2021/067) Let a,b,c be three distinct integers and P be a polynomial with integer coefficients. Show that in this case the conditions P(a)=b,P(b)=c,P(c)=a cannot be satisfied simultaneously.

The polynomial is P(x)

we have m-n divides  P(m) - p(n)

Let the given condition is true

So $a-b | P(a) - p(b)$

or $a-b | b- c$

Similarly

$b - c | c- a$

And $ c- a | a - b$

From above 3 have

$a-b | b-c | c- a | a-b$

So all are same hence a = b= c which is a contradiction.

Or they are -1/+1 and this is also a contradiction 

so condition can not be satisfied simultaneously