Putnam and Beyond by Răzvan Gelca & Titu Andreescu

Author:Răzvan Gelca & Titu Andreescu
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

while the product of the second pair is

We can see that the first of these expressions exceeds the second by mk . This proves that if the permutation has an inversion, then the product is not minimal. The only permutation without inversions is the identity permutation. By Sturm’s principle, it is the permutation for which the minimum is attained. This minimum is , as claimed.

146. Order the numbers and call the expression from the statement . Note that , which shows that as the variables tend to infinity, so does the expression. This means that the minimum exists. Assume that the minimum is attained at the point . If then there exist indices i and j , , such that are still distinct integers. When substituting these numbers into E the denominator stays constant while the numerator changes by , a negative number, decreasing the value of the expression. This contradicts the minimality. We now look at the case with no gaps: . Then there exists a such that , , . We have

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