China (Regional)

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Let $n$ be a positive integer. Show that $$\Big(1+\frac{1}{3}\Big)\Big(1+\frac{1}{3^2}\Big)\cdots\Big(1+\frac{1}{3^n}\Big) < 2$$

Let $\alpha$ and $\beta$ be the two roots of the equation $x^2 -x - 1=0$. If $$a_n = \frac{\alpha^n - \beta^n}{\alpha -\beta}\quad(n=1, 2, \cdots)$$ Show that - For any positive integer $n$, it always hold $a_{n+2}=a_{n+1}+a_n$ - Find all positive integers $a, b$ $( a < b )$ satisfying $b\mid a_n-2na^n$ holds for any positive integer $n$