# Understand the problem

Find all the polynomials of a degree with real non-negative coefficients such that , .

##### Source of the problem

Albanian BMO TST 2009

##### Topic

Algebra

##### Difficulty Level

Easy

##### Suggested Book

An Excursion in Mathematics

# Start with hints

Do you really need a hint? Try it first!

This problem is all about non-negative real numbers. The first thing that should come to your mind is “standard inequalities!”.

Write . Using the Cauchy-Schwarz inequality, show that .

Note that hint 2 along with the hypothesis in the problem implies that . Hence equality holds in hint 2.

As equality holds in CS, it means that for all satisfying , is a constant. This is absurd, hence there can be at most one such . Hence, only monomials can satisfy the given inequality.

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