IMO Problems 2019

IMO Problems 2019

Day 1 16/July/2019

IMO 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)).

IMO 2019 Problem 2

In triangle ABC, point A₁ lies on side BC and point B₁ lies on side AC. Let P and Q be points on segments A₁ and B₁ respectively, such that PQ is parallel to AB. Let P₁ be a point on line PB₁ such that B₁ lies strictly between P and P₁ and angle(PP₁C) = angle(BAC). Similarly let Q₁ be the point on line QA₁, such that A₁ lies strictly between Q and Q₁ and angle(CQ₁Q) = angle(CBA). Prove that the points PQP₁Q₁ are concyclic.

IMO 2019 Problem 3

A social network has 2019 users, some pairs of whom are friends. Whenever user A is friends with user B, user B is also friends with user A. Events of the following kind may happen repeatedly one-at-a-time.
3 users A, B, and C such that A is friends with both B and C, but B and C are not friends, change their friendship statuses such that B and C are now friends, but A is no longer friends with B, and no longer friends with C. All other friendship statuses are unchanged.
initially 1010 users have 1009 friends each, and 1009 users have 1010 friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.

Day 2 17/July/2019

IMO 2019 Problem 4

Find all pairs (k, n) of positive integers such that k! = (2ⁿ – 1)(2ⁿ – 2)(2ⁿ – 4)…(2ⁿ – 2ⁿ⁻¹).

IMO 2019 Problem 5

The Bank of Bath issues coins with an H on one side and a T on other side. Harry has n of these coins arranged in a line from left to right. He repeatedly performs the following operations: if there are exactly k > 0 coins showing H, then he turns over the k-th coin from the left, otherwise all coins show T and he stops. For example, if n = 3 the process is starting with the configuration THT would be THT → HHT → HTT → TTT, which stops after three operations.
(a) Show that for each initial configuration, Harry stops after the finite number of operations.
(b) For each initial configuration C, Let L(C) be the number of operations before Harry stops. For example, L(THT) = 3 and L(TTT) = 0. Determine the average value of L(C) over all 2ⁿ possible initial configurations of C.

IMO 2019 Problem 6

Let I be the incentre of acute triangle ABC with AB ≠ AC. The incircle ω of ABC is tangent to sides BC, CA, and AB at D, E, and F. The line through D perpendicular to EF meets ω at R. Line AR meets ω again at P. The circumcircles of triangle PCE and PBF meet again at Q. Prove that the lines DI and PQ meet on the line through A perpendicular to AI.