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 $P(P(2))$.
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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 $P(b)=P(P(2))$.
George Lowther
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1Several people seem to be answering this one at the same time (how to select a single accepted answer?). Actually, it's a common question that keeps popping up on various maths forums. – George Lowther Aug 26 '10 at 22:40
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For what it's worth, Steve got there a few seconds before me. – George Lowther Aug 26 '10 at 22:43
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1Perhaps you can wait for the 3rd upvote and then delete. You will get a badge :-) – Aryabhata Aug 26 '10 at 22:48
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Yeah, I saw that you deleted your answer, but I don't see any issue with having two the same. I just voted Steve up and pointed out that he did get there before before me in the comment, to be fair to him. – George Lowther Aug 26 '10 at 22:50
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1Oh, I like badges :) – George Lowther Aug 26 '10 at 22:51
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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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2I don't "have" to, I want to :). I do it on all the other forums, so force of habit. – Aug 27 '10 at 19:08