BALKAN-SL 2015

Let $a,b,c$ be sidelengths of a triangle and $r,R,s$ be the inradius, the circumradius and the semiperimeter respectively of the same triangle. Prove that: $$\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c} \leq \frac{r}{16Rs}+\frac{s}{16Rr} + \frac{11}{8s}$$ (Albania)

Let a $,b,c$ be sidelengths of a triangle and $m_a,m_b,m_c$ the medians at the corresponding sides. Prove that $$m_a\left(\frac{b}{a}-1\right)\left(\frac{c}{a}-1\right)+ m_b\left(\frac{a}{b}-1\right)\left(\frac{c}{b}-1\right) +m_c\left(\frac{a}{c}-1\right)\left(\frac{b}{c}-1\right)\geq 0.$$ (FYROM)

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$(x+y)f(2yf(x)+f(y))=x^{3}f(yf(x)), \ \ \ \forall x,y\in \mathbb{R}^{+}.$$ (Albania)

Let $m, n$ be positive integers and $a, b$ positive real numbers different from $1$ such thath $m > n$ and $$\frac{a^{m+1}-1}{a^m-1} = \frac{b^{n+1}-1}{b^n-1} = c$$ . Prove that $a^m c^n > b^n c^{m}$ (Turkey)

For a polynomials $P\in \mathbb{R}[x]$ , denote $f(P)=n$ if $n$ is the smallest positive integer for which is valid $$(\forall x\in \mathbb{R})(\underbrace{P(P(\ldots P}_{n}(x))\ldots )>0),$$ and $f(P)=0$ if such n doeas not exist. Exists polyomial $P\in \mathbb{R}[x]$ of degree $2014^{2015}$ such that $f(P)=2015$ ?

(Serbia)

Isaak and Jeremy play the following game. Isaak says to Jeremy that he thinks a few $2^n$ integers $k_1,..,k_{2^n}$ . Jeremy asks questions of the form: ''Is it true that $k_i<k_j$ ?'' in which Isaak answers by always telling the truth. After $n2^{n-1}$ questions, Jeramy must decide whether numbers of Isaak are all distinct each other or not. Prove that Jeremy has bo way to be ''sure'' for his final decision.

(UK)

A chessboard $1000 \times 1000$ is covered by dominoes $1 \times 10$ that can be rotated. We don't know which is the cover, but we are looking for it. For this reason, we choose a few $N$ cells of the chessboard, for which we know the position of the dominoes that cover them. Which is the minimum $N$ such that after the choice of $N$ and knowing the dominoed that cover them, we can be sure and for the rest of the cover?

(Bulgaria)

In an acute angled triangle $ABC$ , let $BB'$ and $CC'$ be the altitudes. Ray $C'B'$ intersects the circumcircle at $B''$ andl let $\alpha_A$ be the angle $\widehat{ABB''}$ . Similarly are defined the angles $\alpha_B$ and $\alpha_C$ . Prove that $$\sin \alpha _A \sin \alpha _B \sin \alpha _C\leq \frac{3\sqrt{6}}{32}$$ (Romania)

Let $ABC$ be a triangle with circumcircle $\omega$ . Point $D$ lies on the arc $BC$ of $\omega$ and is different than $B,C$ and the midpoint of arc $BC$ . Tangent of $\Gamma$ at $D$ intersects lines $BC$ , $CA$ , $AB$ at $A',B',C'$ , respectively. Lines $BB'$ and $CC'$ intersect at $E$ . Line $AA'$ intersects the circle $\omega$ again at $F$ . Prove that points $D,E,F$ are collinear.

(Saudi Arabia)

A set of points of the plane is called obtuse-angled 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 obtuse-angled set can be extended to an infinite obtuse-angled set?

(UK)

Let $AB$ be a diameter of a circle $(\omega)$ with centre $O$ . From an arbitrary point $M$ on $AB$ such that $MA < MB$ we draw the circles $(\omega_1)$ and $(\omega_2)$ with diameters $AM$ and $BM$ respectively. Let $CD$ be an exterior common tangent of $(\omega_1), (\omega_2)$ such that $C$ belongs to $(\omega_1)$ and $D$ belongs to $(\omega_2)$ . The point $E$ is diametrically opposite to $C$ with respect to $(\omega_1)$ and the tangent to $(\omega_1)$ at the point $E$ intersects $(\omega_2)$ at the points $F, G$ . If the line of the common chord of the circumcircles of the triangles $CED$ and $CFG$ intersects the circle $(\omega)$ at the points $K, L$ and the circle $(\omega_2)$ at the point $N$ (with $N$ closer to $L$ ), then prove that $KC = NL$ .

Let scalene triangle $ABC$ have orthocentre $H$ and circumcircle $\Gamma$ . $AH$ meets $\Gamma$ at $D$ distinct from $A$ . $BH$ and $CH$ meet $CA$ and $AB$ at $E$ and $F$ respectively, and $EF$ meets $BC$ at $P$ . The tangents to $\Gamma$ at $B$ and $C$ meet at $T$ . Show that $AP$ and $DT$ are concurrent on the circumcircle of $AFE$ .

Let $d$ be an even positive integer. John writes the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$ on the blackboard and then chooses three of them, let them be ${a_1}, {a_2}, {a_3}$ , erases them and writes the number $1+ \sum_{1\le i<j\leq 3} |{a_i} -{a_j}|$ He continues until two numbers remain written on on the blackboard. Prove that the sum of squares of those two numbers is different than the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$ .

(Albania)

Sequence $(a_n)_{n\geq 0}$ is defined as $a_{0}=0, a_1=1, a_2=2, a_3=6$ , and $a_{n+4}=2a_{n+3}+a_{n+2}-2a_{n+1}-a_n, n\geq 0$ . Prove that $n^2$ divides $a_n$ for infinite $n$ .

(Romania)

Let $a$ be a positive integer. For all positive integer n, we define $a_n=1+a+a^2+\ldots+a^{n-1}.$ Let $s,t$ be two different positive integers with the following property: If $p$ is prime divisor of $s-t$ , then $p$ divides $a-1$ . Prove that number $\frac{a_{s}-a_{t}}{s-t}$ is an integer.

(FYROM)

Find all pairs of positive integers $(x,y)$ with the following property: If $a,b$ are relative prime and positive divisors of $x^3 + y^3$ , then $a+b - 1$ is divisor of $x^3+y^3$ .

(Cyprus)

For a positive integer $s$ , denote with $v_2(s)$ the maximum power of $2$ that divides $s$ . Prove that for any positive integer $m$ that: $$v_2\left(\prod_{n=1}^{2^m}\binom{2n}{n}\right)=m2^{m-1}+1.$$ (FYROM)

Positive integer $m$ shall be called anagram of positive $n$ if every digit $a$ appears as many times in the decimal representation of $m$ as it appears in the decimal representation of $n$ also. Is it possible to find $4$ different positive integers such that each of the four to be anagram of the sum of the other $3$ ?

(Bulgaria)