1998 BMO Problems/Problem 2
(Bogdan Enescu, Romania) If is an integer and are real numbers, prove the inequality:
Let us denote , and let us denote . We note at first that
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.