| A1. Find all possible values for the sum of the digits of a square. |
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| A2. n > 1. Find all solutions in real numbers x1, x2, ... , xn+1 all at least 1 such that: (1) x11/2 + x21/3 + x31/4 + ... + xn1/(n+1) = n xn+11/2; and (2) (x1 + x2 + ... + xn)/n = xn+1. |
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| A3. L and L' are two perpendicular lines not in the same plane. AA' is perpendicular to both lines, where A belongs to L and A' belongs to L'. S is the sphere with diameter AA'. For which points P on S can we find points X on L and X' on L' such that XX' touches S at P? | |
| B1. ABCD is an n x n board. We call a diagonal row of cells a positive diagonal if it is parallel to AC. How many coins must be placed on an n x n board such that every cell either has a coin or is in the same row, column or positive diagonal as a coin? |
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| B2. The incircle of the triangle ABC touches the sides BC, CA, AB at D, E, F respectively. AD meets the circle again at X and AX = XD. BX meets the circle again at Y and CX meets the circle again at Z. Show that EY = FZ. | |
| B3. f is a function defined on the positive integers with positive integer values. Use f m(n) to mean f(f( ... f(n) ....)) = n where f is taken m times, so that f 2(n) = f(f(n)), for example. Find the largest possible 0 < k < 1 such that for some function f, we have f m(n) ≠ n for m = 1, 2, ... , [kn], but f m(n) = n for some m (which may depend on 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
1 July 2002
Last corrected/updated 22 Jan 04