"Those countries did not exist forty years ago."
Translation:Ty země před čtyřiceti lety neexistovaly.
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lety here is instrumental plural and not nominative, because that case is required after the preposition před.
The rule that genitive is used after numbers higer than 4 only holds for nominative and accusative. In oblique cases (any case other than nom/voc/acc), both the numeral and the noun take the plural form of the corresponding case (notice here that čtyřiceti is an inflected form, the base form being čtyřicet).