Problem

Source: Spanish MO 2022 P2

Tags: Spain, algebra



Let $a,b,c,d$ be four positive real numbers. If they satisfy \[a+b+\frac{1}{ab}=c+d+\frac{1}{cd}\quad\text{and}\quad\frac1a+\frac1b+ab=\frac1c+\frac1d+cd\]then prove that at least two of the values $a,b,c,d$ are equal.