| 1. ABCDEF is a convex hexagon. Consider the following statements. (1) AB is parallel to DE, (2) BC is parallel to EF, (3) CD is parallel to FA, (4) AE = BD, (5) BF = CE, (6) CA = DF. Show that if any five of these statements are true then the hexagon is cyclic. | |
| 2. Find the smallest positive value taken by a3 + b3 + c3 - 3abc for positive integers a, b, c. Find all a, b, c which give the smallest value. | |
| 3. x, y are positive reals such that x + y = 2. Show that x3y3(x3 + y3) ≤ 2. | |
| 4. Do there exist 100 lines in the plane, no three concurrent, such that they intersect in exactly 2002 points? | |
| 5. Do there exist distinct positive integers a, b, c such that a, b, c, -a+b+c, a-b+c, a+b-c, a+b+c form an arithmetic progression (in some order). | |
| 6. The numbers 1, 2, 3, ... , n2 are arranged in an n x n array, so that the numbers in each row increase from left to right, and the numbers in each column increase from top to bottom. Let aij be the number in position i, j. Let bj be the number of possible value for ajj. Show that b1 + b2 + ... + bn = n(n2-3n+5)/3. |
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
30 March 2004
Last corrected/updated 30 Mar 04