Problem: (1999 JBMO – #1) Let be five real numbers such that , , and . If are all distinct numbers, prove that their sum is zero.
Solution: Consider the polynomial . Since it is a third degree polynomial, it can have at most three real roots. But we are given that
so we know that it in fact has exactly three distinct real roots and no other ones. But by Vieta’s Formulas, we know that the sum of the roots is the coefficient of the term, which is zero. Thus
as desired. QED.
Comment: A pretty simple problem to be on an olympiad, even if it is the Junior Balkan Math Olympiad (JBMO). The solution is almost immediate if you have worked with polynomials a lot.
Practice Problem: (2000 JBMO – #1) Let and be positive reals such that
Show that .
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