BALKAN-SL 2018

Let $a, b, c$ be positive real numbers such that $a b c=\frac{2}{3}$. Prove that

$$\frac{a b}{a+b}+\frac{b c}{b+c}+\frac{c a}{c+a} \geqslant \frac{a+b+c}{a^{3}+b^{3}+c^{3}}$$

(FYR Macedonia)

Two ants start at the same point in the plane. Each minute they choose whether to walk due north, east, south or west. They each walk 1 meter in the first minute. In each subsequent minute the distance they walk is multiplied by a rational number $q>0$. They meet after a whole number of minutes, but have not taken exactly the same route within that time. Determine all possible values of $q$. (United Kingdom)

Show that for every positive integer $n$ we have:

$$\sum_{k=0}^{n}\left(\frac{2 n+1-k}{k+1}\right)^{k}=\left(\frac{2 n+1}{1}\right)^{0}+\left(\frac{2 n}{2}\right)^{1}+\cdots+\left(\frac{n+1}{n+1}\right)^{n} \leqslant 2^{n}$$

Let $a, b, c$ be positive real numbers such that $a b c=1$. Prove that the following inequality holds:

$$2\left(a^{2}+b^{2}+c^{2}\right)\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}\right) \geqslant 3(a+b+c+a b+b c+c a)$$

(Romania)

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a concave function and let $g: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Given that

$$f(x+y)+f(x-y)-2 f(x)=g(x) y^{2}$$

for all $x, y \in \mathbb{R}$, prove that $f$ is a quadratic function. (Bulgaria)

Let $n$ be a positive integer and let $x_{1}, \ldots, x_{n}$ be real numbers. Show that

$$\sum_{i=1}^{n} x_{i}^{2} \geqslant \frac{1}{n+1}\left(\sum_{i=1}^{n} x_{i}\right)^{2}+\frac{12\left(\sum_{i=1}^{n} i x_{i}\right)^{2}}{n(n+1)(n+2)(3 n+1)}$$

Let $N \geqslant 3$ be an odd integer. $N$ tennis players take part in a league. Before the league starts, a committee ranks the players in some order based on perceived skill. During the league, each pair of players plays exactly one match, and each match has one winner. A match is considered an upset if the winner had a lower initial ranking than the loser. At the end of the league, the players are ranked according to number of wins, with the initial ranking used to rank players with the same number of wins. It turns out that the final ranking is the same as the initial ranking. What is the largest possible number of upsets? (United Kingdom)

Alice and Bob play the following game: They start with two non-empty piles of coins. Taking turns, with Alice playing first, they choose a pile with an even number of coins and move half of the coins of this pile to the other pile. The game ends if a player cannot move, or if we reach a previously reached position. In the first case, the player who cannot move loses. In the second case, the game is declared a draw. Determine all pairs $(a, b)$ of positive integers such that if initially the two piles have $a$ and $b$ coins respectively, then Bob has a winning strategy. (Cyprus)

An open necklace can contain rubies, emeralds and sapphires. At every step we can perform any of the following operations: $\left(1^{\circ}\right)$ We can replace two consecutive rubies with an emerald and a sapphire, where the emerald is on the left of the sapphire. $\left(2^{\circ}\right)$ We can replace three consecutive emeralds with a sapphire and a ruby, where the sapphire is on the left of the ruby. (3 ) If we find two consecutive sapphires then we can remove them. (4 ) If we find consecutively and in this order a ruby, an emerald, and a sapphire, then we can remove them.

Furthermore we can also reverse all of the above operations. For example, by reversing $\left(3^{\circ}\right)$ we can put two consecutive sapphires on any position we wish. Initially the necklace has one sapphire (and no other precious stones). Decide, with proof, whether there is a finite sequence of steps such that at the end of this sequence the necklace contains one emerald (and no other precious stones). Remark. A necklace is open if its precious stones are on a line from left to right. We are not allowed to move a precious stone from the rightmost position to the leftmost as we would be able to do if the necklace was closed. (Cyprus)

In an acute triangle $A B C$, the midpoint of the side $B C$ is $M$ and the centers of the excircles relative to $M$ of the triangles $A M B$ and $A M C$ are $D$ and $E$ respectively. The circumcircle of the triangle $A B D$ meets line $B C$ at $B$ and $F$. The circumcircle of the triangle $A C E$ meets line $B C$ at $C$ and $G$. Prove that $B F=C G$. (Romania)

Let $A B C$ be a triangle inscribed in circle $\Gamma$ with center $O$ and let $H$ its orthocenter and $K$ be the midpoint of $O H$. The tangent of $\Gamma$ at $B$ meets the perpendicular bisector of $A C$ meets at $L$ and the tangent of $\Gamma$ at $C$ meets the perpendicular bisector of $A B$ at $M$. Prove that $A K \perp L M$. (Greece)

Let $P$ be a point inside a triangle $A B C$ and let $a, b, c$ be the side lengths and $p$ the semi-perimeter of the triangle. Find the maximum value of

$$\min \left(\frac{P A}{p-a}, \frac{P B}{p-b}, \frac{P C}{p-c}\right)$$

over all possible choices of triangle $A B C$ and point $P$.

Let $A B C$ be an acute-angled triangle with $A B<A C<B C$ and let $D$ be an arbitrary point on the extension of $B C$ beyond $C$. The circle $\gamma(A, A D)$ intersects the rays $A C$, $A B, C B$ at points $E, F, G$, respectively. The circumcircle $\omega_{1}$ of triangle $A F G$ intersects the lines $F E, B C, G E, D F$ again at points $J, H, H^{\prime}, J^{\prime}$. The circumcircle $\omega_{2}$ of triangle $A D E$ intersects the lines $F E, B C, G E, D F$ again at points $I, K, K^{\prime}, I^{\prime}$. Prove that the quadrilaterals $H I J K$ and $H^{\prime} I^{\prime} J^{\prime} K^{\prime}$ are cyclic and that their circumcenters coincide. (Greece)

In a triangle $A B C$ with $A B=A C, \omega$ is the circumcircle and $O$ its center. Let $D$ be a point on the extension of $B A$ beyond $A$. The circumcircle $\omega_{1}$ of triangle $O A D$ intersects the line $A C$ and the circle $\omega$ again at points $E$ and $G$, respectively. Point $H$ is such that $D A E H$ is a parallelogram. Line $E H$ meets circle $\omega_{1}$ again at point $J$. The line through $G$ perpendicular to $G B$ meets $\omega_{1}$ again at point $N$ and the line through $G$ perpendicular to $G J$ meets $\omega$ again at point $L$. Prove that the points $L, N, H, G$ lie on a circle. (Cyprus)

For positive integers $m$ and $n$, let $d(m, n)$ be the number of distinct primes that divide both $m$ and $n$. For instance, $d(60,126)=d\left(2^{2} \times 3 \times 5,2 \times 3^{2} \times 7\right)=2$. Does there exist a sequence $\left(a_{n}\right)$ of positive integers such that: (i) $a_{1} \geqslant 2018^{2018}$; (ii) $a_{m} \leqslant a_{n}$ whenever $m \leqslant n$; (iii) $d(m, n)=d\left(a_{m}, a_{n}\right)$ for all positive integers $m \neq n$ ? (United Kingdom)

Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that

$$n!+f(m)!\mid f(n)!+f(m!)$$

for all $m, n \in \mathbb{N}$.

Find all primes $p$ and $q$ such that $3 p^{q-1}+1$ divides $11^{p}+17^{p}$.

Let $P(x)=a_{d} x^{d}+\cdots+a_{1} x+a_{0}$ be a non-constant polynomial with nonnegative integer coefficients having $d$ rational roots. Prove that

$$\operatorname{lcm}(P(m), P(m+1), \ldots, P(n)) \geqslant m\binom{n}{m}$$

for all positive integers $n>m$.

Let $x$ and $y$ be positive integers. If for each positive integer $n$ we have that

$$(n y)^{2}+1 \mid x^{\varphi(n)}-1$$

prove that $x=1$.