Question

已知abc=1,a+b+c=2,a²+b²+c²=3,则1\ab+c-1+1\bc+a-1+1\c

a+b-1的值为

Answer 1

abc = 1
a+b+c=2
a^2 + b^2 + c^2 =3
1=(a+b+c)^2-(a^2+b^2+c^2)
=2(ab+bc+ac)
所以
ab+bc+ac=1/2
abc = 1
a+b+c=2

[1/(ab+c-1)]+[1/(bc+a-1)]+[1/(ca+b-1)]

a+b+c=2
c-1=1-a-b
ab+c-1=ab+1-a-b=(a-1)(b-1)

[1/(ab+c-1)]+[1/(bc+a-1)]+[1/(ca+b-1)]
=1/[(a-1)(b-1)]+1/[(b-1)(c-1)]+1/[(c-1)(a-1)]
=[(a-1)+(b-1)+(c-1)]/[(a-1)(b-1)(c-1)]
=[a+b+c-3]/[abc-(ab+bc+ac)+(a+b+c)-1]
=(-1)/[1-1/2+2-1]
=(-1)/(3/2)
=-2/3

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