Two Cool FEs

For discussing Olympiad Level Algebra (and Inequality) problems
Nirjhor
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Two Cool FEs

Unread post by Nirjhor » Sat Sep 06, 2014 4:15 pm

1. Find all polynomials \(P:\mathbb{R}-\{0\}\to\mathbb{R}\) satisfying \[P(x)~P\left(\dfrac{1}{x}\right)=P(x)+P\left(\dfrac{1}{x}\right).\]
2. Consider all functions \(f:\mathbb{R}-\{0\}\to\mathbb{R}\) satisfying \[f\left(x+\dfrac 1 x\right)~f\left(x^3+\dfrac{1}{x^3}\right)-f\left(x^2+\dfrac{1}{x^2}\right)^2=\left(x-\dfrac 1 x\right)^2.\] Find the sum of all possible values of \(f(2014)\). Hint Below.
This one is not as hard as it looks, in fact, is pretty funny. Just concentrate on what's asked to do, not anything else.
- What is the value of the contour integral around Western Europe?

- Zero.

- Why?

- Because all the poles are in Eastern Europe.


Revive the IMO marathon.

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