INMO – 1986 | A Problem on Polynomials

Q) If $P(x)$ is a polynomial with integer coefficients and $a,b,c$ three distinct integers, then show that it is impossible to have $P(a)=b, P(b)=c, P(c)=a$

Solution:

The solution revolves around an important property of integer coefficient Polynomials,

For any 2 integers $x,y$, $P(x)-P(y)$ is always divisible by $x-y$.

So from this we have $a-b$ divides $P(a)-P(b)$ which implies $a-b$ divides $b-c$, $b-c$ divides $c-a$, $c-a$ divides $a-b$.

Once we have this, the remaining solution is trivial. We see that the above scenario is possible when $|a-b|=|b-c|=|c-a|$. Let’s assume $a>b>c$, thus we have

$a-b=b-c=a-c$, which violates the condition that $a \ne b\ne c$.