International Competitions 2022

The 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 $k \leq 2n$, Marianne repeatedly performs the following operation: she identifies the longest chain containing the $k^{\text{th}}$ coin from the left, and moves all coins in that chain to the left end of the row. For example, if $n = 4$ and $k = 4$, the process starting from the ordering $AABBBABA$ would be

$$AAB\underline{B}BABA \rightarrow BBB\underline{A}AABA \rightarrow AAA\underline{B}BBBA \rightarrow BBB\underline{B}AAAA \rightarrow BBB\underline{B}AAAA \rightarrow \cdots.$$

Find all pairs $(n,k)$ with $1 \leq k \leq 2n$ such that for every initial ordering, at some moment during the process, the leftmost $n$ coins will all be of the same type.

Let $\mathbb{R}^+$ 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

$$xf(y) + yf(x) \leq 2.$$

Let $k$ 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 $x$.

Let $ABCDE$ 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. Assume that the points $P, B, A, Q$ occur on their line in that order. Let line $AE$ intersect lines $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R, E, A, S$ occur on their line in that order. Prove that the points $P, S, Q, R$ lie on a circle.

Find all triples $(a, b, p)$ of positive integers with $p$ prime and

$$a^p = b! + p.$$

Let $n$ 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. Two different cells are considered adjacent if they share a common side. 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 $n$, the smallest possible total number of uphill paths in a Nordic square.

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that

$$f (x + f (x + y)) = x + f (f (x) + y)$$

holds for all $x, y \in \mathbb{R}$.

Let $n$ be a positive integer. Anna and Beatrice play a game with a deck of $n$ cards labelled with the numbers $1, 2, \ldots, n$. Initially, the deck is shuffled. The players take turns, starting with Anna. At each turn, if $k$ denotes the number written on the topmost card, then the player first looks at all the cards and then rearranges the $k$ topmost cards. If, after rearranging, the topmost card shows the number $k$ again, then the player has lost and the game ends. Otherwise, the turn of the other player begins. Determine, depending on the initial shuffle, if either player has a winning strategy, and if so, who does.

Let $ABCD$ be a parallelogram with $\angle DAB < 90^{\circ}$. Let $E \neq B$ be the point on the line $BC$ such that $AE = AB$ and let $F \neq D$ be the point on the line $CD$ such that $AF = AD$. The circumcircle of the triangle $CEF$ intersects the line $AE$ again in $P$ and the line $AF$ again in $Q$. Let $X$ be the reflection of $P$ over the line $DE$ and $Y$ the reflection of $Q$ over the line $BF$. Prove that $A, X$ and $Y$ lie on the same line.

Initially, two positive integers $a$ and $b$ with $a \neq b$ are written on a blackboard. At each step, Andrea picks two numbers $x$ and $y$ on the blackboard with $x \neq y$ and writes the number

$$\gcd(x, y) + \operatorname{lcm}(x, y)$$

on the blackboard as well. Let $n$ be a positive integer. Prove that, regardless of the values of $a$ and $b$, Andrea can perform a finite number of steps such that a multiple of $n$ appears on the blackboard.

Remark. If $x$ and $y$ are two positive integers, then $\gcd(x, y)$ denotes their greatest common divisor and $\operatorname{lcm}(x, y)$ their least common multiple.

Given a pair $(a_0, b_0)$ of real numbers, we define two sequences $a_0, a_1, a_2, \ldots$ and $b_0, b_1, b_2, \ldots$ of real numbers by $$a_{n+1} = a_n + b_n \quad \text{and} \quad b_{n+1} = a_n \cdot b_n$$ for all $n = 0, 1, 2, \ldots$. Find all pairs $(a_0, b_0)$ of real numbers such that $a_{2022} = a_0$ and $b_{2022} = b_0$.

Let $k$ be a positive integer and $a_1, a_2, \ldots, a_k$ be nonnegative real numbers. Initially, there is a sequence of $n \geq k$ zeros written on a blackboard. At each step, Nicole chooses $k$ consecutive numbers written on the blackboard and increases the first number by $a_1$, the second one by $a_2$, and so on, until she increases the $k$-th one by $a_k$. After a positive number of steps, Nicole managed to make all the numbers on the blackboard equal. Prove that all the nonzero numbers among $a_1, a_2, \ldots, a_k$ are equal.

Let $n$ be a positive integer. There are $n$ purple and $n$ white cows queuing in a line in some order. Tim wishes to sort the cows by colour, such that all purple cows are at the front of the line. At each step, he is only allowed to swap two adjacent groups of equally many consecutive cows. What is the minimal number of steps Tim needs to be able to fulfill his wish, regardless of the initial alignment of the cows?

Example. For instance, Tim can perform the following three swaps: $$W\underline{PW}\overline{PP}W \longrightarrow \underline{W}\overline{P}PPWW \longrightarrow P\underline{WP}\overline{PW}W \longrightarrow PPWWPW.$$

Let $n$ be a positive integer. We are given a $2n \times 2n$ table. Each cell is coloured with one of $2n^2$ colours such that each colour is used exactly twice. Jana stands in one of the cells. There is a chocolate bar lying in one of the other cells. Jana wishes to reach the cell with the chocolate bar. At each step, she can only move in one of the following two ways. Either she walks to an adjacent cell or she teleports to the other cell with the same colour as her current cell. (Jana can move to an adjacent cell of the same colour by either walking or teleporting.) Determine whether Jana can fulfill her wish, regardless of the initial configuration, if she has to alternate between the two ways of moving and has to start with a teleportation.

Remark. Two cells are adjacent if they share a common edge.

Let $\Omega$ be the circumcircle of a triangle $ABC$ with $\angle CAB = 90^{\circ}$. The medians through $B$ and $C$ meet $\Omega$ again at $D$ and $E$, respectively. The tangent to $\Omega$ at $D$ intersects the line $AC$ at $X$ and the tangent to $\Omega$ at $E$ intersects the line $AB$ at $Y$. Prove that the line $XY$ is tangent to $\Omega$.

Let $ABCD$ be a convex quadrilateral such that $AC = BD$ and the sides $AB$ and $CD$ are not parallel. Let $P$ be the intersection point of the diagonals $AC$ and $BD$. Points $E$ and $F$ lie, respectively, on segments $BP$ and $AP$ such that $PC = PE$ and $PD = PF$. Prove that the circumcircle of the triangle determined by the lines $AB$, $CD$ and $EF$ is tangent to the circumcircle of the triangle $ABP$.

Let $\mathbb{N}$ denote the set of positive integers. Determine all functions $f\colon \mathbb{N}\to \mathbb{N}$ such that $f(1)\leq f(2)\leq f(3)\leq \ldots$ and the numbers $f(n) + n + 1$ and $f(f(n)) - f(n)$ are both perfect squares for every positive integer $n$.

We call a positive integer cheesy if we can obtain the average of the digits in its decimal representation by putting a decimal separator after the leftmost digit. Prove that there are only finitely many cheesy numbers.

Example. For instance, 2250 is cheesy, as the average of the digits is 2.250.