[Beatmania]

3. Algebra
ให่ $\frac{a}{b} =x$ $\frac{b}{c} =y$ $\frac{c}{a} =z$
จากจะได้ $x+y+z=xy+yz+zx$
$1-xy-yz-zx+x+y+z-1=0$
$(1-x)(1-y)(1-z)=0$
$\frac{a-b}{b} \frac{b-c}{c} \frac{c-a}{a} =0$
ได้ $a=b$ $b=c$ หรือ $c=a$
WLOG ให้ $a=b$
จะได้
$(\frac{a}{b} )^c+(\frac{b}{c} )^a+(\frac{c}{a} )^b=1+(\frac{a}{c} )^b+(\frac{c}{b} )^a=(\frac{b}{a} )^c+(\frac{c}{b} )^a+(\frac{a}{c} )^b$