Nemotron-IMO-Bench 2026 C9

Let $a_1,\ldots, a_n$ be positive integers such that $$\frac{1}{a_1}+\ldots+\frac{1}{a_n}\leq \frac{1}{2}.$$ The government of Optimistica publishes each year an annual report, with $n$ economic indicators, the values of the $i$th indicator being in the set $\{1,\ldots, a_i\}$. We say the report is optimistic if at least $n-1$ indicators have equal or higher values than in the previous report. Prove that the government can publish annual reports in an infinitely long sequence.