2014 IMO Shortlist A2

Define the function $f:(0,1) \rightarrow(0,1)$ by $$f(x)= \begin{cases}x+\frac{1}{2} & \text { if } x<\frac{1}{2} \\ x^{2} & \text { if } x \geq \frac{1}{2}\end{cases}$$ Let $a$ and $b$ be two real numbers such that $0<a<b<1$. We define the sequences $a_{n}$ and $b_{n}$ by $a_{0}=a, b_{0}=b$, and $a_{n}=f\left(a_{n-1}\right), b_{n}=f\left(b_{n-1}\right)$ for $n>0$. Show that there exists a positive integer $n$ such that $$\left(a_{n}-a_{n-1}\right)\left(b_{n}-b_{n-1}\right)<0 .$$ (Denmark)

定义函数 $f:(0,1) \rightarrow(0,1)$ $$f(x)= \begin{cases}x+\frac{1}{2} & \text { if } x<\frac{1}{2} \\ x^{2} & \text { if } x \geq \frac{1}{2}\end{cases}$$ 设 $a$ 和 $b$ 为两个实数，使得 $0<a<b<1$。我们通过 $a_{0}=a、b_{0}=b$ 和 $a_{n}=f\left(a_{n-1}\right)、b_{n}=f\left(b_{n-1}\right)$(对于 $n>0$)定义序列 $a_{n}$ 和 $b_{n}$。证明存在一个正整数 $n$ 使得 $$\left(a_{n}-a_{n-1}\right)\left(b_{n}-b_{n-1}\right)<0 。$$(丹麦)