1989 USAMO Problems/Problem 3
Let be a polynomial in the complex variable , with real coefficients . Suppose that . Prove that there exist real numbers and such that and .
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.
|1989 USAMO (Problems • Resources)|
|1 • 2 • 3 • 4 • 5|
|All USAMO Problems and Solutions|