IMO 2003


A1. S is the set {1, 2, 3, ... , 1000000}. Show that for any subset A of S with 101 elements we can find 100 distinct elements x_i of S, such that the sets x_i + A are all pairwise disjoint. [Note that x_i + A is the set {a + x_i \| a is in A} ].
A2. Find all pairs (m, n) of positive integers such that m²/(2mn² - n³ + 1) is a positive integer.
A3. A convex hexagon has the property that for any pair of opposite sides the distance between their midpoints is (√3)/2 times the sum of their lengths. Show that all the hexagon's angles are equal.
B1. ABCD is cyclic. The feet of the perpendicular from D to the lines AB, BC, CA are P, Q, R respectively. Show that the angle bisectors of ABC and CDA meet on the line AC iff RP = RQ.
B2. Given n > 2 and reals x₁ ≤ x₂ ≤ ... ≤ x_n, show that (∑_i,j \|x_i - x_j\| )² ≤ (2/3) (n² - 1) ∑_i,j (x_i - x_j)². Show that we have equality iff the sequence is an arithmetic progression.
B3. Show that for each prime p, there exists a prime q such that n^p - p is not divisible by q for any positive integer n.

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