Question here: https://www.codechef.com/practice/course/4-star-difficulty-problems/DIFF1900/problems/SEGDIV?tab=statement

I'm intrigued about this submission over here: https://www.codechef.com/viewsolution/82916450

I have no idea why it works. The poster also just left out the proof part and simply wrote, "It's difficult to prove it". Is there any proof of this or is just conveniently satisfies for the given question's constraints? (n <= 500)