VTMO 2026 - Grade 8 - Semifinal - SET 06 - Self-study Question 7

Partition $1,2,\ldots,100$ into disjoint sets $A=\{a_1,\ldots,a_{50}\}$ and $B=\{b_1,\ldots,b_{50}\}$, where $a_1\lt \cdots\lt a_{50}$ and $b_1>\cdots>b_{50}$. Prove $\sum_{i=1}^{50}|a_i-b_i|=50^2$.