Problem

Source: INMO 2024/6

Tags: floor function, INMO 2024, INMO



For each positive integer $n \ge 3$, define $A_n$ and $B_n$ as \[A_n = \sqrt{n^2 + 1} + \sqrt{n^2 + 3} + \cdots + \sqrt{n^2+2n-1}\]\[B_n = \sqrt{n^2 + 2} + \sqrt{n^2 + 4} + \cdots + \sqrt{n^2 + 2n}.\]Determine all positive integers $n\ge 3$ for which $\lfloor A_n \rfloor = \lfloor B_n \rfloor$. Note. For any real number $x$, $\lfloor x\rfloor$ denotes the largest integer $N\le x$. Anant Mudgal and Navilarekallu Tejaswi