International Mathematical Olympiad 1985 Problem 6

AlgebraInequalitiesRecurrencesSeq

For every real number x1x_1, construct the sequence x1,x2,…x_1, x_2, \ldots by setting xn+1=xn(xn+1n) for each n≥1.x_{n+1} = x_n\left(x_n + \frac{1}{n}\right) \text{ for each } n \geq 1. Prove that there exists exactly one value of x1x_1 for which 0<xn<xn+1<10 < x_n < x_{n+1} < 1 for every nn.