IMO 1959–1966 - Longlisted Problems (individual years unknown) Question 2

(GDR) Given $n$ positive real numbers $a_1,a_2,\ldots,a_n$ such that $a_1a_2\cdots a_n=1$, prove that
$$(1+a_1)(1+a_2)\cdots(1+a_n)\ge2^n.$$