BALKAN-SL 2016

Let $a, b, c$ be positive real numbers. Prove that

$$\sqrt{a^{3} b+a^{3} c}+\sqrt{b^{3} c+b^{3} a}+\sqrt{c^{3} a+c^{3} b} \geq \frac{4}{3}(a b+b c+c a)$$

For all $x, y, z>0$ satisfying $\frac{x}{y z}+\frac{y}{z x}+\frac{z}{x y} \leq x+y+z$, prove that

$$\frac{1}{x^{2}+y+z}+\frac{1}{y^{2}+z+x}+\frac{1}{z^{2}+x+y} \leq 1$$

Find all monotonic functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the condition: for every real number $x$ and every natural number $n$

$$\left|\sum_{i=1}^{n} i\left(f(x+i+1)-f^{2}(x+i)\right)\right|<C$$

where $C>0$ is independent of $x$ and $f^{2}(x)=f(f(x))$.

The positive real numbers $a, b, c$ satisfy the equality $a+b+c=1$. For every natural number $n$ find the minimal possible value of the expression

$$E=\frac{a^{-n}+b}{1-a}+\frac{b^{-n}+c}{1-b}+\frac{c^{-n}+a}{1-c}$$

Let $a, b, c$ and $d$ be real numbers such that $a+b+c+d=2$ and $a b+b c+c d+d a+a c+b d=0$. Find the minimum value and the maximum value of the product $a b c d$.

Prove that there is no function from positive real numbers to itself, $f:(0,+\infty) \rightarrow(0,+\infty)$ such that:

$$f(f(x)+y)=f(x)+3 x+y f(y) \text {,for every } x, y \in(0,+\infty)$$

Find all integers $n \geq 2$ for which there exist the real numbers $a_{k}, 1 \leq k \leq n$, which are satisfying the following conditions:

$$\sum_{k=1}^{n} a_{k}=0, \sum_{k=1}^{n} a_{k}^{2}=1 \text { and } \sqrt{n} \cdot\left(\sum_{k=1}^{n} a_{k}^{3}\right)=2(b \sqrt{n}-1), \text { where } b=\max _{1 \leq k \leq n}\left\{a_{k}\right\}$$

Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ for which $f(g(n))-g(f(n))$ is independent on $n$ for any $g: \mathbb{Z} \rightarrow \mathbb{Z}$.

Let positive integers $K$ and $d$ be given. Prove that there exists a positive integer $n$ and a sequence of $K$ positive integers $b_{1}, b_{2}, \ldots, b_{K}$ such that the number $n$ is a $d$-digit palindrome in all number bases $b_{1}, b_{2}, \ldots, b_{K}$.

There are 2016 costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.) Find the maximal $k$ such that the following holds: There are $k$ customers such that either all of them were in the shop at a specific time instance or no two of them were both in the shop at any time instance.

The plane is divided into unit squares by means of two sets of parallel lines. The unit squares are coloured in 1201 colours so that no rectangle of perimeter 100 contains two squares of the same colour. Show that no rectangle of size $1 \times 1201$ contains two squares of the same colour.

The point $M$ lies on the side $A B$ of the circumscribed quadrilateral $A B C D$. The points $I_{1}, I_{2}$, and $I_{3}$ are the incenters of $\triangle M B C, \triangle M C D$, and $\triangle M D A$. Show that the points $M, I_{1}, I_{2}$, and $I_{3}$ lie on a circle.

Let $A B C D$ be a cyclic quadrilateral, with $A B<C D$, whose diagonals intersect at the point $F$ and $A D, B C$ intersect at the point $E$. Let also $K, L$ be the projections of $F$ onto the sides $A D, B C$ respectively, and $M, S, T$ be the midpoints of $E F, C F, D F$. Prove that the second intersection point of the circumcircles of the triangles $M K T, M L S$ lies on the side $C D$.

Given that $A B C$ is a triangle where $A B<A C$. On the half-lines $B A$ and $C A$ we take points $F$ and $E$ respectively such that $B F=C E=B C$. Let $M, N$ and $H$ be the mid-points of the segments $B F, C E$ and $B C$ respectively and $K$ and $O$ be the circumcircles of the triangles $A B C$ and $M N H$ respectively. We assume that $O K$ cuts $B E$ and $H N$ at the points $A_{1}$ and $B_{1}$ respectively and that $C_{1}$ is the point of intersection of $H N$ and $F E$. If the parallel line from $A_{1}$ to $O C_{1}$ cuts the line $F E$ at $D$ and the perpendicular from $A_{1}$ to the line $D B_{1}$ cuts $F E$ at the point $M_{1}$, prove that $E$ is the orthocenter of the triangle $A_{1} O M_{1}$.

Find all natural numbers $n$ for which $1^{\phi(n)}+2^{\phi(n)}+\ldots+n^{\phi(n)}$ is coprime with $n$.

Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$ $(d(n)$ is the number of divisors of the number $n$ including 1 and $n$ ).

Find all the integer solutions $(x, y, z)$ of the equation

$$(x+y+z)^{5}=80 x y z\left(x^{2}+y^{2}+z^{2}\right)$$

Find all monic polynomials $f$ with integer coefficients satisfying the following condition: There exists a positive integer $N$ such that for every prime $p>N, p$ divides $2(f(p))!+1$.

A positive integer $n$ is downhill if its decimal representation $\overline{a_{k} a_{k-1} \ldots a_{0}}$ satisfies $a_{k} \geq a_{k-1} \geq \ldots \geq a_{0}$. A real-coefficient polynomial $P$ is integer-valued if $P(n)$ is an integer for all integer $n$, and downhill-integervalued if $P(n)$ is an integer for all downhill positive integers $n$. Is it true that every downhill-integer-valued polynomial is also integer-valued?