| 1. p(x1, x2, ... , xn) is a polynomial with integer coefficients. For each positive integer r, k(r) is the number of n-tuples (a1, a2, ... , an) such that 0 ≤ ai ≤ r-1 and p(a1, a2, ... , an) is prime to r. Show that if u and v are coprime then k(u·v) = k(u)·k(v), and if p is prime then k(ps) = pn(s-1) k(p). | |
| 2. Given a point p inside a convex polyhedron P. Show that there is a face F of P such that the foot of the perpendicular from p to F lies in the interior of F. |
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| 3. Two players play alternately. The first player is given a pair of positive integers (x1, y1). Each player must replace the pair (xn, yn) that he is given by a pair of non-negative integers (xn+1, yn+1) such that xn+1 = min(xn, yn) and yn+1 = max(xn, yn) - k·xn+1 for some positive integer k. The first player to pass on a pair with yn+1 = 0 wins. Find for which values of x1/y1 the first player has a winning strategy. |
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| 4. Given points A1 (x1, y1, z1), A2 (x2, y2, z2), ... , An (xn, yn, zn) let P (x, y, z) be the point which minimizes ∑ ( |x - xi| + |y - yi| + |z - zi| ). Give an example (for each n > 4) of points Ai for which the point P lies outside the convex hull of the points Ai. |
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| 5. A and B wish to divide a cake into two pieces. Each wants the largest piece he can get. The cake is a triangular prism with the triangular faces horizontal. A chooses a point P on the top face. B then chooses a vertical plane through the point P to divide the cake. B chooses which piece to take. Which point P should A choose in order to secure as large a slice as possible? |
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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
12 July 2003
Last corrected/updated 3 Apr 04