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. See also IMO problems statistics (eternal)

Let a1, a2, a3, . . . be an infinite sequence of positive integers, and let positive integer. Suppose that, for each number of times of the sequences. be a n > N, the number an is equal to the an 1 appears in the list (a1; a2; : : : ; an 1). Prove that at least one a1, a3, a5, . . . and a2, a4, a6, . . . is eventually periodic.

2024 IMO problems and solutions. The test will take place in July 2024 in Bath, United Kingdom. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

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,

Language versions of problems are not complete. Please send relevant PDF files to the webmaster: [email protected].

Mar 24, 2025 · This page contains problems and solutions to the International Math Olympiad and several USA contests, and a few others. Check the AoPS contest index for even more problems and solutions, including most of the ones below. International Math Olympiad# IMO 1997 ; IMO 1998 ; IMO 1999 ; IMO 2000 ; IMO 2001

Solution 1. We will determine all pairs of positive integers such that for all . First, works with . Now for any solution : \begin{lemma} or . \end{lemma} \begin{proof} Since divides both and , it divides their difference . Similarly, divides . Thus divides , so divides . Hence divides , a contradiction unless divides both and . \end{proof}

Problems–solutions 7 Solutions Day 1 Problem 1. Determineallrealnumbers αsuchthat,foreverypositiveinteger n,the integer tαu`t2αu`¨¨¨`tnαu is a multiple of n. (Note that tzu denotes the greatest integer less than or equal to z. For …

This repository is a collection of solutions for International Mathematical Olympiad (IMO) 2024 problems: Formalized solutions for problems 1, 2, and 6 from DeepMind-AlphaProof . Solutions in natural language for all problems from here .

solutions of all of the problems ever set in the IMO, together with many problems proposed for the contest. … serves as a vast repository of problems at the Olympiad level, useful both to students … and to faculty looking for hard elementary problems.