lean_workbook_plus_64811

prove $\left( {\frac {a}{c}}+{\frac {c}{b}}+{\frac {b}{d}}+{\frac {d}{e}}+{\frac {e}{a}} \right) \left( {\frac {b}{a}}+{\frac {a}{d}}+{\frac {d}{c}}+{\frac {c}{e}}+{\frac {e}{b}} \right)\left( {\frac {d}{b}}+{\frac {b}{a}}+{\frac {a}{c}}+{\frac {c}{e}}+{\frac {e}{d}} \right) \left( {\frac {c}{b}}+{\frac {b}{d}}+{\frac {d}{a}}+{\frac {a}{e}}+{\frac {e}{c}} \right) \left( {\frac {d}{c}}+{\frac {c}{a}}+{\frac {a}{b}}+{\frac {b}{e}}+{\frac {e}{d}} \right)\left( {\frac {a}{b}}+{\frac {b}{c}}+{\frac {c}{d}}+{\frac {d}{e}}+{\frac {e}{a}} \right) \left( {\frac {b}{d}}+{\frac {d}{c}}+{\frac {c}{a}}+{\frac {a}{e}}+{\frac {e}{b}} \right) \left( {\frac {d}{a}}+{\frac {a}{b}}+{\frac {b}{c}}+{\frac {c}{e}}+{\frac {e}{d}} \right) \left( {\frac {d}{b}}+{\frac {b}{c}}+{\frac {c}{a}}+{\frac {a}{e}}+{\frac {e}{d}} \right) \left( {\frac {d}{c}}+{\frac {c}{b}}+{\frac {b}{a}}+{\frac {a}{e}}+{\frac {e}{d}} \right) \left( {\frac {b}{d}}+{\frac {d}{a}}+{\frac {a}{c}}+{\frac {c}{e}}+{\frac {e}{b}} \right) \left( {\frac {c}{d}}+{\frac {d}{a}}+{\frac {a}{b}}+{\frac {b}{e}}+{\frac {e}{c}} \right) \left( {\frac {c}{a}}+{\frac {a}{b}}+{\frac {b}{d}}+{\frac {d}{e}}+{\frac {e}{c}} \right) \left( {\frac {b}{c}}+{\frac {c}{a}}+{\frac {a}{d}}+{\frac {d}{e}}+{\frac {e}{b}} \right)\left( {\frac {a}{c}}+{\frac {c}{d}}+{\frac {d}{b}}+{\frac {b}{e}}+{\frac {e}{a}} \right) \left( {\frac {b}{c}}+{\frac {c}{d}}+{\frac {d}{a}}+{\frac {a}{e}}+{\frac {e}{b}} \right) +\left( {\frac {a}{b}}+{\frac {b}{d}}+{\frac {d}{c}}+{\frac {c}{e}}+{\frac {e}{a}} \right) \left( {\frac {d}{a}}+{\frac {a}{c}}+{\frac {c}{b}}+{\frac {b}{e}}+{\frac {e}{d}} \right) \left( {\frac {c}{d}}+{\frac {d}{b}}+{\frac {b}{a}}+{\frac {a}{e}}+{\frac {e}{c}} \right) \geq 281474976710656\, \left( {\frac {a}{b+c+d+e}}+{\frac {b}{c+d+e+a}}+{\frac {c}{d+e+a+b}}+{\frac {d}{e+a+b+c}}+{\frac {e}{a+b+c+d}} \right) ^{24}$