Competition problem — Algebra — Olympiad Maths Prep

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Algebra Difficulty 5.9 AIME, harder Prove it Ukraine

Olesya was given a homework to add two canonical fractions $\frac{a}{b}$ and $\frac{c}{d}$ . Her classmate Andriy who missed the class asked her by phone about the homework, and due to bad connection he heard that they need to add $\frac{b}{a}$ and $\frac{d}{c}$ . After he added them, Andriy asked Olesya for the answer. It turns out that the answers are the same. Was Andriy right in his calculations, if Olesya got an excellent mark, and all 4 fractions, they have been working with are distinct?

Solution

Suppose his calculations are correct, then the following equality holds: $\frac{a}{b} + \frac{c}{d} = \frac{b}{a} + \frac{d}{c}$ or $\frac{ad+bc}{bd} = \frac{bc+ad}{ac}$ . Therefore, we have $bd = ac$ or $\frac{b}{a} = \frac{c}{d}$ , which contradicts to the assumption that all fractions are distinct.

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