Problem:
There is a smallest positive real number such that there exists a positive real number such that all the roots of the polynomial are real. In fact, for this value of the value of is unique. What is this value of
Answer Choices:
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Solution:
Because and are positive, all the roots must be positive. Let the roots be , and . Then .
Therefore . The Arithmetic Mean-Geometric Mean Inequality implies that , from which . Furthermore, equality is achieved if and only if . In this case .
The problems on this page are the property of the MAA's American Mathematics Competitions