Extreme workout with Polynomial

Let $P(x)=a_5x^5+a_4x^4+\cdots+a_0$ , then $P'(x)=5a_5x^4+4a_4x^3+\cdots+a_1$ , calculate the composite function: $P'(P(x)) = 5a_5(a_5x^5+\cdots )^4+\cdots$ $P(P'(x))=a_5(5a_5x^4+\cdots)^5+\cdots$

Since the leading term of these two is only $5a_5^5 x^{20}$ and $5^5 a_5^6x^{20}$ , they must be equal, so we ended up $a_5=\dfrac1{625}$ , the desired reciprocal is $\boxed{625}$ .

• This problem is inspired by the book Putnam and Beyond Question No.166.

• Using similar process, any polynomial with degree $n\geq 2$ must have leading coefficient $a_n=\dfrac1{n^{n-1}}$ .