Question

已知a,b,c为实数，且ab/a+b=1/3，bc/b+c=1/4,ca/c+a=1/3,求abc/a+b+c的值

Answer 1

因为ab/(a+b)=1/3，bc/(b+c)=1/4,ca/(c+a)=1/3,
取倒数，得1/a+1/b=3,1/a+1/c=4,1/c+1/a=3
所以1/a=1,1/b=1/c=2
所以abc/(a+b+c)的倒数=4
因此abc/(a+b+c)=1/4

Answer 2

ab/(a+b)=1/3，bc/(b+c)=1/4,ca/(c+a)=1/3,
1/a+1/b=3,1/b+1/c=4,1/c+1/a=3
1/a=1 1/b=2 1/c=2
a=1 b=1/2 c=1/2
abc/(a+b+c)=1*1/2*1/2/(1+1/2+1/2)=1/8

Answer 3

b=1/6，c=1/8，a=1/6.所以abc/a+b+c=1/48+1/6+1/8=15/48=5/16

Answer 4

1/3=20/60 1/4=15/60 1/5=12/60 20+15+12=47 ab/a+b=20/60 bc/b+c ca/c+a=12/60 ab/a+b + bc/b+c + ca/c+a = 47/60 ab+bc+ca/a+b+c=47/30 abc/ab+bc+ca=235/470=1/2