IMO Problems 2011

IMO Problems 2011

Day 1

IMO 2011 Problem 1

Given any set A = {a₁, a₂, a₃, a₄} of four distinct positive integers, we denote the sum a₁ + a₂ + a₃ + a₄ by S. Let n denotes the number of pairs (i, j) with 1 ≤ i < j ≤ 4 for which aᵢ + aⱼ divides S. Find all sets A of four distinct positive integers which achieve the largest possible value of n.

IMO 2011 Problem 2

Let S be a finite set of at least two points in the plane. Assume that no 3 points of S are collinear. A windmill is a process that starts with a line l going through a single point P ∈ S. The line rotates clockwise about the private P until the first time that the line meets some other point belonging to S. This point Q takes over as the new pivot and the line now rotates clockwise about Q until it next meet a point of S. This process continues in indefinitely. Show that we can choose a point P in S and a line l going through P such that the resulting windmill uses each point of S as a pivot infinitely many times.

IMO 2011 Problem 3

Let f: R → R be a real valued function defined on a set of real numbers that satisfies f(x + y) ≤ yf(x) + f(f(x)), for all real numbers x and y.
Prove that f(x) = 0 for all x ≤ 0.

Day 2

IMO 2011 Problem 4

Let n > 0 be an integer, we are given a balance and n weights of weight 2⁰, 2¹, 2², …, 2ⁿ⁻¹. We are to place each of the n weights on the balance one after another in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance and place it on either the left pan or the right pan until all the all of the weights have been placed. Determine the number of ways in which this can be done.

IMO 2011 Problem 5

Let f be a function from the set of integers to the set of positive integers. Suppose that for any two integers m and n the difference of f(m) – f(n) is divisible by f(m – n). Prove that for all integers m and n with f(m) ≤ f(n), the number f(n) is divisible by f(m).

IMO 2011 Problem 6

Let ABC be an acute triangle with circumcircle Γ. Let l the tangent to Γ and let la, la, la be the lines obtained by reflecting l in the lines BC, CA ,and AB respectively. Show that the circumcircle of the triangle determined by the lines la, la, la is tangent to the circle Γ.