Problem Solving Set 6

09 July 2012

1. Let f be a polynomial of degree 2 with integer coefficients. Suppose that f(k) is divisible by 5 for every integer k. Prove that all coefficients of f are divisible by 5.

2. (a) Given a triangle ABC show that there is a unique triangle P QR with the points P, Q, R on the sides BC, CA, AB, and the edges QR, RP, P Q parallel to BC, CA, AB, respectively.

(b) Suppose we inscribe a third triangle in the same way inside P QR, and a fourth triangle inside this one, and so on. Show that the areas of the triangles form a geometric sequence.