# 2014-15 an equation

Let $$\theta$$ be a fixed constant. Characterize all functions $$f:\mathcal R\to \mathcal R$$ such that $$f”(x)$$ exists for all real $$x$$ and for all real $$x,y$$, $f(y)=f(x)+(y-x)f'(x)+ \frac{(y-x)^2}{2} f”(\theta y + (1-\theta) x).$

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# 2013-13 Functional equation

Find all continuous functions $$f : \mathbb{R} \to \mathbb{R}$$ satisfying
$f(x) = f(x^2 + \frac{x}{3} + \frac{1}{9} )$
for all $$x \in \mathbb{R}$$.

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# 2013-02 Functional equation

Let $$\mathbb{Z}^+$$ be the set of positive integers. Suppose that $$f : \mathbb{Z}^+ \to \mathbb{Z}^+$$ satisfies the following conditions.

i) $$f(f(x)) = 5x$$.

ii) If $$m \geq n$$, then $$f(m) \geq f(n)$$.

iii) $$f(1) \neq 2$$.

Find $$f(256)$$.

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