ELMO 2022

Let $n > 1$ be an integer. The numbers $1, 2, \ldots, n$ are written on a board. Aliceurill and Bobasaur take turns circling an uncircled number on the board, with Aliceurill going first. When the product of the circled numbers becomes a multiple of $n$ , the game ends and the last player to have circled a number loses. For which values of $n$ can Bobasaur guarantee victory?

Max Lu

Find all monic nonconstant polynomials $P$ with integer coefficients for which there exist positive integers $a$ and $m$ such that for all positive integers $n\equiv a\pmod m$ , $P(n)$ is nonzero and $$2022\cdot\frac{(n+1)^{n+1} - n^n}{P(n)}$$ is an integer.

Jaedon Whyte, Luke Robitaille, and Pitchayut Saengrungkongka

Let $\mathcal{P}$ be a regular $2022$ -gon with area $1$ . Find a real number $c$ such that, if points $A$ and $B$ are chosen independently and uniformly at random on the perimeter of $\mathcal{P}$ , then the probability that $AB \geq c$ is $\tfrac{1}{2}$ .

Espen Slettnes

Let $ABCDE$ be a convex pentagon such that $\triangle ABE$ , $\triangle BEC$ , and $\triangle EDB$ are similar (with vertices in order). Lines $BE$ and $CD$ intersect at point $T$ . Prove that line $AT$ is tangent to the circumcircle of $\triangle ACD$ .

Holden Mui

Let $n\ge 3$ be a positive integer. There are $n^3$ users on a social media network called Everyone Likes Meeting Online (ELMO), and some pairs of these users are buddies. A set of at least three ELMO users forms an ELMOClub if and only if all pairs of members of the set are buddies. It is known that among every $n$ users, some three form an ELMOclub. Prove that there is an ELMOclub with five members.

Luke Robitaille

Find all functions $f\colon \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $m$ and $n$ , $$f(f(m)-n)+f(f(n)-m)=f(m+n).$$ Espen Slettnes and Luke Robitaille