| A1. Find all real-valued functions f on the reals such that (x-y)f(x+y) - (x+y)f(x-y) = 4xy(x2-y2) for all x, y. | |
| A2. For n > 1 and positive reals x1, x2, ... , xn, show that x12/(x12+x2x3) + x22/(x22+x3x4) + ... + xn2/(xn2+x1x2) ≤ n-1. | |
| A3. In a tournament there are n players. Each pair of players play each other just once. There are no draws. Show that either (1) one can divide the players into two groups A and B, such that every player in A beat every player in B, or (2) we can label the players P1, P2, ... , Pn such that Pi beat Pi+1 for i = 1, 2, ... n (where we use cyclic subscripts, so that Pn+1 means P1). | |
| B1. A triangle with each side length at least 1 lies inside a square side 1. Show that the center of the square lies inside the triangle. | |
| B2. a1, a2, a3, ... is a sequence of positive integers such that limn→∞ n/an = 0. Show that we can find k such that there are at least 1990 squares between a1 + a2 + ... + ak and a1 + a2 + ... + ak+1. | |
| B3. Show that ∑k=0[n/3] (-1)k nC3k is a multiple of 3 for n > 2. (nCm is the binomial coefficient) |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
20 March 2004
Last corrected/updated 25 Mar 04