2014 IMO Shortlist N1

Let $n \geq 2$ be an integer, and let $A_{n}$ be the set $$A_{n}=\left\{2^{n}-2^{k} \mid k \in \mathbb{Z}, 0 \leq k<n\right\} .$$ Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_{n}$. (Serbia)

令 $n \geq 2$ 为整数，并令 $A_{n}$ 为集合 $$A_{n}=\left\{2^{n}-2^{k} \mid k \in \mathbb{Z}, 0 \leq k<n\right\} 。$$ 确定不能写为 $A_{n}$ 的一个或多个(不一定不同)元素之和的最大正整数。 (塞尔维亚)