# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

• ## Meta

Legend of Max. Topic: Inequalities. Level: Olympiad. February 27th, 2006

Problem: (1999 USAMO – #4) Let () be real numbers such that

and .

Prove that .

Solution: To simply things, let . We wish to show that there exists an .

Our conditions become

.

.

Assume for the sake of contradiction that for all and let . We have

, or

.

But we have , so

.

However, since and , we have (looking at the parabola it is easy to see; has zeros at ). Therefore

.

,

so that gives us a contradiction. Hence our assumption must be false and there exists an as desired. QED.

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Comment: This wasn't a particularly difficult inequality, but it has some key ideas. Using all parts of the question is important (in this case is actually relevant). Another note is that this didn’t really require any of the classical inequalities, just algebraic manipulation. Lastly, the crucial step was setting converting the real to positive which makes things a whole lot nicer.

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Practice Problem: Go take the 2006 Mock AIME 5 below.