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RU-ALLRUSSIAN 1969 Problem 4

    Given positive numbers a, b, c, d prove that at least one of the inequalities does not hold: a+\mathrm{a + } b<c+d\mathrm{b< c + d} ; (a+b)(c+d)<ab+cd(\mathrm{a + b})(\mathrm{c + d})< \mathrm{ab + cd} ; (a+b)cd<ab(c+d)(\mathrm{a + b})\mathrm{cd}< \mathrm{ab}(\mathrm{c + d}) .

    Dataset text NuminaMath olympiads. Quoted with source.

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