MathDB

Let

a_1, . . . , a_n

be positive integers satisfying the inequality

\sum_{i=1}^{n}\frac{1}{a_n}\le \frac{1}{2}

. Every year, the government of Optimistica publishes its Annual Report with n economic indicators. For each

i = 1, . . . , n

,the possible values of the

i-th

indicator are

1, 2, . . . , 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.

Problem Statement