1. Given any set A = {a_1,a_2,a_3,a_4} of four distinct positive integers, we
denote the sum a_1 + a_2 + a_3 + a_4 by s_A. Let n_A denote the number of pairs
(i,j) with 1 ≦ i ＜ j ≦ 4 for which a_i + a_j divides s_A. Find all sets A of
four distinct positive integers which achieve the largest possible value of
n_A.
2. Let S be a finite set of at least two points in the plane. Assume that no
three 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 pivot 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 meets a point of S. This process
continues 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.
3. Let f : R → R be a real-valued function defined on the 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.