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max of \frac{n!}{n_1!n_2!n_3!n_4!}2^{ {n_1 \choose 2}+...}

Source: Romania IMO TST 1993 2.4

February 17, 2020

combinatoricsCombinationsBinomialmaxinequalities

Problem Statement

For each integer

n > 3

find all quadruples

(n_1,n_2,n_3,n_4)

of positive integers with

n_1 +n_2 +n_3 +n_4 = n

which maximize the expression

\frac{n!}{n_1!n_2!n_3!n_4!}2^{ {n_1 \choose 2}+{n_2 \choose 2}+{n_3 \choose 2}+{n_4 \choose 2}+n_1n_2+n_2n_3+n_3n_4}