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Omni-MATH / Let n be an even positive integer. Let p be a monic, real polynomial of degree 2n; that is to say, p(x)=x^(2n)+a_(2n−1)x^(2n−1)+⋯+a₁x+a₀ for some real coefficients a₀,…,a_(2n−1). S…

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problem

Let nn be an even positive integer. Let pp be a monic, real polynomial of degree 2n2n; that is to say, p(x)=x2n+a2n−1x2n−1+⋯+a1x+a0p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0 for some real coefficients a0,…,a2n−1a_0, \dots, a_{2n-1}. Suppose that p(1/k)=k2p(1/k) = k^2 for all integers kk such that 1≤|k|≤n1 \leq |k| \leq n. Find all other real numbers xx for which p(1/x)=x2p(1/x) = x^2.
Plain-text mathematical notation (without MathML)
Let n be an even positive integer. Let p be a monic, real polynomial of degree 2n; that is to say, p(x)=x^(2n)+a_(2n−1)x^(2n−1)+⋯+a₁x+a₀ for some real coefficients a₀,…,a_(2n−1). Suppose that p(1/k)=k² for all integers k such that 1≤|k|≤n. Find all other real numbers x for which p(1/x)=x².
Original LaTeX notation
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n$; that is to say, $p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0$ for some real coefficients $a_0, \dots, a_{2n-1}$. Suppose that $p(1/k) = k^2$ for all integers $k$ such that $1 \leq |k| \leq n$. Find all other real numbers $x$ for which $p(1/x) = x^2$.

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task_id
f2c8f681-b01c-5ebc-98c2-860733ca37cd
task_key
test--f2c8f681-b01c-5ebc-98c2-860733ca37cd
task_revision_id
1
config / split
/ test

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