20th USAMO 1991


1. An obtuse angled triangle has integral sides and one acute angle is twice the other. Find the smallest possible perimeter.
2. For each non-empty subset of {1, 2, ... , n} take the sum of the elements divided by the product. Show that the sum of the resulting quantities is n² + 2n - (n + 1)s_n, where s_n = 1 + 1/2 + 1/3 + ... + 1/n.
3. Define the function f on the natural numbers by f(1) = 2, f(n) = 2^f(n-1). Show that f(n) has the same residue mod m for all sufficiently large n.
4. a and b are positive integers and c = (a^a+1 + b^b+1)/(a^a + b^b). By considering (xⁿ - nⁿ)/(x - n) or otherwise, show that c^a + c^b ≥ a^a + b^b.
5. X is a point on the side BC of the triangle ABC. Take the other common tangent (apart from BC) to the incircles of ABX and ACX which intersects the segments AB and AC. Let it meet AX at Y. Show that the locus of Y, as X varies, is the arc of a circle.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.