1989 USAMO Problems/Problem 3
Problem
Let be a polynomial in the complex variable , with real coefficients . Suppose that . Prove that there exist real numbers and such that and .
Solution
Let be the (not necessarily distinct) roots of , so that Since all the coefficients of are real, it follows that if is a root of , then , so , the complex conjugate of , is also a root of .
Since it follows that for some (not necessarily distinct) conjugates and , Let and , for real . We note that Thus Since , these real numbers satisfy the problem's conditions.
See Also
1989 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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.