5th USAMO 1976

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1.  The squares of a 4 x 7 chess board are colored red or blue. Show that however the coloring is done, we can find a rectangle with four distinct corner squares all the same color. Find a counter-example to show that this is not true for a 4 x 6 board.
2.  AB is a fixed chord of a circle, not a diameter. CD is a variable diameter. Find the locus of the intersection of AC and BD.
3.  Find all integral solutions to a2 + b2 + c2 = a2b2.
4.  A tetrahedron ABCD has edges of total length 1. The angles at A (BAC etc) are all 90o. Find the maximum volume of the tetrahedron.
5.  The polynomials a(x), b(x), c(x), d(x) satisfy a(x5) + x b(x5) + x2c(x5) = (1 + x + x2 + x3 + x4) d(x). Show that a(x) has the factor (x -1).

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
5 May 2002