# 2022 IMO Problems

## Problem 1

The Bank of Oslo issues two types of coin: aluminium (denoted A) and bronze (denoted B). Marianne has aluminium coins and bronze coins, arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer , Marianne repeatedly performs the following operation: she identifies the longest chain containing the coin from the left, and moves all coins in that chain to the left end of the row. For example, if and , the process starting from the ordering AABBBABA would be

AABBBABA → BBBAAABA → AAABBBBA → BBBBAAAA → BBBBAAAA → ...

Find all pairs with such that for every initial ordering, at some moment during the process, the leftmost coins will all be of the same type.

## Problem 2

Let denote the set of positive real numbers. Find all functions such that for each , there is exactly one satisfying

.

## Problem 3

Let be a positive integer and let be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of around a circle such that the product of any two neighbours is of the form for some positive integer .

## Problem 4

Let be a convex pentagon such that . Assume that there is a point inside with , and . Let line intersect lines and at points and , respectively. Assume that the points occur on their line in that order. Let line intersect lines and at points and , respectively. Assume that the points occur on their line in that order. Prove that the points lie on a circle.

## Problem 5

Find all triples of positive integers with prime and

## Problem 6

Let be a positive integer. A Nordic square is an board containing all the integers from to so that each cell contains exactly one number. Two different cells are considered adjacent if they share an edge. Every cell that is adjacent only to cells containing larger numbers is called a valley. An uphill path is a sequence of one or more cells such that:

(i) the first cell in the sequence is a valley,

(ii) each subsequent cell in the sequence is adjacent to the previous cell, and

(iii) the numbers written in the cells in the sequence are in increasing order.

Find, as a function of , the smallest possible total number of uphill paths in a Nordic square.