# Prosenjit Sarkar ## EGMO Problems and Solutions 2018

Problem 1: Let ABC be a triangle with CA = CB and ∠ACB = 120◦, and let, M be the midpoint of AB. Let P be a variable point on the circumcircle of ABC, and let Q be the point on the segment CP such that QP = 2QC. It is given that the line …

## EGMO Problems and Solutions 2020

Problem 1. The positive integers a0, a1, a2, …, a3030 satisfy 2an+2 = an+1 + 4an for n = 0, 1, 2, …, 3028. Prove that at least one of the numbers a0, a1, a2, …, a3030 is divisible by 22020. Problem 2. Find all lists (x1, x2, …, x2020) of non-negative real numbers such …

## EGMO Problems and Solutions 2021

Problem 1: The number 2021 is fantabulous. For any positive integer m, if any element of the set {m, 2m + 1, 3m} is fantabulous, then all the elements are fantabulous. Does it follow that the number 20212021 is fantabulous? Problem 2. Find all functions f : ℚ → ℚ such that the equation f(xf(x) …

## APMO Problems and Solutions 2017

Problem 1: We call a 5-tuple of integers arrangeable if its elements can be labeled a, b, c, d, e in some order so that a – b + c – d + e = 29. Determine all 2017-tuples of integers n1, n2, …, n2017such that if we place them in a circle in clockwise …

## APMO Problems and Solutions 2018

Problem 1: Let H be the orthocenter of the triangle ABC. Let M and N be the midpoints of the sides AB and AC, respectively. Assume that H lies inside the quadrilateral BMNC and that the circumcircles of triangles BMH and CNH are tangent to each other. The line through H parallel to BC intersects …

## APMO Problems and Solutions 2019

Problem 1. Let ℤ+ be the set of positive integers. Determine all functions f : ℤ+→ ℤ+ such that a2 + f(a)f(b) is divisible by f(a) + b for all positive integers a and b. Problem 2. Let, m be a fixed positive integer. The infinite sequence {an}n≥1 is defined in the following way: a1 …

## APMO Problems and Solutions 2020

Problem 1. Let Γ be the circumcircle of ∆ABC. Let D be a point on the side BC. The tangent to Γ at A intersects the parallel line to BA through D at point E. The segment CE intersects Γ again at F. Suppose B, D, F, E are concyclic. Prove that AC, BF, DE …

## 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 …