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.