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Greek salad : functions and ... a question !!!

Source: Greek Olympiad - 2006 , Senior level

February 25, 2006

functionalgebra proposedalgebra

Problem Statement

Does there exist a function

f : \mathbb{R} \rightarrow \mathbb{R}

, which satisfies both conditions :
a)

f( x + y + z) \leq 3(xy + yz + zx)

for all real numbers

x , y , z

and
b) there exist function

g

and natural number

n

, such that

g(g(x)) = x ^ {2n + 1}

and

f(g(x)) = (g(x)) ^2

for every real number

x

?

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