5th USAMO 1976


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 90^o. 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.