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Omni-MATH / Let b(x)=x²+x+1. The polynomial x²⁰¹⁵+x²⁰¹⁴+⋯+x+1 has a unique "base b(x) " representation x²⁰¹⁵+x²⁰¹⁴+⋯+x+1=∑_(k=0)^(N)a_(k)(x)b(x)^(k) where each "digit" a_(k)(x) is either the z…

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problem

Let b(x)=x2+x+1b(x)=x^{2}+x+1. The polynomial x2015+x2014+⋯+x+1x^{2015}+x^{2014}+\cdots+x+1 has a unique "base b(x)b(x) " representation x2015+x2014+⋯+x+1=∑k=0Nak(x)b(x)kx^{2015}+x^{2014}+\cdots+x+1=\sum_{k=0}^{N} a_{k}(x) b(x)^{k} where each "digit" ak(x)a_{k}(x) is either the zero polynomial or a nonzero polynomial of degree less than degb=2\operatorname{deg} b=2; and the "leading digit aN(x)a_{N}(x) " is nonzero. Find aN(0)a_{N}(0).
Plain-text mathematical notation (without MathML)
Let b(x)=x²+x+1. The polynomial x²⁰¹⁵+x²⁰¹⁴+⋯+x+1 has a unique "base b(x) " representation x²⁰¹⁵+x²⁰¹⁴+⋯+x+1=∑_(k=0)^(N)a_(k)(x)b(x)^(k) where each "digit" a_(k)(x) is either the zero polynomial or a nonzero polynomial of degree less than degb=2; and the "leading digit a_(N)(x) " is nonzero. Find a_(N)(0).
Original LaTeX notation
Let $b(x)=x^{2}+x+1$. The polynomial $x^{2015}+x^{2014}+\cdots+x+1$ has a unique "base $b(x)$ " representation $x^{2015}+x^{2014}+\cdots+x+1=\sum_{k=0}^{N} a_{k}(x) b(x)^{k}$ where each "digit" $a_{k}(x)$ is either the zero polynomial or a nonzero polynomial of degree less than $\operatorname{deg} b=2$; and the "leading digit $a_{N}(x)$ " is nonzero. Find $a_{N}(0)$.

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Source and history

task_id
898b7964-a568-5aaa-a2b5-413ba3c27415
task_key
test--898b7964-a568-5aaa-a2b5-413ba3c27415
task_revision_id
1
config / split
/ test

Official source

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