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• 10 If $f: \mathbb{N} \to \mathbb{N}$ and $\forall x \in \mathbb{N},\,(f \circ f )(x) = x^{4}$, is $f$ polynomial?
• 7 How to prove that $\sum _{n=0}^{\infty }\:\frac{(x^n)'}{(n-1)!} = e^{x}(x-1)$
• 6 If $H$ is a subgroup of infinite index and $G = H \cup H_1 \cup H_2 \cup \cdots \cup H_p$, show that $G = H_1 \cup H_2 \cup \cdots \cup H_p$.
• 6 If $\forall n \in \mathbb{N}^*, \ a^n - 1 \mid b^n - 1$ therefore $\exists p \in \mathbb{N}^*,\ b=a^p$.
• 6 If $f$ is continuous and bounded, and $\text{span}\{x \mapsto f(x+k) \mid k \in \Bbb Z\}$ is finite-dimensional, what can we say about $f$?
• 6 Describe $n$ x $n$ matrix $A$ , such that it follows $A+A^T = 2A^{-1}$?
• 5 Let $|f(x)-f(y)| \leqslant (x-y)^2$ for all $x,y\in \mathbb{R}.$ Show that $f$ is a constant.
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