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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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 \( x \in \mathbb{R} \).

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 \( x_1 + \dots + x_n = 1\).

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}.

\]

Find all continuous functions \(f : \mathbb{R} \to \mathbb{R}\) satisfying

\[

f(x) = f(x^2 + \frac{x}{3} + \frac{1}{9} )

\]

for all \( x \in \mathbb{R} \).