BxMO 2025 Problem 4

Let $a_{0},a_{1},\ldots ,a_{10}$ be integers such that, for each $i\in \{0,1,\ldots ,2047\}$ , there exists a subset $S\subseteq \{0,1,\ldots ,10\}$ with

$$\sum_{j\in S}a_{j}\equiv i\pmod {2048}.$$

Show that for each $i\in \{0,1,\ldots ,10\}$ , there is exactly one $j\in \{0,1,\ldots ,10\}$ such that $a_{j}$ is divisible by $2^{i}$ but not by $2^{i + 1}$ .

Note: $\sum_{j\in S}a_{j}$ is the summation notation, for instance, $\sum_{j\in \{2,5\}}a_{j} = a_{2} + a_{5}$ , while, for the empty set $\varnothing$ , one defines $\sum_{j\in \varnothing}a_{j} = 0$ .