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 $P(x)$ , then clearly these terms are not constant. Thus, $a_1 = 1$ and our second solution is $P(x) = x^k$ .

~bigbrain123

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1975 Canadian MO Problems/Problem 8

Problem 8

Solution 1