| A1. Find all triples of positive integers (a, b, c) such that a2 + 1 and b2 + 1 are prime and (a2 + 1)(b2 + 1) = c2 + 1. | |
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A2. ABC is an acute-angled triangle. BCKL, ACPQ are rectangles on the outside of two of the sides and have equal area. Show that the midpoint of PK lies on the line through C and the circumcenter.
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| A3. Three non-negative integers are written on a blackboard. A move is to replace two of the integers by their sum and (non-negative) difference. Can we always get two zeros by a sequence of moves? | |
| B1. Given any finite sequence x1, x2, ... , xn of at least 3 positive integers, show that either ∑1n xi/(xi+1 + xi+2) ≥ n/2 or ∑ 1n xi/(xi-1 + xi-2) ≥ n/2. (We use the cyclic subscript convention, so that xn+1 means x1 and x-1 means xn-1 etc). | |
| B2. ABC is a triangle. A sphere does not intersect the plane of ABC. There are 4 points K, L, M, P on the sphere such that AK, BL, CM are tangent to the sphere and AK/AP = BL/BP = CM/CP. Show that the sphere touches the circumsphere of ABCP. | |
| B3. k is a positive integer. The sequence a1, a2, a3, ... is defined by a1 = k+1, an+1 = an2 - kan + k. Show that am and an are coprime (for m ≠ n). |
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
6 March 2004
Last corrected/updated 6 Mar 04