MathNet 2025 — Part 02 (100 complete bilingual questions) Question 89

A subset $S$ of $\{1,2,3,\ldots ,2025\}$ is called balanced if for all elements $a$ and $b$ both in $S$, there exists an element $c$ in $S$ such that $2025$ divides $a + b - 2c$. Compute the number of nonempty balanced subsets.