# 1998 BMO Problems/Problem 2

Revision as of 00:08, 31 December 2007 by Boy Soprano II (talk | contribs) (problem and Karamata solution)

## Problem

(*Bogdan Enescu, Romania*)
If is an integer and are real numbers, prove the inequality:

## Solution

Let us denote , and let us denote . We note at first that

**Lemma.**

*Proof.* Evidently, the two sequences have sum . Let be the least integer such that . We now note that for integers ,
and for integers ,
which is the sum of the smallest terms of the second sequence. Equality holds when . Therefore the first sequence majorizes the second, as was to be proven.

Now, since the function is strictly concave and our two sequences are not equal, by Karmata's Inequality or as desired.