HMIC 2025 Problem 5

Compute the smallest positive integer $k > 45$ for which there exists a sequence $a_{1}$ , $a_{2}$ , $a_{3}$ , ..., $a_{k - 1}$ of positive integers satisfying the following conditions:

$a_{i} = i$ for all integers $1\leq i\leq 45$ $a_{k - i} = i$ for all integers $1\leq i\leq 45$ , and for any odd integer $1\leq n\leq k - 45$ , the sequence $a_{n}$ , $a_{n + 1}$ , ..., $a_{n + 44}$ is a permutation of $\{1,2,\ldots ,45\}$ .