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max N such not all of the sums a_i+a_j are integers

Source: 2002 Estonia National Olympiad Final Round grade 11 p4

March 14, 2020

Integersnumber theorySummax

Problem Statement

Let

a_1, ... ,a_5

be real numbers such that at least

N

of the sums

a_i+a_j

(

i < j

) are integers. Find the greatest value of

N

for which it is possible that not all of the sums

a_i+a_j

are integers.