Q55 Matemática  (IMO Longlists 1988)

Suponha ${\alpha }_i>0,{\beta }_i>0$ para $1\le i\le n,n>1$ e que $$\sum _{i=1}^n{\alpha }_i=\sum _{i=1}^n{\beta }_i=\pi .$$ Prove que $$\sum _{i=1}^n\frac{\cos ({\beta }_i)}{\mathrm{sen}({\alpha }_i)}\le \sum _{i=1}^n\mathrm{cotg}({\alpha }_i).$$