| 1. Show that n! is divisible by (1 + 2 + ... + n) iff n+1 is not an odd prime. |
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| 2. Show that x(x - z)2 + y(y - z)2 ≥ (x - z)(y - z)(x + y - z) for all non-negative reals x, y, z. When does equality hold? |
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| 3. ABCD is a square. X is a point on the side AB, and Y is a point on the side CD. AY meets DX at R, and BY meets CX at S. How should X and Y be chosen to maximise the area of the quadrilateral XRYS? |
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| 4. Find all real solutions to x2(x + 1)2 + x2 = 3(x + 1)2. |
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| 5. There are 2n+1 cards. There are two cards with each integer from 1 to n and a joker. The cards are arranged in a line with the joker in the center position (with n cards each side of it). For which n < 11 can we arrange the cards so that the two cards with the number k have just k-1 cards between them (for k = 1, 2, ... , n)? |
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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
18 Aug 2002