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MATH / Let P(x) be a polynomial of degree 2011 such that P(1)=0, P(2)=1, P(4)=2, …, P(2²⁰¹¹)=2011. Then the coefficient of x in P(x) can be expressed in the form a−(1)/(b^(c)),where a, b,…

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problem

Let P(x)P(x) be a polynomial of degree 2011 such that P(1)=0,P(1) = 0, P(2)=1,P(2) = 1, P(4)=2,P(4) = 2, …,\dots, P(22011)=2011.P(2^{2011}) = 2011. Then the coefficient of xx in P(x)P(x) can be expressed in the form a−1bc,a - \frac{1}{b^c},where a,a, b,b, cc are positive integers, and bb is prime. Find a+b+c.a + b + c.
Plain-text mathematical notation (without MathML)
Let P(x) be a polynomial of degree 2011 such that P(1)=0, P(2)=1, P(4)=2, …, P(2²⁰¹¹)=2011.  Then the coefficient of x in P(x) can be expressed in the form
a−(1)/(b^(c)),where a, b, c are positive integers, and b is prime.  Find a+b+c.
Original LaTeX notation
Let $P(x)$ be a polynomial of degree 2011 such that $P(1) = 0,$ $P(2) = 1,$ $P(4) = 2,$ $\dots,$ $P(2^{2011}) = 2011.$  Then the coefficient of $x$ in $P(x)$ can be expressed in the form
\[a - \frac{1}{b^c},\]where $a,$ $b,$ $c$ are positive integers, and $b$ is prime.  Find $a + b + c.$

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
2ff704cc-9d3a-5aa3-bd3b-e0f3592a0087
task_key
intermediate~5falgebra--test--2ff704cc-9d3a-5aa3-bd3b-e0f3592a0087
task_revision_id
1
config / split
intermediate_algebra / test

Official source

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