# IMO

## IMO Problems and Solutions 2017

Problem 1. For each integer a0 > 1, define the sequence a0, a1, a2, … … by: Determine all values of a0 for which there is a number A such that an = A for infinitely many values of n. Problem 2. Let ℝ be the set of real numbers. Determine all functions f : …

## IMO Problems and Solutions 2018

Problem 1. Let Γ be the circumcircle of acute-angled triangle ABC. Points D and E lie on segments AB and AC, respectively, such that AD = AE. The perpendicular bisectors of BD and CE intersect the minor arcs AB and AC of Γ at points F and G, respectively. Prove that the lines DE and …

## IMO Problems and Solutions 2019

Problem 1. Let ℤ be the set of integers. Determine all functions f : ℤ → ℤ such that, for all integers a and b, f(2a) + 2f(b) = f(f(a + b)). Problem 2. In triangle ABC, point A1 lies on side BC and point B1 lies on side AC. Let P and Q be …

## IMO Problems and Solutions 2020

Problem 1. Consider the convex quadrilateral ABCD. The point P is in the interior of ABCD. The following ratio equalities hold: ∠PAD : ∠PBA : ∠DPA = 1 : 2 : 3 = ∠CBP : ∠BAP : ∠BPC. Prove that the following three lines meet in a point: the internal bisectors of angles ∠ADP and …

## IMO Problems and Solutions 1959

Problem 1: Prove that the fraction  is irreducible for every natural number n:     Problem 2: For what real values of x is given (a) A = √2, (b) A = 1, (c) A = 2; where only non-negative real numbers are admitted for square roots? Problem 3: Let a, b, c be real …