17th USAMO 1988


1. The repeating decimal 0.ab ... k pq ... u = m/n, where m and n are relatively prime integers, and there is at least one decimal before the repeating part. Show that n is divisible by 2 or 5 (or both). [For example, 0.01136 = 0.01136363636 ... = 1/88 and 88 is divisible by 2.]
2. The cubic x³ + ax² + bx + c has real coefficients and three real roots r ≥ s ≥ t. Show that k = a² - 3b ≥ 0 and that √k <= r - t.
3. Let X be the set {1, 2, ... , 20} and let P be the set of all 9-element subsets of X. Show that for any map f: P → X we can find a 10-element subset Y of X, such that f(Y - {k}) ≠ k for any k in Y.
4. ABC is a triangle with incenter I. Show that the circumcenters of IAB, IBC, ICA lie on a circle whose center is the circumcenter of ABC.
5. Let p(x) be the polynomial (1 - x)^a (1 - x²)^b (1 - x³)^c ... (1 - x³²)^k, where a, b, ..., k are integers. When expanded in powers of x, the coefficient of x¹ is -2 and the coefficients of x², x³, ... , x³² are all zero. Find k.

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