IMO Problems 2010

IMO Problems 2010

Day 1

IMO 2010 Problem 1

Find all function f: R → R such that for all real numbers x and y the following equality holds
f([x]y) = f(x)[f(y)], here [a] is the greatest integer not greater than a.

IMO 2010 Problem 2

Given a triangle ABC with I as its in centre and Γ as it’s a circumcircle. AI intersect Γ again at D. Let E be a point on the Arc BDC and F be a point on the segment BC such that angle(BAF) = angle(CAE) < (angle(BAC))/2. If G is the midpoint of IF, prove that the meeting point of the lines EI and DG lies on Γ.

IMO 2010 Problem 3

Find all functions f: N → N, such that f((m) + n)f((n) + m) is a perfect square for all m, n belongs to N.

Day 2

IMO 2010 Problem 4

Let P be a point interior to a triangle ABC with CA ≠ CB. The lines AP, BP, and CP meet again it’s circumcircle Γ at K, L, and M respectively. The tangent line at C to Γ meets the line AB at S. Show that to from SC = SP follows MK = ML.

IMO 2010 Problem 5

Each of the 6 boxes B₁, B₂, B₃, B₄, B₅, B₆ initially contains one coin. The following operations are allowed:
(1) Choose a non empty box Bⱼ, 1 ≤ j ≤ 5, remove one coin from Bⱼ and add two coins Bⱼ₊₁.
(2) Choose a non empty box Bₖ, 1 ≤ k ≤ 4, remove one coin from Bₖ and swap the contents (may be empty) of the boxes Bₖ₊₁ and Bₖ₊₂.

Determine if there exists a finite sequence of operations of the allowed the types, such that the five boxes B₁, B₂, B₃, B₄, B₅ become an empty, while box B₆ contains exactly (20102010)2010 coins.

IMO 2010 Problem 6

Let a₁, a₂, a₃, … be a sequence of positive real numbers, and s be a positive integer, such that
aₙ = max{aₖ + aₙ₋ₖ | 1 ≤ k ≤ n – 1} for all n > s.
Prove that there exist a positive integer l ≤ s and N, such that aₙ = aₙ – aₙ₋ₗ for all n ≥ N.