Show that if a polynomial \(P(x)\) has an integer root (i.e: a root that is a whole number), then it can’t be that \(P(0)\) and \(P(1)\) are both odd.
Solve the equation \[\left(x^2-3x+3\right)^2-3\left(x^2-3x+3\right)+3=x\]
(USO 1974) Let \(a,b,c\) be three distinct integers, and let \(P(x)\) be a polynomial whose coefficients are all integers. Prove that it is not possible that the following three conditions hold at the same time: \(P(a)=b, P(b)=c,\) and \(P(c)=a\).
For a polynomial \(P(x)=ax^2+bx+c\), consider the following two kinds of transformations:
Swap coefficients \(a\) and \(c\). Hence the polynomial \(P(x)\) becomes \(cx^2+bx+a\) after this transformation.
For any number \(t\) of your choice, change the variable \(x\) into \(x+t\). For example, with the choice of \(t=1\), after this transformation, the polynomial \(x^2+x+1\) becomes \((x+1)^2+(x+1)+1=x^2+3x+3\).
Is it possible, using only a sequence of these two transformations, to change the polynomial \(x^2-x-2\) into the polynomial \(x^2-x-1\)?