IMO Problems 2015

IMO Problems 2015

Day 1

IMO 2015 Problem 1

We say that a finite set S of points in the plane is balanced, if for any two different points A and B in S, there is a point C in S such that AC = BC. We say that S is centre free if for any three different points A, B and C in S, there is no point P in S such that PA = PB = PC.

(A) Show that for all integers n ≥ 3, there exists a balanced set consisting of n points.
(B) Determine all integers n ≥ 3, for which there exists a balanced centre free set consisting of n points.

IMO 2015 Problem 2

Find all positive integers (a, b, c) such that:

ab – c, bc – a, ca – b are all powers of 2.

IMO 2015 Problem 3

let ABC be an acute triangle with AB > AC. let Γ be its circumcircle, H is orthocentre, and F is the foot of altitude from A. Let M be the midpoint of BC and Q be the point of Γ such that angle(HQA) = 90ᵒ. Let K be the point on Γ such that angle (HKQ) = 90ᵒ. Assume that point A, B, C, K, and Q are all different and lie on Γ in this order.

Prove that the circumcircle of triangle KQH and FKM are tangent to each other.

Day 2

IMO 2015 Problem 4

Triangle ABC has circumcircle Ω and circumcenter O. A circle Γ with centre A intersect the segment BC at points D and E, such that B, D, E and C are all different and lie online BC in this order. Let F and G be the points of intersection of Ω and Γ, such that A, F, B, C and G lies on Ω in this order. Let K be the second point of intersection of circumcircle of triangle BDF and segment AB. Let L be the second point of intersection of circumcircle of triangle CGE and the segment CA. Suppose that the line FK and GL are different and intersect at point X. Prove that X lies on the line AO.

IMO 2015 Problem 5

Let R be the set of real numbers. Determine all functions f: R → R that satisfy the equation

f(x + f(x + y)) + f(xy) = x + f(x + y) + yf(x) for all real numbers x and y.

IMO 2015 Problem 6

The sequesnce a₁, a₂, … of integers satisfying the conditions:

(i) 1 ≤ aⱼ ≤ 2015 for all j ≥ 1.
(ii) k + aₖ ≠ m + aₘ for all 1 ≤ k < m.

Prove that there exist two positive integers b and N for which

\left| \sum ^{n}_{j=f+1}\left( a_{j}-b\right) \right| \leq \left( 1007\right) ^{2}

for all integers f and n such that n > f ≥ N.