Does there exist a sequence of positive integers \( a_1, a_2, \ldots, a_{100} \) such that every number from \( 1 \) to \( 100 \) appears exactly once, and for each \( 1 \leq i \leq 100 \), the condition \[ a_{a_i + i} = i \]holds? Here it is assumed that \( a_{k+100} = a_k \) for each \( 1 \leq k \leq 100 \). Proposed by Mykhailo Shtandenko