EGMO 2025 Problem 1

For a positive integer $N$ , let $c_{1}< c_{2}< \dots < c_{m}$ be all the positive integers smaller than $N$ that are coprime to $N$ . Find all $N\geq 3$ such that

$$\gcd (N,c_{i} + c_{i + 1})\neq 1$$

for all $1\leq i\leq m - 1$

Here $\gcd (a,b)$ is the largest positive integer that divides both $a$ and $b$ . Integers $a$ and $b$ are coprime if $\gcd (a,b) = 1$ .