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Irrational numbers an equal sets.

Source: St. Petersburg MO 2000, 11th grade, P7

April 22, 2023
algebrairrational numberSets

Problem Statement

It is known that for irrational numbers α\alpha, β\beta, γ\gamma, δ\delta and for any positive integer nn the following is true: [nα]+[nβ]=[nγ]+[nδ][n\alpha]+[n\beta]=[n\gamma]+[n\delta] Does this mean that sets {α,β}\{\alpha,\beta\} and {γ,δ}\{\gamma,\delta\} are equal? (As usual [x][x] means the greatest integer not greater than xx).
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