Školjka

Let $n$ be a positive integer and let $u_1, u_2, \ldots u_n$ be positive integers not larger than $2^k$ for some integer $k \geq 3$. A $\emph{representation}$ of a non-negative integer $t$ is a sequence of non-negative integers $a_1, a_2, \ldots, a_n$ such that $$t = a_1u_1 + a_2u_2 + \ldots + a_nu_n$$ Prove that if a non-negative integer $t$ has a representation, then it also has a representation where less than $2k$ of the numbers $a_1, a_2, \ldots, a_n$ are non-zero.