已知x、y、z均为实数,若X+Y+Z≠0,a=X/X+Y,b=Y/Z+X,c=Z/X+Y,求a/a+1,b/b+1,c/c+1的值

题目

已知x、y、z均为实数,若X+Y+Z≠0,a=X/X+Y,b=Y/Z+X,c=Z/X+Y,求a/a+1,b/b+1,c/c+1的值

答案

a/a+1=1/(1+1/a)=1/(1+(z+y)/x)=x/(x+y+z)
b/b+1=1/(1+1/b)=1/(1+(x+z)/y)=y/(x+y+z)
c/c+1=1/(1+1/c)=1/(1+(x+y)/z)=z/(x+y+z)
a/a+1+b/b+1+c/c+1=1