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(324) 1

Part of 1992 Tournament Of Towns

Problems(1)

TOT 324 1992 Spring A J1 crowded collections of numbers

Source:

6/9/2024
A collection of n>2n > 2 numbers is called crowded if each of them is less than their sum divided by n−1n - 1 . Let {a,b,c,,...}\{a, b, c, ,...\} be a crowded collection of nn numbers whose sum equals SS. Prove that:
(a) each of the numbers is positive,
(b) we always have a+b>ca + b > c,
(c) we always have a+b≥Sn−1a + b \ge \frac{S}{n-1} . (Regina Schleifer)
number theoryalgebra