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Problems Algebra A1. Version 1. Let nbe a positive integer, and set N“ 2n. Determine the smallest real number an such that, for all real x, N c x2N `1 2 ď anpx´1q2 `x. Version 2. For every positive integer N, determine the smallest real number bN such that, for all real x, N c x2N `1 2 ď bNpx´1q2 `x. (Ireland) A2.

Problems (with solutions) 59th International Mathematical Olympiad Cluj-Napoca — Romania, 3–14 July 2018

Problems (with solutions) Confidential until 1:30pm on 12 July 2022 (Norwegian time) 62nd International Mathematical Olympiad Saint-Petersburg — Russia, 16th–24th July 2021

Problems (with solutions) 60th International Mathematical Olympiad Bath — UK, 11th–22nd July 2019

Problem proposals for the 63rd International Mathematical Olympad 2022, Oslo, Norway Keywords IMO, International Mathematical Olympiad, problem, solution, shortlist, mathematics, algebra, combinatorics, geometry, number theory

The International Mathematical Olympiad (IMO) is the World Championship Mathematics Competition for High School students and is held annually in a different country. The first IMO was held in 1959 in Romania, with 7 countries participating.

4 IMO 2016 Hong Kong A6. The equation (x 1)(x 2) (x 2016) = (x 1)(x 2) (x 2016) is written on the board. One tries to erase some linear factors from both sides so that each side still has at least one factor, and the resulting equation has no real roots. Find the least number of linear factors one needs to erase to achieve this. A7.

Shortlisted Problems with Solutions 55th International Mathematical Olympiad Cape Town, South Africa, 2014

52nd International Mathematical Olympiad 12-24 July 2011 Amsterdam The Netherlands Problem shortlist with solutions