Alibaba Global Mathematics Competition

1. Yes. We prove that $s=1$ has the property. Denote $(\sqrt{2}+1)^{n}=x_{n}+\sqrt{2} y_{n}$, where $x_{n}, y_{n} \in \mathbb{Z}$. Then $(-\sqrt{2}+1)^{n}=x_{n}-\sqrt{2} y_{n}$ and $x_{n}^{2}-2 y_{n}^{2}=(-1)^{n}$. It follows that $\mid x_{n}+$ $\sqrt{2} y_{n}-2 x_{n}|=| \sqrt{2} y_{n}-x_{n} \left\lvert\,=\frac{\left|2 y_{n}^{2}-x_{n}^{2}\right|}{\sqrt{2} y_{n}+x_{n}} \rightarrow 0\right.$. 2. No. We prove this by contradiction. Assume that there is some real number $s \neq 0$ such that $(\sqrt{2}+3)^{n} s=m_{n}+\epsilon_{n}$, where $\lim _{n \rightarrow \infty} \epsilon_{n}=0$. Denote $\alpha=\sqrt{2}+3, \bar{\alpha}=-\sqrt{2}+3$. Consider the power series: $$\frac{s}{1-\alpha x}=\sum_{n=0}^{\infty} m_{n} x^{n}+\sum_{n=0}^{\infty} \epsilon_{n} x^{n}$$ Since $(1-\alpha x)(1-\bar{\alpha} x)=1-6 x+7 x^{2}$, multiplying both sides of the above equation by $1-6 x+7 x^{2}$ we get $$\begin{equation*} s(1-\bar{\alpha} x)=\left(1-6 x+7 x^{2}\right) \sum_{n=0}^{\infty} m_{n} x^{n}+\left(1-6 x+7 x^{2}\right) \sum_{n=0}^{\infty} \epsilon_{n} x^{n} \tag{9} \end{equation*}$$ Denote $\left(1-6 x+7 x^{2}\right) \sum_{n=0}^{\infty} m_{n} x^{n}=\sum_{n=0}^{\infty} p_{n} x^{n},\left(1-6 x+7 x^{2}\right) \sum_{n=0}^{\infty} \epsilon_{n} x^{n}=\sum_{n=0}^{\infty} \eta_{n} x^{n}$, where $p_{n} \in \mathbb{Z}$, $\lim _{n \rightarrow \infty} \eta_{n}=0$. Because the left hand side of (9) is a polynomial of degree 1it holds that $p_{n}+\eta_{n}=0, n \geq 2$. Since $\lim _{n \rightarrow \infty} \eta_{n}=0$, we get $p_{n}=\eta_{n}=0$ when $n$ is large enough. As a consequnce, $\left(1-6 x+7 x^{2}\right) \sum_{n=0}^{\infty} m_{n} x^{n}$ and $\left(1-6 x+7 x^{2}\right) \sum_{n=0}^{\infty} \epsilon_{n} x^{n}$ are polynomials. So we have $$\sum_{n=0}^{\infty} \epsilon_{n} x^{n}=\frac{G(x)}{1-6 x+7 x^{2}}$$ Write the right hand side as $H(x)+\frac{A}{1-\bar{\alpha} x}+\frac{B}{1-\alpha x}$, where $H(x)$ is a polynomial and $A, B$ are constants. Since $\lim _{n \rightarrow \infty} \epsilon_{n}=0$, the radius of convergence of the power series in the left hand side is at least 1. While $\alpha$ and $\bar{\alpha}$ are larger than $1 A$ and $B$ must be zero. Hence $\epsilon_{n}=0$ for large $n$. It follows that $(\sqrt{2}+3)^{n} s=m_{n} \in \mathbb{Z}$ for large $n$. It's a contradiction!