| A1. Find all solutions in positive integers to a + b + c = xyz, x + y + z = abc. | |
| A2. Fn is the Fibonacci sequence F0 = F1 = 1, Fn+2 = Fn+1 + Fn. Find all pairs m > k ≥ 0 such that the sequence x0, x1, x2, ... defined by x0 = Fk/Fm and xn+1 = (2xn - 1)/(1 - xn) for xn ≠ 1, or 1 if xn = 1, contains the number 1. | |
| A3. PABCDE is a pyramid with ABCDE a convex pentagon. A plane meets the edges PA, PB, PC, PD, PE in points A', B', C', D', E' distinct from A, B, C, D, E and P. For each of the quadrilaterals ABB'A', BCC'B, CDD'C', DEE'D', EAA'E' take the intersection of the diagonals. Show that the five intersections are coplanar. | |
| B1. Define the sequence a1, a2, a3, ... by a1 = 1, an = an-1 + a[n/2]. Does the sequence contain infinitely many multiples of 7? | |
| B2. The points D, E on the side AB of the triangle ABC are such that (AD/DB)(AE/EB) = (AC/CB)2. Show that ∠ACD = ∠BCE. | |
| B3. S is a board containing all unit squares in the xy plane whose vertices have integer coordinates and which lie entirely inside the circle x2 + y2 = 19982. +1 is written in each square of S. An allowed move is to change the sign of every square in S in a given row, column or diagonal. Can we end up with all -1s by a sequence of allowed moves? |
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 March 2004
Last corrected/updated 1 Mar 04