Problem

Source: Spain MO 2024 P4

Tags: algebra, Spain



Let $a,b,c,d$ be real numbers satisfying \[abcd=1\quad \text{and}\quad a+\frac1a+b+\frac1b+c+\frac1c+d+\frac1d=0.\]Prove that at least one of the numbers $ab$, $ac$, $ad$ equals $-1$.