| A1. ABCD is a fixed rhombus. P lies on AB and Q on BC, so that PQ is perpendicular to BD. Similarly P' lies on AD and Q' on CD, so that P'Q' is perpendicular to BD. The distance between PQ and P'Q' is more than BD/2. Show that the perimeter of the hexagon APQCQ'P' depends only on the distance between PQ and P'Q'. |
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| A2. Prove that (n+1)mnm ≥ (n+m)!/(n-m)! ≥ 2mm! for all positive integers n, m with n ≥ m. |
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| A3. Given four concyclic points. For each subset of three points take the incenter. Show that the four incenters from a rectangle. |
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| A4. For which n in the range 1 to 1996 is it possible to divide n married couples into exactly 17 single sex groups, so that the size of any two groups differs by at most one. |
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| A5. A triangle has side lengths a, b, c. Prove that √(a + b - c) + √(b + c - a) + √(c + a - b) ≤ √a + √b + √c. When do you have equality? |
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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
5 April 2002