Nemotron-IMO-Bench 2026 C36

Let $n>1$ be an integer and let $a_1,\ldots, a_n$ be positive integers. Suppose that we have an unlimited supply of coins, each with value in $\{a_1,\ldots, a_n\}$ and that the sum $a_1+a_2+\ldots+a_n$ can be collected in only one way, by taking one coin of each value.\ a) Give an example of such integers $a_1,\ldots, a_n$ such that $a_1+a_2+\ldots+a_n<n\cdot 2^n$.\ b) Prove that $$\min (a_1,\ldots, a_n)\geq 2^{n-1}.$$