MathDB
floor and square roots

Source: INMO 2024/6

January 21, 2024
floor functionINMO 2024INMO

Problem Statement

For each positive integer n≥3n \ge 3, define AnA_n and BnB_n as An=n2+1+n2+3+⋯+n2+2n−1A_n = \sqrt{n^2 + 1} + \sqrt{n^2 + 3} + \cdots + \sqrt{n^2+2n-1} Bn=n2+2+n2+4+⋯+n2+2n.B_n = \sqrt{n^2 + 2} + \sqrt{n^2 + 4} + \cdots + \sqrt{n^2 + 2n}. Determine all positive integers n≥3n\ge 3 for which ⌊An⌋=⌊Bn⌋\lfloor A_n \rfloor = \lfloor B_n \rfloor. Note. For any real number xx, ⌊x⌋\lfloor x\rfloor denotes the largest integer N≤xN\le x.
Anant Mudgal and Navilarekallu Tejaswi
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