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## Main Question or Discussion Point

I was told to move my problem solving questions here instead of cluttering up the Homework Help forums. I will use this thread to post questions related to the Putnam Exam. For all of the problems that follow, please JUST GIVE HINTS.

Problem 170 from Putnam and Beyond:

Let

[tex]P_n(x) = (x^n-1)(x^{n-1}-1)\cdots(x-1), \mbox{ for n \geq 1}[/tex].

Prove that for [itex]\mbox{n \geq 2}, P'_n(x)[/itex] is divisible by

[tex]P_{\lfloor n/2 \rfloor}(x)[/tex]

in the ring of polynomials with integers coefficients.

We can use the division algorithm but then I don't know how to show that the remainder is 0. I'm not really sure whether it is enough to show that all the roots of [tex]P_{\lfloor n/2 \rfloor}(x)[/tex] are also roots of P'_n(x) since we are only working in the Z[x]? The roots of P'_n(x) are just the rth roots of unity for all r <= n. Anyone have a hint?

Problem 170 from Putnam and Beyond:

Let

[tex]P_n(x) = (x^n-1)(x^{n-1}-1)\cdots(x-1), \mbox{ for n \geq 1}[/tex].

Prove that for [itex]\mbox{n \geq 2}, P'_n(x)[/itex] is divisible by

[tex]P_{\lfloor n/2 \rfloor}(x)[/tex]

in the ring of polynomials with integers coefficients.

We can use the division algorithm but then I don't know how to show that the remainder is 0. I'm not really sure whether it is enough to show that all the roots of [tex]P_{\lfloor n/2 \rfloor}(x)[/tex] are also roots of P'_n(x) since we are only working in the Z[x]? The roots of P'_n(x) are just the rth roots of unity for all r <= n. Anyone have a hint?

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