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Question Number 217764 by ArshadS last updated on 20/Mar/25
Let a, b, c be distinct real numbers such that (a/(b−c))+(b/(c−a))+(c/(a−b))=0 then prove that (a/((b−c)^2 ))+(b/((c−a)^2 ))+(c/((a−b)^2 ))=0
Leta,b,cbedistinctrealnumberssuchthatab−c+bc−a+ca−b=0thenprovethata(b−c)2+b(c−a)2+c(a−b)2=0
Answered by vnm last updated on 20/Mar/25
Let P(a,b,c)=a(c−a)(a−b)+b(a−b)(b−c)+c(b−c)(c−a) and Q(a,b,c)=a(c−a)^2 (a−b)^2 +b(a−b)^2 (b−c)^2 +c(b−c)^2 (c−a)^2 . Then Q(a,b,c)=P(a,b,c)(−a^2 −b^2 −c^2 +bc+ca+ab). It′s not obvious, of course, but can be checked. So if P(a,b,c)=0 then Q(a,b,c)=0.
LetP(a,b,c)=a(c−a)(a−b)+b(a−b)(b−c)+c(b−c)(c−a)andQ(a,b,c)=a(c−a)2(a−b)2+b(a−b)2(b−c)2+c(b−c)2(c−a)2.ThenQ(a,b,c)=P(a,b,c)(−a2−b2−c2+bc+ca+ab).It′snotobvious,ofcourse,butcanbechecked.SoifP(a,b,c)=0thenQ(a,b,c)=0.

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