title: "IMO 1983 SL 21" description: "Find the greatest integer less than or equal to 21983" date: "2026-05-29T15:05:12+07:00" tags: ["imo", "shortlist", "mathematics", "olympiad"] categories: ["mathematics"] year: 1983 type: "shortlist" number: 21 origin: "SWE" weight: 198300021 draft: false

IMO 1983 SL 21

Origin: SWE

Problem

Find the greatest integer less than or equal to 21983 k=1 k1/1983−1.

Solution

Using the identity an −bn = (a −b) n−1  m=0 an−m−1bm with a = k1/n and b = (k −1)1/n one obtains 1 <  k1/n −(k −1)1/n nk1−1/n for all integers n > 1 and k \geq1. This gives us the inequality k1/n−1 < n
k1/n −(k −1)1/n if n > 1 and k \geq1. In a similar way one proves that n
(k + 1)1/n −k1/n < k1/n−1 if n > 1 and k \geq1. Hence for n > 1 and m > 1 it holds that n m  k=1  (k + 1)1/n −k1/n < m  k=1 k1/n−1 < n m  k=2  k1/n −(k −1)1/n

• 1, or equivalently, n  (m + 1)1/n −1

< m  k=1 k1/n−1 < n  m1/n −1

The choice n = 1983 and m = 21983 then gives 1983 < 21983  k=1 k1/1983−1 < 1984. Therefore the greatest integer less than or equal to the given sum is 1983.