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A1. How many solutions in non-negative reals are there to the equations:
x1 + xn2 = 4xn x2 + x12 = 4x1 ... xn + xn-12 = 4xn-1? |
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| A2. The triangle ABC has AC = BC. P is a point inside the triangle such that ∠PAB = ∠PBC. M is the midpoint of AB. Show that ∠APM + ∠BPC = 180o. | |
| A3. The sequence a1, a2, a3, ... is defined as follows. a1 and a2 are primes. an is the greatest prime divisor of an-1 + an-2 + 2000. Show that the sequence is bounded. | |
| B1. PA1A2...An is a pyramid. The base A1A2...An is a regular n-gon. The apex P is placed so that the lines PAi all make an angle 60o with the plane of the base. For which n is it possible to find Bi on PAi for i = 2, 3, ... , n such that A1B2 + B2B3 + B3B4 + ... + Bn-1Bn + BnA1 < 2A1P? | |
| B2. For each n ≥ 2 find the smallest k such that given any subset S of k squares on an n x n chessboard we can find a subset T of S such that every row and column of the board has an even number of squares in T. | |
| B3. p(x) is a polynomial of odd degree which satisfies p(x2-1) = p(x)2 - 1 for all x. Show that p(x) = x. |
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