25th Brasil 2003


A1. Find the smallest positive prime that divides n² + 5n + 23 for some integer n.
A2. Let S be a set with n elements. Take a positive integer k. Let A₁, A₂, ... A_k be any distinct subsets of S. For each i take B_i = A_i or S - A_i. Find the smallest k such that we can always choose B_i so that ∪ B_i = S.
A3. ABCD is a parallelogram with perpendicular diagonals. Take points E, F, G, H on sides AB, BC, CD, DA respectively so that EF and GH are tangent to the incircle of ABCD. Show that EH and FG are parallel.
B1. Given a circle and a point A inside the circle, but not at its center. Find points B, C, D on the circle which maximise the area of the quadrilateral ABCD.
B2. f(x) is a real-valued function defined on the positive reals such that (1) if x < y, then f(x) < f(y), (2) f(2xy/(x+y)) ≥ (f(x) + f(y))/2 for all x. Show that f(x) < 0 for some value of x.
B3. A graph G with n vertices is called great if we can label each vertex with a different positive integer ≤ [n²/4] and find a set of non-negative integers D so that there is an edge between two vertices iff the difference between their labels is in D. Show that if n is sufficiently large we can always find a graph with n vertices which is not great.

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