BALKAN-SL 2020

Given are positive reals $a, b, c$ , such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$ . Prove that $\frac{\sqrt{a+\frac{b}{c}}+\sqrt{b+\frac{c}{a}}+\sqrt{c+\frac{a}{b}}}{3}\leq \frac{a+b+c-1}{\sqrt{2}}$ .

Albania

Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$ .

Demetres Christofides, Cyprus

Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$ , $\rho_2$ , $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$ . Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$ , $C(\rho_2, 0)$ , $D(\rho_3, 0)$ . If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$ , find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained.

Brazitikos Silouanos, Greece

Let $s \geq 2$ and $n \geq k \geq 2$ be integes, and let $A$ be a subset of $\{1, 2, . . . , n\}^k$ of size at least $2sk^2n^{k-2}$ such that any two members of $A$ share some entry. Prove that there are an integer $p \leq k$ and $s+2$ members $A_1, A_2, . . . , A_{s+2}$ of $A$ such that $A_i$ and $A_j$ share the $p$ -th entry alone, whenever $i$ and $j$ are distinct.

Miroslav Marinov, Bulgaria

Odin and Evelyn are playing a game, Odin going first. There are initially $3k$ empty boxes, for some given positive integer $k$ . On each player's turn, they can write a non-negative integer in an empty box, or erase a number in a box and replace it with a strictly smaller non-negative integer. However, Odin is only ever allowed to write odd numbers, and Evelyn is only allowed to write even numbers. The game ends when either one of the players cannot move, in which case the other player wins; or there are exactly $k$ boxes with the number $0$ , in which case Evelyn wins if all other boxes contain the number $1$ , and Odin wins otherwise. Who has a winning strategy? $Agnijo \ Banerjee \ , United \ Kingdom$

A strategical video game consists of a map of finitely many towns. In each town there are $k$ directions, labelled from $1$ through $k$ . One of the towns is designated as initial, and one - as terminal. Starting from the initial town the hero of the game makes a finite sequence of moves. At each move the hero selects a direction from the current town. This determines the next town he visits and a certain positive amount of points he receives. Two strategical video games are equivalent if for every sequence of directions the hero can reach the terminal town from the initial in one game, he can do so in the other game, and, in addition, he accumulates the same amount of points in both games. For his birthday John receives two strategical video games - one with $N$ towns and one with $M$ towns. He claims they are equivalent. Marry is convinced they are not. Marry is right. Prove that she can provide a sequence of at most $N +M$ directions that shows the two games are indeed not equivalent.

Stefan Gerdjikov, Bulgaria

Let $G, H$ be the centroid and orthocentre of $\vartriangle ABC$ which has an obtuse angle at $\angle B$ . Let $\omega$ be the circle with diameter $AG$ . $\omega$ intersects  $\odot(ABC)$ again at $L \ne A$ . The tangent to $\omega$ at $L$ intersects  $\odot(ABC)$ at $K \ne L$ . Given that $AG = GH$ , prove $\angle HKG = 90^o$ . Sam Bealing, United Kingdom

Let $ABC$ be a triangle. On the sides $BC$ , $CA$ , $AB$ of the triangle, construct outwardly three squares with centres $O_a$ , $O_b$ , $O_c$ respectively. Let $\omega$ be the circumcircle of $\vartriangle O_aO_bO_c$ . Given that $A$ lies on $\omega$ , prove that the centre of $\omega$ lies on the perimeter of $\vartriangle ABC$ .

Sam Bealing, United Kingdom

Let $MAZN$ be an isosceles trapezium inscribed in a circle $(c)$ with centre $O$ . Assume that $MN$ is a diameter of $(c)$ and let $B$ be the midpoint of $AZ$ . Let $(\epsilon)$ be the perpendicular line on $AZ$ passing through $A$ . Let $C$ be a point on $(\epsilon)$ , let $E$ be the point of intersection of $CB$ with $(c)$ and assume that $AE$ is perpendicular to $CB$ . Let $D$ be the point of intersection of $CZ$ with $(c)$ and let $F$ be the antidiametric point of $D$ on $(c)$ . Let $P$ be the point of intersection of $FE$ and $CZ$ . Assume that the tangents of $(c)$ at the points $M$ and $Z$ meet the lines $AZ$ and $PA$ at the points $K$ and $T$ respectively. Prove that $OK$ is perpendicular to $TM$ .

Theoklitos Parayiou, Cyprus

Let $ABC$ be an isosceles triangle with $AB = AC$ and $\angle A = 45^o$ . Its circumcircle $(c)$ has center $O, M$ is the midpoint of $BC$ and $D$ is the foot of the perpendicular from $C$ to $AB$ . With center $C$ and radius $CD$ we draw a circle which internally intersects $AC$ at the point $F$ and the circle $(c)$ at the points $Z$ and $E$ , such that $Z$ lies on the small arc $BC$ and $E$ on the small arc $AC$ . Prove that the lines $ZE$ , $CO$ , $FM$ are concurrent.

Brazitikos Silouanos, Greece

Determine all positive integers $n$ such that $\frac{a^2+n^2}{b^2-n^2}$ is a positive integer for some $a,b\in \mathbb{N}$ . $Turkey$

A number of $N$ children are at a party and they sit in a circle to play a game of Pass and Parcel. Because the host has no other form of entertainment, the parcel has infinitely many layers. On turn $i$ , starting with $i=1$ , the following two things happen in order:$(1)$ The parcel is passed $i^2$ positions clockwise; and**$(2)$** The child currently holding the parcel unwraps a layer and claims the prize inside. For what values of $N$ will every chidren receive a prize?

$Patrick \ Winter United \ Kingdom$

Given an integer $k\geq 2$ , determine all functions $f$ from the positive integers into themselves such that $f(x_1)!+f(x_2)!+\cdots f(x_k)!$ is divisibe by $x_1!+x_2!+\cdots x_k!$ for all positive integers $x_1,x_2,\cdots x_k$ . $Albania$

Consider an integer $n\geq 2$ and an odd prime $p$ . Let $U$ be the set of all positive integers $($ strictly $)$ less than $p^n$ that are not divisible by $p$ , and let $N$ be the number of elements of $U$ . Does there exist permutation $a_1,a_2,\cdots a_N$ of the numbers in $U$ such that the sum $\sum_{k=1}^N a_ka_{k+1}$ ,where $a_{N+1}=a_1$ , be divisible by $p^{n-1}$ but not by $p^n$ ? $Alexander \ Ivanov Bulgaria$