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CZ-SK-MO 2025 Problem 1

    Real numbers a,b,c,da,b,c,d satisfy a+b+c+d=0,1a+1b+1c+1d=0.a+b+c+d=0, \frac1a+\frac1b+\frac1c+\frac1d=0. How many equalities ab=cd,ac=bd,ad=bcab=cd, ac=bd, ad=bc can simultaneously hold? Determine all possibilities.

    Dataset text AI-MO AoPS. Quoted with source.

    Official text: Czech-Slovak Mathematical Olympiad 2025 official paper, page 1.

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