1975 Canadian MO Problems/Problem 8

Problem 8

Let $k$ be a positive integer. Find all polynomials \[P(x) = a_0+a_1x+\cdots+a_nx^n\] where the $a_i$ are real, which satisfy the equation \[P(P(x)) = \{P(x)\}^k\].

1975 Canadian MO (Problems)
Preceded by
Problem 7
1 2 3 4 5 6 7 8 Followed by
"Last Question"



Solution 1

Let $f(n)$ be the degree of polynomial $n$. We begin by noting that $f(P(x)) = k$. This is because the degree of the LHS is $f(P(x))^{f(P(x))}$ and the RHS is $f(P(x))^k$. Now we split $P(x)$ into two cases.

In the first case, $P(x)$ is a constant. This means that $c = c^k \Longrightarrow c\in \{0,1\}$ or $c\in \{0,1,-1\}$ if $k$ is even.

In the second case, $P(x)$ is nonconstant with coefficients of $a_1,a_2,\hdots a_{k+1}$. If we divide by $P(x)$ on both sides, then we have that $a_1 + \frac{a_2}{P(x)} + \frac{a_2}{P(x)^2} \hdots \frac{a_{k+1}}{P(x)^{k+1}} = 1$. This can only be achieved if $a_2,a_3, \hdots a_{k+1} = 0$. This is because if we factor out a $\frac{1}{P(x)}$, then clearly these terms are not constant. Thus, $a_1 = 1$ and our second solution is $P(x) = x^k$.

~bigbrain123

Invalid username
Login to AoPS