Given pairwise different real numbers $a,b,c,d,e$ such that $$ \left\{ \begin{array}{ll} ab + b = ac + a, \\ bc + c = bd + b, \\ cd + d = ce + c, \\ de + e = da + d. \end{array} \right. $$Prove that $abcde=1$.
Source: Polish Math Olympiad 2nd stage 2023 P4
Tags: algebra, system of equations
Given pairwise different real numbers $a,b,c,d,e$ such that $$ \left\{ \begin{array}{ll} ab + b = ac + a, \\ bc + c = bd + b, \\ cd + d = ce + c, \\ de + e = da + d. \end{array} \right. $$Prove that $abcde=1$.