2000 JBMO Problems/Problem 1
Let and be positive reals such that Show that .
Rearranging the equation yields If in the large equation, then must be a factor of the large equation. Note that we can rewrite the large equation as We can factor the difference of cubes in the first part and factor in the second part, resulting in Finally, we can factor by grouping, which results in By the Zero Product Property, either or However, since and are both positive, can not equal zero, so we have proved that
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