| A1. How many subsets of {1, 2, ... , 2n} do not contain two numbers with sum 2n+1? | |
| A2. The diagonals of a convex pentagon divide it into a small pentagon and ten triangles. What is the largest number of the triangles that can have the same area? | |
| A3. p ≥ 5 is prime. The sequence a0, a1, a2, ... is defined by a0 = 1, a1 = 1, ... , ap-1 = p-1 and an = an-1 + an-p for n ≥ p. Find ap3 mod p. | |
| B1. The positive reals x1, x2, ... , xn have harmonic mean 1. Find the smallest possible value of x1 + x22/2 + x33/3 + ... + xnn/n. | |
| B2. An urn contains n balls labeled 1, 2, ... , n. We draw the balls out one by one (without replacing them) until we obtain a ball whose number is divisible by k. Find all k such that the expected number of balls removed is k. | |
| B3. PA, PB, PC are three rays in space. Show that there is just one pair of points B', C' with B' on the ray PB and C' on the ray PC such that PC' + B'C' = PA + AB' and PB' + B'C' = PA + AC'. |
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
1 March 2004
Last corrected/updated 1 Mar 04