2006
Problem - 3610
Find all pairs of real numbers $(a, b)$ so that there exists a polynomial $P(x)$ with real coefficients and $P(P(x))=x^4-8x^3+ax^2+bx+40$.
Let $P(x)=x^2 + rx +s$. Then $P(P(x))=x^4+2rx^3+(r^2+r+2s)x^2+(2rs+r^2)x+(rs+s+s^2)$. Then
$$2r=-8\implies r=-4$$
and
$$rs+s+s^2=40\implies (s-8)(s+5)=0$$
If $s=-5$ the $(a,b)=(2,56)$. Otherwise, if $s=8$, then $(a,b)=(28,-48)$