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From a St. Petersburg school olympiad, 11th grade.

Prove or disprove: a non constant polynomial $P$ with non-negative integer coefficients is uniquely determined by its values $P(2)$ and $P(P(2))$.

2 Answers 2

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True. If $P(x)=a_nx^n+\cdots+a_1x+a_0$ then each of the coefficients are less then $b\equiv P(2)$. Each of these coefficients can then be read off from the base-b expansion of $P(b)=P(P(2))$.

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    Oh, I like badges :)2010-08-26
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Look at $P(P(2))$ in base $P(2)$. The nth place is the coefficient of $x^n$.

Steve

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    I don't "have" to, I want to :). I do it on all the other forums, so force of habit.2010-08-27