2024 Switzerland Team Selection Test

May 4, 2024 - Day 1

3

Determine all monic polynomial with integer coefficient P such that for every integer a,b there exists integer c so that P(a)P(b)=P(c)

May 5, 2024 - Day 2

5

Let a1,,an,b1,,bn be 2n positive integers such that the n+1 products a1a2a3an,b1a2a3an,b1b2a3an,,b1b2b3bnform a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

6

Let n be a positive integer. Paul has a 1\times n^2 rectangular strip consisting of n^2 unit squares, where the i^{\text{th}} square is labelled with i for all 1\leqslant i\leqslant n^2. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then translate (without rotating or flipping) the pieces to obtain an n\times n square satisfying the following property: if the unit square in the i^{\text{th}} row and j^{\text{th}} column is labelled with a_{ij}, then a_{ij}-(i+j-1) is divisible by n. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

May 18, 2024 - Day 3

7

Let m and n be positive integers greater than 1. In each unit square of an m\times n grid lies a coin with its tail side up. A move consists of the following steps. select a 2\times 2 square in the grid; flip the coins in the top-left and bottom-right unit squares; flip the coin in either the top-right or bottom-left unit square. Determine all pairs (m,n) for which it is possible that every coin shows head-side up after a finite number of moves. Thanasin Nampaisarn, Thailand

8

Let \mathbb R_{>0} be the set of positive real numbers. Determine all functions f \colon \mathbb R_{>0} \to \mathbb R_{>0} such that x \left(f(x) + f(y)\right) \geqslant \left(f(f(x)) + y\right) f(y)for every x, y \in \mathbb R_{>0}.

May 19, 2024 - Day 4

10

Let ABC be a triangle with AC > BC, let \omega be the circumcircle of \triangle ABC, and let r be its radius. Point P is chosen on \overline{AC} such taht BC=CP, and point S is the foot of the perpendicular from P to \overline{AB}. Ray BP mets \omega again at D. Point Q is chosen on line SP such that PQ = r and S,P,Q lie on a line in that order. Finally, let E be a point satisfying \overline{AE} \perp \overline{CQ} and \overline{BE} \perp \overline{DQ}. Prove that E lies on \omega.

12

Determine all functions f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0} such that, for all positive integers a and b, f^{bf(a)}(a+1)=(a+1)f(b).