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sum x_i<=3, sum x_i^2 >=1, 3 may be chosen with sum >=1

Source: Argentina 1998 OMA L3 p6

May 13, 2024

inequalitiesalgebra

Problem Statement

Given

n

non-negative real numbers,

n\geq 3

, such that the sum of the

n

numbers is less than or equal to

3

and the sum of the squares of the

n

numbers is greater than or equal to

1

, prove that among the

n

numbers three can be chosen whose sum is greater than or equal to

1