若abc=1,求证：ab+a+1分之1+bc+b+1分之1+ca+c+1分之1=1

Question

若abc=1,求证：ab+a+1分之1+bc+b+1分之1+ca+c+1分之1=1

Answer 1

反复将abc替换为1,或用1替换abc
具体解法：
1/(ab+a+1)+1/(bc+b+1)+1/(ca+c+1)
=1/(ab+a+abc)+1/(bc+b+1)+1/(ca+c+abc)
=1/a(b+1+bc)+1/(bc+b+1)+1/c(a+1+ab)
=1/a(b+1+bc)+1/(bc+b+1)+1/c(a+abc+ab)
=1/a(b+1+bc)+1/(bc+b+1)+1/ac(1+bc+b)
=abc/a(b+1+bc)+abc/(bc+b+1)+abc/ac(1+bc+b)
=bc/(b+1+bc)+abc/(bc+b+1)+b/(1+bc+b)
=bc/(b+1+bc)+1/(bc+b+1)+b/(1+bc+b)
=(bc+1+b)/(b+1+bc)
=1