International Mathematical Olympiad - IMO official
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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) 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
Shortlisted problems 5 C6. In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of flights. The distance between two cities is defined to be the least possible number of flights required to go from one of them to the other. It is known
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.
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
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= 2 - IMO official
Problem 1. Determine all real numbers α such that, for every positive integer n, the integer ⌊α⌋+⌊2α⌋+···+⌊nα⌋ is a multiple of n. (Note that ⌊z⌋ denotes the greatest integer less than or equal to z. For example, ⌊−π⌋ = −4 and ⌊2⌋ = ⌊2.9⌋ = 2.) Problem 2.
Shortlisted Problems with Solutions 55th International Mathematical Olympiad Cape Town, South Africa, 2014
64th International Mathematical Olympiad Chiba, Japan, 2nd–13th July 2023 SHORTLISTED PROBLEMS WITH SOLUTIONS