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Least number n for which there exists permutations

Source: Romanian IMO Team Selection Test TST 1987, problem 5

September 25, 2005

inequalitiesrearrangement inequalityalgebra proposedalgebra

Problem Statement

Let

A

be the set

\{1,2,\ldots,n\}

n\geq 2

. Find the least number

n

for which there exist permutations

\alpha

\beta

\gamma

\delta

of the set

A

with the property:

\sum_{i=1}^n \alpha(i) \beta (i) = \dfrac {19}{10} \sum^n_{i=1} \gamma(i)\delta(i) .

Marcel Chirita