18th Iberoamerican 2003


A1. Let A, B be two sets of N consecutive integers. If N = 2003, can we form N pairs (a, b) with a ∈ A, b ∈ B such that the sums of the pairs are N consecutive integers? What about N = 2004?
A2. C is a point on the semicircle with diameter AB. D is a point on the arc BC. M, P, N are the midpoints of AC, CD and BD. The circumcenters of ACP and BDP are O, O'. Show that MN and OO' are parallel.
A3. Pablo was trying to solve the following problem: find the sequence x₀, x₁, x₂, ... , x₂₀₀₃ which satisfies x₀ = 1, 0 ≤ x_i ≤ 2 x_i-1 for 1 ≤ i ≤ 2003 and which maximises S. Unfortunately he could not remember the expression for S, but he knew that it had the form S = ± x₁ ± x₂ ± ... ± x₂₀₀₂ + x₂₀₀₃. Show that he can still solve the problem.
B1. A ⊆ {1, 2, 3, ... , 49} does not contain six consecutive integers. Find the largest possible value of \|A\|. How many such subsets are there (of the maximum size)?
B2. ABCD is a square. P, Q are points on the sides BC, CD respectively, distinct from the endpoints such that BP = CQ. X, Y are points on AP, AQ respectively. Show that there is a triangle with side lengths BX, XY, YD.
B3. The sequences a₀, a₁, a₂, ... and b₀, b₁, b₂, ... are defined by a₀ = 1, b₀ = 4, a_n+1 = a_n²⁰⁰¹ + b_n, b_n+1 = b_n²⁰⁰¹ + a_n. Show that no member of either sequence is divisible by 2003.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.