Tessellate S.T.E.M.S - Mathematics - School - Set 1 - Problem 3

Suppose $\{ a_1,a_2, \cdots a_n \}$ be a permutation (rearrangement) of the numbers $\{ 1,2, \cdots n \}, n \geq 3$ such that $a_i \neq i$ , for all $i$ . Which of the following options are correct?

$(a)$ $\sum_{i=1}^n \dfrac{a_i}{|a_i-i|} \leq \dfrac{n(n+1)}{2} - \dfrac{1}{2}$ for only finitely many integers $n \geq 3$

$(b)$ $\sum_{i=1}^n \dfrac{a_i}{|a_i-i|} \leq \dfrac{n(n+1)}{2} - \dfrac{1}{2}$ for no integer $n \geq 3$

$(c)$ $\sum_{i=1}^n \dfrac{a_i}{|a_i-i|} \leq \dfrac{n(n+1)}{2} - \dfrac{1}{2}$ for infinitely many integers $n \geq 3$

$(d)$ $\sum_{i=1}^n \dfrac{a_i}{|a_i-i|} \leq \dfrac{n(n+1)}{2} - \dfrac{1}{2}$ for all integers $n \geq 3$

This problem is a part of Tessellate S.T.E.M.S.