1988 USAMO Problems
Problems from the 1988 USAMO.
Problem 1
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.)
Problem 2
The cubic polynomial has real coefficients and three real roots . Show that and that .
Problem 3
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 .
Problem 4
is a triangle with incenter . Show that the circumcenters of , , and lie on a circle whose center is the circumcenter of .
Problem 5
Let be the polynomial , where are integers. When expanded in powers of , the coefficient of is and the coefficients of , , ..., are all zero. Find .
See Also
1988 USAMO (Problems • Resources) | ||
Preceded by 1987 USAMO |
Followed by 1989 USAMO | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.