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LIMIT 2020 P4

Source: LIMIT 2020

April 11, 2020
pigeonhole principle

Problem Statement

color=#000 Let a1,a2,...,ana_1,a_2,...,a_n be n real numbers. Show that there exists some real number α\alpha such that a1+α,a2+α,...,an+αa_1+\alpha,a_2+\alpha,...,a_n+\alpha are all irrational. \\ color=#000 Prove that such a satetement is not valid if all these are rquired to be rational. \\ Hint (given in question) : Use Pigeon Hole Principle
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