Select Page

# Understand the problem

Find all the polynomials $P(x)$ of a degree $\leq n$ with real non-negative coefficients such that $P(x) \cdot P(\frac{1}{x}) \leq [P(1)]^2$ , $\forall x>0$.
##### Source of the problem
Albanian BMO TST 2009
Algebra
Easy
##### Suggested Book
An Excursion in Mathematics

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 $P(x)=\Sigma a_kx^k$. Using the Cauchy-Schwarz inequality, show that $P(x)P(1/x)\ge (P(1))^2$.
Note that hint 2 along with the hypothesis in the problem implies that $P(x)P(1/x)=(P(1))^2$. Hence equality holds in hint 2.
As equality holds in CS, it means that for all $k$ satisfying $a_k\neq 0$, $\frac{a_kx^k}{a_kx^{-k}=x^{2k}$ is a constant. This is absurd, hence there can be at most one such $k$. Hence, only monomials can satisfy the given inequality.

# Connected Program at Cheenta

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

# Similar Problems

## Problem related to Money | AMC 8, 2002 | Problem 25

Try this beautiful problem from Algebra based on Number theory fro AMC-8(2002) problem no 25.You may use sequential hints to solve the problem.

## Divisibility Problem | PRMO 2019 | Question 8

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2015 based on Smallest Perimeter of Triangle.

## Area of Trapezoid | AMC 10A, 2018 | Problem 9

Try this beautiful problem from AMC 10A, 2018 based on area of trapezoid. You may use sequential hints to solve the problem.

## Problem on Series and Sequences | SMO, 2012 | Problem 23

Try this beautiful problem from Singapore Mathematics Olympiad, 2012 based on Series and Sequences. You may use sequential hints to solve the problem.

## Theory of Equations | AIME 2015 | Question 10

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2015 based on Theory of Equations.

## Trigonometry Problem | AIME 2015 | Question 13

Try this beautiful problem number 13 from the American Invitational Mathematics Examination, AIME, 2015 based on Trigonometry.

## Problem from Probability | AMC 8, 2004 | Problem no. 21

Try this beautiful problem from Probability from AMC 8, 2004, Problem 21.You may use sequential hints to solve the problem.

## Smallest Perimeter of Triangle | AIME 2015 | Question 11

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2015 based on Smallest Perimeter of Triangle.

## Intersection of two Squares | AMC 8, 2004 | Problem 25

Try this beautiful problem from Geometry based on Intersection of two Squares AMC-8, 2004,Problem-25. You may use sequential hints to solve the problem.

## Probability | AMC 8, 2004 | Problem no. 22

Try this beautiful problem from Probability from AMC-8, 2004 Problem 22. You may use sequential hints to solve the problem.