KöMaL 2016 A674

Let $a_1, \ldots, a_n>1$ be integers satisfying the inequality $\sum_{i=1}^{n} \frac{1}{a_{i}+1} \le \frac{7}{12}.$ Every year, the government of Optimistica publishes its Annual Report with $n$ economic indicators. For each $i=1,\ldots,n$, the possible values of the $i$-th indicator are $1, 2, \ldots, a_i$. The Annual Report is said to be optimistic if at least $n - 1$ indicators have higher values than in the previous report. Prove that the government can publish optimistic Annual Reports in an infinitely long sequence.