Q1 Matemática  (Tournament Of Towns 1990)

Para cada número natural $n$ prove que $${\left(1+\frac{1}{2}+...+\frac{1}{n}\right)}^2+{\left(\frac{1}{2}+...+\frac{1}{n}\right)}^2+...+{\left(\frac{1}{n-1}+\frac{1}{2}\right)}^2+{\left(\frac{1}{n}\right)}^2=2n-\left(1+\frac{1}{2}+...+\frac{1}{n}\right)$$(S. Manukian, Yerevan)