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. -- 没有伴随着痛苦的教训是没意义的。 人如果不牺牲一些东西，就无法得到任何东西。 但是当超越了障碍，并且把得到的东西变成属於自己的东西时... 人应该就能够得到无法取代的钢之心灵吧。 <Fullmetal Alchemist 钢之链金术师> --

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