| A1. The angle bisectors of the angles A, B, C in the triangle ABC meet the circumcircle again at K, L, M. Show that |AK| + |BL| + |CM| > |AB| + |BC| + |CA|. | |
| A2. For given n, we choose k and m at random subject to 0 ≤ k ≤ m ≤ 2n. Let pn be the probability that the binomial coefficient mCk is even. Find limn→∞ pn. | |
| A3. Q is a point inside the n-gon P1P2...Pn which does not lie on any of the diagonals. Show that if n is even, then Q must lie inside an even number of triangles PiPjPk. | |
| B1. Given a real numbers x ∈ (0,1) and a positive integer N, prove that there exist positive integers a, b, c, d such that (1) a/b < x < c/d, (2) c/d - a/b < 1/n, and (3) qr - ps = 1. | |
B2. There is a piece in each square of an m x n rectangle on an infinite chessboard. An allowed move is to remove two pieces which are adjacent horizontally or vertically and to place a piece in an empty square adjacent to the two removed and in line with them (as shown below)
X X . to . . X, or . to X
X .
X .
Show that if mn is a multiple of 3, then it is not possible to end up with only one piece after a sequence of moves.
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| B3. Show that if the positive integers a, b, c, d satisfy ab = cd, then we have gcd(a,c) gcd(a,d) = a gcd(a,b,c,d). |
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 23 Mar 04