Problem 167 (difficulty: 9/10)

Define recursively the sequence \(\displaystyle x_{n+1}=x_n\left(x_n+\frac1n\right)\) for any \(\displaystyle x_1\). Show that there is exactly one \(\displaystyle x_1\) for which \(\displaystyle 0<x_n<x_{n+1}<1\) for any \(\displaystyle n\).

(IMO 1985/6)

Give me another random problem!