ELMO 2020

Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that $$f^{f^{f(x)}(y)}(z)=x+y+z+1$$ for all $x,y,z \in \mathbb{N}$ .

Proposed by William Wang.

Define the Fibonacci numbers by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n\geq 3$ . Let $k$ be a positive integer. Suppose that for every positive integer $m$ there exists a positive integer $n$ such that $m \mid F_n-k$ . Must $k$ be a Fibonacci number?

Proposed by Fedir Yudin.

Janabel has a device that, when given two distinct points $U$ and $V$ in the plane, draws the perpendicular bisector of $UV$ . Show that if three lines forming a triangle are drawn, Janabel can mark the orthocenter of the triangle using this device, a pencil, and no other tools.

Proposed by Fedir Yudin.

Let acute scalene triangle $ABC$ have orthocenter $H$ and altitude $AD$ with $D$ on side $BC$ . Let $M$ be the midpoint of side $BC$ , and let $D'$ be the reflection of $D$ over $M$ . Let $P$ be a point on line $D'H$ such that lines $AP$ and $BC$ are parallel, and let the circumcircles of $\triangle AHP$ and $\triangle BHC$ meet again at $G \neq H$ . Prove that $\angle MHG = 90^\circ$ .

Proposed by Daniel Hu.

Let $m$ and $n$ be positive integers. Find the smallest positive integer $s$ for which there exists an $m \times n$ rectangular array of positive integers such that

• each row contains $n$ distinct consecutive integers in some order,
• each column contains $m$ distinct consecutive integers in some order, and
• each entry is less than or equal to $s$ .

Proposed by Ankan Bhattacharya.

For any positive integer $n$ , let

• $\tau(n)$ denote the number of positive integer divisors of $n$ ,
• $\sigma(n)$ denote the sum of the positive integer divisors of $n$ , and
• $\varphi(n)$ denote the number of positive integers less than or equal to $n$ that are relatively prime to $n$ .

Let $a,b > 1$ be integers. Brandon has a calculator with three buttons that replace the integer $n$ currently displayed with $\tau(n)$ , $\sigma(n)$ , or $\varphi(n)$ , respectively. Prove that if the calculator currently displays $a$ , then Brandon can make the calculator display $b$ after a finite (possibly empty) sequence of button presses.

Proposed by Jaedon Whyte.