RMM 2020

Let $ABC$ be a triangle with a right angle at $C$ . Let $I$ be the incentre of triangle $ABC$ , and let $D$ be the foot of the altitude from $C$ to $AB$ . The incircle $\omega$ of triangle $ABC$ is tangent to sides $BC$ , $CA$ , and $AB$ at $A_1$ , $B_1$ , and $C_1$ , respectively. Let $E$ and $F$ be the reflections of $C$ in lines $C_1A_1$ and $C_1B_1$ , respectively. Let $K$ and $L$ be the reflections of $D$ in lines $C_1A_1$ and $C_1B_1$ , respectively.

Prove that the circumcircles of triangles $A_1EI$ , $B_1FI$ , and $C_1KL$ have a common point.

Let $N \geq 2$ be an integer, and let $\mathbf a$ $= (a_1, \ldots, a_N)$ and $\mathbf b$ $= (b_1, \ldots b_N)$ be sequences of non-negative integers. For each integer $i \not \in \{1, \ldots, N\}$ , let $a_i = a_k$ and $b_i = b_k$ , where $k \in \{1, \ldots, N\}$ is the integer such that $i-k$ is divisible by $n$ . We say $\mathbf a$ is $\mathbf b$ -harmonic if each $a_i$ equals the following arithmetic mean: $$a_i = \frac{1}{2b_i+1} \sum_{s=-b_i}^{b_i} a_{i+s}.$$ Suppose that neither $\mathbf a$ nor $\mathbf b$ is a constant sequence, and that both $\mathbf a$ is $\mathbf b$ -harmonic and $\mathbf b$ is $\mathbf a$ -harmonic.

Prove that at least $N+1$ of the numbers $a_1, \ldots, a_N,b_1, \ldots, b_N$ are zero.

Let $n\ge 3$ be an integer. In a country there are $n$ airports and $n$ airlines operating two-way flights. For each airline, there is an odd integer $m\ge 3$ , and $m$ distinct airports $c_1, \dots, c_m$ , where the flights offered by the airline are exactly those between the following pairs of airports: $c_1$ and $c_2$ ; $c_2$ and $c_3$ ; $\dots$ ; $c_{m-1}$ and $c_m$ ; $c_m$ and $c_1$ .

Prove that there is a closed route consisting of an odd number of flights where no two flights are operated by the same airline.

Let $\mathbb N$ be the set of all positive integers. A subset $A$ of $\mathbb N$ is sum-free if, whenever $x$ and $y$ are (not necessarily distinct) members of $A$ , their sum $x+y$ does not belong to $A$ . Determine all surjective functions $f:\mathbb N\to\mathbb N$ such that, for each sum-free subset $A$ of $\mathbb N$ , the image $\{f(a):a\in A\}$ is also sum-free.

Note: a function $f:\mathbb N\to\mathbb N$ is surjective if, for every positive integer $n$ , there exists a positive integer $m$ such that $f(m)=n$ .

A lattice point in the Cartesian plane is a point whose coordinates are both integers. A lattice polygon is a polygon all of whose vertices are lattice points.

Let $\Gamma$ be a convex lattice polygon. Prove that $\Gamma$ is contained in a convex lattice polygon $\Omega$ such that the vertices of $\Gamma$ all lie on the boundary of $\Omega$ , and exactly one vertex of $\Omega$ is not a vertex of $\Gamma$ .

For each integer $n \geq 2$ , let $F(n)$ denote the greatest prime factor of $n$ . A strange pair is a pair of distinct primes $p$ and $q$ such that there is no integer $n \geq 2$ for which $F(n)F(n+1)=pq$ .

Prove that there exist infinitely many strange pairs.