| A1. There are n ≥ 2 points in a square side 1. Show that one can label the points P1, P2, ... , Pn such that ∑i=1n |Pi-1 - Pi|2 ≤ 4, where we use cyclic subscripts, so that P0 means Pn. | |
| A2. A regular n-gon is inscribed in a circle radius 1. Let X be the set of all arcs PQ, where P, Q are distinct vertices of the n-gon. 5 elements L1, L2, ... , L5 of X are chosen at random (so two or more of the Li can be the same). Show that the expected length of L1 ∩ L2 ∩ L3 ∩ L4 ∩ L5 is independent of n. | |
| A3. w(x) is a polynomial with integral coefficients. Let pn be the sum of the digits of the number w(n). Show that some value must occur infinitely often in the sequence p1, p2, p3, ... . | |
| B1. Let S be the set of all tetrahedra which satisfy (1) the base has area 1, (2) the total face area is 4, and (3) the angles between the base and the other three faces are all equal. Find the element of S which has the largest volume. | |
| B2. Find the smallest n such that n2-n+11 is the product of four primes (not necessarily distinct). | |
| B3. A plane is tiled with regular hexagons of side 1. A is a fixed hexagon vertex. Find the number of paths P such that (1) one endpoint of P is A, (2) the other endpoint of P is a hexagon vertex, (3) P lies along hexagon edges, (4) P has length 60, and (5) there is no shorter path along hexagon edges from A to the other endpoint of P. |
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
20 March 2004
Last corrected/updated 25 Mar 04