Ta có :
`x/{y+z+t} = y/{z+t+x} = z/{t+x+y} = t/{x+y+z}`
$⇒$ `x/{y+z+t} + 1 = y/{z+t+x} + 1 = z/{t+x+y} + 1 = t/{x+y+z} + 1`
$⇔$ `{x+y+z+t}/{y+z+t} = {x+y+z+t}/{z+t+x} = {x+y+z+t}/{t+x+y} ={x+y+z+t}/{x+y+z} `
Nếu $x+y+z+t=0$ thì:
$x+y = -(z+t)$ $⇒$ `{x+y}/{z+t} = {z+t}/{x+y} = -1`
$y+z = - (t+x)$ $⇒$ `{y+z}/{t+x} = {t+x}/{y+z} = -1`
$⇒$ $P = -1 + (-1) + (-1) + (-1) = -4$
$⇒$ $P ∈ Z$ ($1$)
Nếu $x+y+z+t \neq 0$ thì:
`y+z+t = z+t+x= t+x+y= x+y+z`
$⇒ x = y= z = t$
$⇒$ `{x+y}/{z+t} = {z+t}/{x+y} = {y+z}/{t+x} = {t+x}/{y+z} = 1`
$⇒$ $P = 1 + 1 + 1 + 1 = 4$
$⇒$ $P ∈ Z$ ($2$)
Từ ($1$);($2$) $⇒$ $P ∈ Z$ ($đ.p.c.m$)