CN-SMO 2016

The sequence $(a_n)$ is defined by $a_1=1,a_2=\frac{1}{2}$ , $$n(n+1) a_{n+1}a_{n}+na_{n}a_{n-1}=(n+1)^2a_{n+1}a_{n-1}(n\ge 2).$$ Prove that $$\frac{2}{n+1}<\sqrt[n]{a_n}<\frac{1}{\sqrt{n}}(n\ge 3).$$

Given any integer $n\geq 3$ . A finite series is called $n$ -series if it satisfies the following two conditions $1)$ It has at least $3$ terms and each term of it belongs to $\{ 1,2,...,n\}$ $2)$ If series has $m$ terms $a_1,a_2,...,a_m$ then $(a_{k+1}-a_k)(a_{k+2}-a_k)<0$ for all $k=1,2,...,m-2$ How many $n$ -series are there $?$

Toss the coin $n$ times, assume that each time, only appear only head or tail Let $a(n)$ denote number of way that head appear in multiple of $3$ times among $n$ times Let $b(n)$ denote numbe of way that head appear in multiple of $6$ times among $n$ times $(1)$ Find $a(2016)$ and $b(2016)$ $(2)$ Find the number of positive integer $n\leq 2016$ that $2b(n)-a(n)\geq 0$

Let $\{ a_n\}$ be a series consisting of positive integers such that $n^2 \mid \sum_{i=1}^{n}{a_i}$ and $a_n\leq (n+2016)^2$ for all $n\geq 2016$ . Define $b_n=a_{n+1}-a_n$ . Prove that the series $\{ b_n\}$ is eventually constant.