BALTIC-WAY 2021

Let $n$ be a positive integer. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the equation

$$(f(x))^{n} f(x+y)=(f(x))^{n+1}+x^{n} f(y)$$

for all $x, y \in \mathbb{R}$.

Let $a, b, c$ be the side lengths of a triangle. Prove that

$$\sqrt[3]{\left(a^{2}+b c\right)\left(b^{2}+c a\right)\left(c^{2}+a b\right)}>\frac{a^{2}+b^{2}+c^{2}}{2} .$$

Determine all infinite sequences $\left(a_{1}, a_{2}, \ldots\right)$ of positive integers satisfying

$$a_{n+1}^{2}=1+(n+2021) a_{n}$$

for all $n \geq 1$.

BALTIC WAY

REYKJAVÍK$\cdot$ 2021

Let $\Gamma$ be a circle in the plane and $S$ be a point on $\Gamma$. Mario and Luigi drive around the circle $\Gamma$ with their go-karts. They both start at $S$ at the same time. They both drive for exactly 6 minutes at constant speed counterclockwise around the track. During these 6 minutes, Luigi makes exactly one lap around $\Gamma$ while Mario, who is three times as fast, makes three laps.

While Mario and Luigi drive their go-karts, Princess Daisy positions herself such that she is always exactly in the middle of the chord between them. When she reaches a point she has already visited, she marks it with a banana.

How many points in the plane, apart from $S$, are marked with a banana by the end of the race?

Let $x, y \in \mathbb{R}$ be such that $x=y(3-y)^{2}$ and $y=x(3-x)^{2}$. Find all possible values of $x+y$.

Let $n$ be a positive integer and $t$ be a non-zero real number. Let $a_{1}, a_{2}, \ldots, a_{2 n-1}$ be real numbers (not necessarily distinct). Prove that there exist distinct indices $i_{1}, i_{2}, \ldots, i_{n}$ such that, for all $1 \leq k, l \leq n$, we have $a_{i_{k}}-a_{i_{l}} \neq t$.

Let $n>2$ be an integer. Anna, Edda and Magni play a game on a hexagonal board tiled with regular hexagons, with $n$ tiles on each side. The figure shows a board with 5 tiles on each side. The central tile is marked.

We are given a collection of $2^{2^{k}}$ coins, where $k$ is a non-negative integer. Exactly one coin is fake. We have an unlimited number of service dogs. One dog is sick but we do not know which one. A test consists of three steps: select some coins from the collection of all coins; choose a service dog; the dog smells all of the selected coins at once. A healthy dog will bark if and only if the fake coin is amongst them. Whether the sick dog will bark or not is random.

Devise a strategy to find the fake coin, using at most $2^{k}+k+2$ tests, and prove that it works.

We are given 2021 points on a plane, no three of which are collinear. Among any 5 of these points, at least 4 lie on the same circle. Is it necessarily true that at least 2020 of the points lie on the same circle?

John has a string of paper where $n$ real numbers $a_{i} \in[0,1]$, for all $i \in\{1, \ldots, n\}$, are written in a row. Show that for any given $k<n$, he can cut the string of paper into $k$ non-empty pieces, between adjacent numbers, in such a way that the sum of the numbers on each piece does not differ from any other sum by more than 1 .

A point $P$ lies inside a triangle $A B C$. The points $K$ and $L$ are the projections of $P$ onto $A B$ and $A C$, respectively. The point $M$ lies on the line $B C$ so that $K M=L M$, and the point $P^{\prime}$ is symmetric to $P$ with respect to $M$. Prove that $\angle B A P=\angle P^{\prime} A C$.

Let $I$ be the incentre of a triangle $A B C$. Let $F$ and $G$ be the projections of $A$ onto the lines $B I$ and $C I$, respectively. Rays $A F$ and $A G$ intersect the circumcircles of the triangles $C F I$ and $B G I$ for the second time at points $K$ and $L$, respectively. Prove that the line $A I$ bisects the segment $K L$.

Let $D$ be the foot of the $A$-altitude of an acute triangle $A B C$. The internal bisector of the angle $D A C$ intersects $B C$ at $K$. Let $L$ be the projection of $K$ onto $A C$. Let $M$ be the intersection point of $B L$ and $A D$. Let $P$ be the intersection point of $M C$ and $D L$. Prove that $P K \perp A B$.

Let $A B C$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. Denote by $M$ the midpoint of $B C$. The point $D$ is the reflection of $A$ over $B C$, and the point $E$ is the intersection of $\Gamma$ and the ray $M D$. Let $S$ be the circumcentre of the triangle $A D E$. Prove that the points $A, E, M, O$, and $S$ lie on the same circle.

For which positive integers $n \geq 4$ does there exist a convex $n$-gon with side lengths $1,2, \ldots, n$ (in some order) and with all of its sides tangent to the same circle?

Show that no non-zero integers $a, b, x, y$ satisfy

$$\left\{\begin{array}{l} a x-b y=16, \\ a y+b x=1 . \end{array}\right.$$

Distinct positive integers $a, b, c, d$ satisfy

$$\left\{\begin{array}{l} a \mid b^{2}+c^{2}+d^{2}, \\ b \mid a^{2}+c^{2}+d^{2}, \\ c \mid a^{2}+b^{2}+d^{2}, \\ d \mid a^{2}+b^{2}+c^{2}, \end{array}\right.$$

and none of them is larger than the product of the three others. What is the largest possible number of primes among them?

Find all integer triples $(a, b, c)$ satisfying the equation

$$5 a^{2}+9 b^{2}=13 c^{2}$$

Find all polynomials $p$ with integer coefficients such that the number $p(a)-p(b)$ is divisible by $a+b$ for all integers $a, b$, provided that $a+b \neq 0$.

Let $n \geq 2$ be an integer. Given numbers $a_{1}, a_{2}, \ldots, a_{n} \in\{1,2,3, \ldots, 2 n\}$ such that $\operatorname{lcm}\left(a_{i}, a_{j}\right)>2 n$ for all $1 \leq i<j \leq n$, prove that

$$a_{1} a_{2} \cdots a_{n} \mid(n+1)(n+2) \cdots(2 n-1)(2 n) .$$