International Mathematical Olympiad · 2017

Problems

Problem 1For each integer $a_0>1$ , define the sequence $a_0,a_1,a_2,\ldots$ for $n\ge 0$ by $a_{n+1}=\begin{cases}\sqrt{a_n} & \text{if }\sqrt{a_n}\text{ is an integer},\\ a_n+3 & \text{otherwise}.\end{cases}$ Determine all values of $a_0$ for which there exists a number $A$ such that $a_n=A$ for infinitely many values of $n$ .Solutions: 1
Problem 2Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that, for all real numbers $x$ and $y$ , $f(f(x)f(y)) + f(x+y) = f(xy).$ Solutions: 1
Problem 3A 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 $A_{n-1}$ and the hunter is at $B_{n-1}$ . In the $n$ th round, three things occur in order: (i) the rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1$ ; (ii) a tracking device reports a point $P_n$ to the hunter, with the only guarantee that the distance between $P_n$ and $A_n$ is at most $1$ ; (iii) the hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly $1$ . 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 $100$ ?Solutions: 1
Problem 4Let $R$ 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 $RT$ . 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 line $KT$ is tangent to $\Gamma$ .Solutions: 1
Problem 5An 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: no one stands between the two tallest players, no one stands between the third and fourth tallest players, $\ldots$ , no one stands between the two shortest players. Show that this is always possible.Solutions: 1
Problem 6An ordered pair $(x,y)$ of integers is called a primitive point if $\gcd(x,y)=1$ . 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 every $(x,y)\in S$ , we have $a_0x^n+a_1x^{n-1}y+a_2x^{n-2}y^2+\cdots+a_{n-1}xy^{n-1}+a_ny^n=1$ .Solutions: 1