Title: IMO2022 Shortlisted Problems with Solutions Author: Dávid Kunszenti-Kovács, Alexander Betts, Márton Borbényi, James Cranch, Elisa Lorenzo García, Karl Erik Holter, Maria-Romina Ivan, Johannes Kleppe, Géza Kós, Dmitry Krachun, Charles Leytem, Sofia Lindqvist, Arnaud Maret, Waldemar Pompe, Paul Vaderlind

The Organising Committee and the Problem Selection Committee of IMO 2020 thank the following 39 countries for contributing 149 problem proposals: Armenia, Australia, Austria, Belgium, Brazil, Canada, Croatia, Cuba,

64th International Mathematical Olympiad Chiba, Japan, 2nd–13th July 2023 SHORTLISTED PROBLEMS WITH SOLUTIONS

Problems from the IMO Shortlists, by year: There was no IMO in 1980.

To the current moment, there is only a single IMO problem that has two distinct proposing countries: The if-part of problem 1994/2 was proposed by Australia and its only-if part by Armenia.

Turbo makes a series of attempts to go from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then repeatedly moves to an orthogonal neighbor. (He is allowed to return to a previously visited cell.)

Find all functions f : (0, ∞) → (0, ∞) such that. for all p, q, r, s > 0 with pq = rs. Solution. Let f satisfy the given condition. Setting p = q = hence f(1) = 1. Now take any x > 0 and set p = x, q = 1, r. xf(x)2 + x = x2f(x) + f(x), xf(x) − 1 f(x) − x = 0. then the condition of the problem is satisfied.

Oct 21, 2020 · IMO 2019 Shortlisted Problems (with solutions) 60 th International Mathematical Olympiad. Bath — UK, 11th–22nd July 2019

1.1 The Fiftieth IMO Bremen, Germany, July 10–22, 2009 1.1.1 Contest Problems First Day (July 15) 1. Let n be a positive integer and let a1, ..., ak (k ≥2) be distinct integers in the set {1,...,n} such that n divides ai(ai+1 −1) for i =1,...,k−1. Prove that n does not divide ak(a1 −1). 2. Let ABC be a triangle with circumcenter O.

This is a compilation of solutions for the 2019 IMO. The ideas of the solution are a mix of my own work, the solutions provided by the competition organizers, and solutions found by the community. However, all the writing is maintained by me. These notes will tend to be a bit more advanced and terse than the “oficial” solutions from the organizers.