha}(-b+c+a)+c^{2+\alpha}(b-c+a)$. Determine $\min{|f(\alpha)-g(\alpha)|}$ and $\max{|f(\alpha)-g(\alpha)|}$, if they are exists.

Let $H$ be the orthocenter of an obtuse triangle $ABC$ and $A_1B_1C_1$ arbitrary points on the sides $BC,AC,AB$ respectively.Prove that the tangents drawn from $H$ to the circles with diametrs $AA_1,BB_1,CC_1$ are equal.

Professor David proposed to his wife to calculate the steps of an escalator that worked in a shopping mall, asking him to walk up counting the steps that rise from the bottom to the end. The teacher, in turn, left with his wife, but walking the twice as fast, so that the woman advanced one step each time her husband advanced $2$. When The lady arrived at the top reported that she had counted $21$ steps, while the teacher counted $28$ of them. How many steps are there in sight on the ladder at any given time? [hide=original wording]El profesor David propuso a su senora calcular los escalones de una escalera mecanica que funcionaba en un centro comercial, pidiendole que caminara hacia arriba contando los escalones que subiera desde la base hasta el final. El profesor a su vez, partio junto a su senora, pero caminando el doble de rapido, de modo que la senora avanzaba un escalon cada vez que su marido avanzaba 2. Cuando la senora llego arriba informo que habıa contado 21 escalones, mientras que el profesor conto 28 de ellos, ¿Cuantos escalones hay a la vista en la escalera en un instante cualquiera?[/hide]

S is the set of all ($a$, $b$, $c$, $d$, $e$, $f$) where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides abcdef for all members of $S$.

Let $a,b$ and $c$ be positive real numbers. Prove that $$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $$

For which of the following values of $ k$ does the equation $ \frac{x\minus{}1}{x\minus{}2}\equal{}\frac{x\minus{}k}{x\minus{}6}$ have no solution for $ x$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

The numbers $1, 2, 3, . . . , 624$ are paired in such a way that the sum of the two numbers in each pair is $625$. For example $1$ and $624$ form a pair, and $30$ and $595$ form a pair. In how many of the $312$ pairs does the smaller number evenly divide the larger?

A number $N$ that is not divisible by $81$ can be represented as a sum of squares of three integers divisible by $3$. Prove that it is also representable as the sum of the squares of three integers not divisible by $3$.

The incircle of triangle $ ABC $ has centre $I$ and touches the sides $ BC $, $ CA $, $ AB $ at points $ A_1 $, $ B_1 $, $ C_1 $, respectively. Let $ I_a $, $ I_b $, $ I_c $ be excentres of triangle $ ABC $, touching the sides $ BC $, $ CA $, $ AB $ respectively. The segments $ I_aB_1 $ and $ I_bA_1 $ intersect at $ C_2 $. Similarly, segments $ I_bC_1 $ and $ I_cB_1 $ intersect at $ A_2 $, and the segments $ I_cA_1 $ and $ I_aC_1 $ at $ B_2 $. Prove that $ I $ is the center of the circumcircle of the triangle $ A_2B_2C_2 $. [i]L. Emelyanov, A. Polyansky[/i]

Find all real solutions $(x,y)$ of the system $x^2+y=12=y^2+x$.

Each edge of a complete graph with $101$ vertices is marked with $1$ or $-1$. It is known that absolute value of the sum of numbers on all the edges is less than $150$. Prove that the graph contains a path visiting each vertex exactly once such that the sum of numbers on all edges of this path is zero. [i](Y. Caro, A. Hansberg, J. Lauri, C. Zarb)[/i]

Let $\mathbb{R}_{\neq 0}$ be the set of nonzero real numbers. Find all functions $f:\mathbb{R}_{\neq 0}\rightarrow\mathbb{R}_{\neq 0}$ such that, for all $x,y\in\mathbb{R}_{\neq 0}$, \[(x-y)f(y^2)+f\left(xy\,f\left(\frac{x^2}{y}\right)\right)=f(y^2f(y)).\]

Let $F:\mathbb{R}^2\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be twice continuously differentiable functions with the following properties: • $F(u,u)=0$ for every $u\in\mathbb{R};$ • for every $x\in\mathbb{R},g(x)>0$ and $x^2g(x)\le 1;$ • for every $(u,v)\in\mathbb{R}^2,$ the vector $\nabla F(u,v)$ is either $\mathbf{0}$ or parallel to the vector $\langle g(u),-g(v)\rangle.$ Prove that there exists a constant $C$ such that for every $n\ge 2$ and any $x_1,\dots,x_{n+1}\in\mathbb{R},$ we have \[\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.\]

Determine all solutions to the system of equations \begin{align*}x+y+z=2 \\ x^2-y^2-z^2=2 \\ x-3y^2+z=0\end{align*}

A list of $2018$ numbers is created using the following procedure: the first number is $47$, the second number is $74$, and from there, each number is equal to the number formed by the last two digits of the sum of the two previous numbers:$$47, 74, 21, 95, 16, 11, \dots$$ Bruno squares each of the $2018$ numbers and sums them. Determine the remainder when this sum is divided by $8$.