1988 USAMO Problems
Problems from the 1988 USAMO.
The repeating decimal , where and are relatively prime integers, and there is at least one decimal before the repeating part. Show that is divisible by 2 or 5 (or both). (For example, , and 88 is divisible by 2.)
The cubic polynomial has real coefficients and three real roots . Show that and that .
Let be the set and let be the set of all 9-element subsets of . Show that for any map we can find a 10-element subset of , such that for any in .
is a triangle with incenter . Show that the circumcenters of , , and lie on a circle whose center is the circumcenter of .
Let be the polynomial , where are integers. When expanded in powers of , the coefficient of is and the coefficients of , , ..., are all zero. Find .
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.