International Mathematical Olympiad · 2022

Problems

Problem 1The Bank of Oslo issues two types of coin: aluminium (denoted $A$ ) and bronze (denoted $B$ ). Marianne has $n$ aluminium coins and $n$ 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 $k\le2n$ , Marianne repeatedly performs the following operation: she identifies the longest chain containing the $k$ -th coin from the left, and moves all coins in that chain to the left end of the row. Find all pairs $(n,k)$ with $1\le k\le2n$ such that for every initial ordering, at some moment during the process, the leftmost $n$ coins will all be of the same type.Solutions: 1
Problem 2Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for each $x\in\mathbb{R}^+$ , there is exactly one $y\in\mathbb{R}^+$ satisfying $xf(y)+yf(x)\le2$ Solutions: 1
Problem 3Let $k$ be a positive integer and let $S$ 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 $S$ around a circle such that the product of any two neighbours is of the form $x^2+x+k$ for some positive integer $x$ .Solutions: 1
Problem 4Let $ABCDE$ be a convex pentagon such that $BC=DE$ . Assume that there is a point $T$ inside $ABCDE$ with $TB=TD$ , $TC=TE$ and $\angle ABT=\angle TEA$ . Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$ , respectively, with $P,B,A,Q$ in that order on the line. Let line $AE$ intersect lines $CD$ and $DT$ at points $R$ and $S$ , respectively, with $R,E,A,S$ in that order on the line. Prove that the points $P$ , $S$ , $Q$ , $R$ lie on a circle.Solutions: 1
Problem 5Find all triples $(a,b,p)$ of positive integers with $p$ prime and $a^p=b!+p$ Solutions: 1
Problem 6Let $n$ be a positive integer. A Nordic square is an $n\times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. An uphill path is a sequence of one or more cells such that: (a) the first cell in the sequence is a valley, meaning the number written is less than all its orthogonal neighbours; (b) each subsequent cell in the sequence is orthogonally adjacent to the previous cell; and (c) the numbers written in the cells in the sequence are in increasing order. Find, as a function of $n$ , the smallest possible total number of uphill paths in a Nordic square.Solutions: 1