The real numbers $a,b,c,d$ satisfy the equations: $$\begin{cases} a =\sqrt{45-\sqrt{21-a}} \\ b =\sqrt{45+\sqrt{21-b}}\\ c =\sqrt{45-\sqrt{21+c}}\ \\ d=\sqrt{45+\sqrt{21+d}} \end {cases}$$Prove that $abcd = 2004$.
Source: Switzerland - Swiss TST 2004 p7
Tags: system of equations, Product, algebra
The real numbers $a,b,c,d$ satisfy the equations: $$\begin{cases} a =\sqrt{45-\sqrt{21-a}} \\ b =\sqrt{45+\sqrt{21-b}}\\ c =\sqrt{45-\sqrt{21+c}}\ \\ d=\sqrt{45+\sqrt{21+d}} \end {cases}$$Prove that $abcd = 2004$.