1977 USAMO Problems/Problem 3
Problem
If and are two of the roots of , prove that is a root of .
Solution
Given the roots of the equation .
First, Vieta's relations give .
Then and .
The other coefficients give or .
Let and , so (1).
Second, is a root, and is a root, .
Multiplying: or .
Solving .
In (1): .
or .
Conclusion: is a root of .
See Also
1977 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
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