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I would appreciate if somebody could help me with the following problem

Q: Find conti-function $f(x)=?$ $$4(1-x)^{2} f \left({1-x\over 2} \right)+16f \left({1+x\over 2} \right)=16(1-x)-(1-x)^{4}$$

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Change $x$ to $-x$ and then solve it as a system of two equations for the unknown $f$ values.

EDIT

As noted by Blatter, below,

We end up with $f(x)=x^2-1$ or $f(x)=1-x^2$

only $f(x)=1-x^2$ works in the original equation!

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3  
This only gives necessary conditions for $f$. At the end you have to check which of the obtained $f$'s actually fulfill the original condition. – Christian Blatter Jul 30 '13 at 8:52
    
(+1) nice hint. – Mhenni Benghorbal Jul 30 '13 at 13:43

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