31st USAMO 2002


A1. Let S be a set with 2002 elements and P the set of all its subsets. Prove that for any n (in the range from zero to \|P\|) we can color n elements of P white, and the rest black, so that the union of any two elements of P with the same color has the same color.
A2. The triangle ABC satisfies the relation cot²A/2 + 4 cot²B/2 + 9 cot²C/2 = 9(a+b+c)²/(49r²), where r is the radius of the incircle (and a = \|BC\| etc, as usual). Show that ABC is similar to a triangle whose sides are integers and find the smallest set of such integers.
A3. p(x) is a polynomial of degree n with real coefficients and leading coefficient 1. Show that we can find two polynomials q(x) and r(x) which both have degree n, all roots real and leading coefficient 1, such that p(x) = q(x)/2 + r(x)/2.
B1. Find all real-valued functions f on the reals such that f(x² - y²) = x f(x) - y f(y) for all x, y.
B2. Show that we can link any two integers m, n greater than 2 by a chain of positive integers m = a₁, a₂, ... , a_k+1 = n, so that the product of any two consecutive members of the chain is divisible by their sum. [For example, 7, 42, 21, 28, 70, 30, 6, 3 links 7 and 3.]
B3. A tromino is a 1 x 3 rectangle. Trominoes are placed on an n x n board. Each tromino must line up with the squares on the board, so that it covers exactly three squares. Let f(n) be the smallest number of trominoes required to stop any more being placed. Show that for all n > 0, n²/7 + hn ≤ f(n) ≤ n²/5 + kn for some reals h and k.

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