ILL 1987 Problem 14

Given $n$ real numbers $0 < t_1 \leq t_2 \leq \cdots \leq t_n < 1$ , prove that $$(1-t_n^2) \left( \frac{t_1}{(1-t_1^2)^2}+\frac{t_2}{(1-t_2^3)^2}+\cdots +\frac{t_n}{(1-t_n^{n+1})^2} \right) < 1.$$