IMO 2017

For each integer $a_0 > 1$ , define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} a_n + 3 & \text{otherwise.} \end{cases}$$ Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$ .

Proposed by Stephan Wagner, South Africa

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for any real numbers $x$ and $y$ , $$f(f(x)f(y)) + f(x+y) = f(xy).$$

Proposed by Dorlir Ahmeti, Albania

A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, $A_0,$ and the hunter's starting point, $B_0$ are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}.$ In the $n^{\text{th}}$ round of the game, three things occur in order:

[list=i] []The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1.$ []A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most $1.$ [*]The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly $1.$ [/list] Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds, she can ensure that the distance between her and the rabbit is at most $100?$ Proposed by Gerhard Woeginger, Austria

Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$ . Point $T$ is such that $S$ is the midpoint of the line segment $RT$ . Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$ . Line $AJ$ meets $\Omega$ again at $K$ . Prove that the line $KT$ is tangent to $\Gamma$ .

Proposed by Charles Leytem, Luxembourg

An integer $N \ge 2$ is given. A collection of $N(N + 1)$ soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove $N(N - 1)$ players from this row leaving a new row of $2N$ players in which the following $N$ conditions hold: ( $1$ ) no one stands between the two tallest players, ( $2$ ) no one stands between the third and fourth tallest players, $\;\;\vdots$ ( $N$ ) no one stands between the two shortest players.

Show that this is always possible.

Proposed by Grigory Chelnokov, Russia

An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$ . Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$ , we have: $$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$$ Proposed by John Berman, United States