# 2022-16 Identity for continuous functions

For a positive integer $$n$$, find all continuous functions $$f: \mathbb{R} \to \mathbb{R}$$ such that
$\sum_{k=0}^n \binom{n}{k} f(x^{2^k}) = 0$
for all $$x \in \mathbb{R}$$.

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# 2020-19 Continuous functions

Let $$n$$ be a positive integer. Determine all continuous functions $$f: [0, 1] \to \mathbb{R}$$ such that
$f(x_1) + \dots + f(x_n) =1$
for all $$x_1, \dots, x_n \in [0, 1]$$ satisfying $$x_1 + \dots + x_n = 1$$.

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# 2017-05 Inequality for a continuous function

Suppose that $$f : (2, \infty) \to (-2, 2)$$ is a continuous function and there exists a positive constant $$m$$ such that $$| 1 + xf(x) + (f(x))^2 | \leq m$$ for any $$x > 2$$. Prove that, for any $$x > 2$$,
$\left| f(x) – \frac{\sqrt{x^2 -4}-x}{2} \right| \leq 6 \sqrt{m}.$

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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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