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Omni-MATH / Find all real-coefficient polynomials f(x) which satisfy the following conditions: [b]i.[/b] f(x)=a₀x^(2n)+a₂x^(2n−2)+⋯+a_(2n−2)x²+a_(2n),a₀>0; [b]ii.[/b] …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Find all real-coefficient polynomials f(x)f(x) which satisfy the following conditions: [b]i.[/b] f(x)=a0x2n+a2x2n−2+⋯+a2n−2x2+a2n,a0>0f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0; [b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}$; [b]iii.[/b] All the roots of f(x)f(x) are imaginary numbers with no real part.
Plain-text mathematical notation (without MathML)
Find all real-coefficient polynomials f(x) which satisfy the following conditions:

[b]i.[/b] f(x)=a₀x^(2n)+a₂x^(2n−2)+⋯+a_(2n−2)x²+a_(2n),a₀>0;
[b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left(
\begin{array}{c}
2n\\
n\end{array} \right) a_0 a_{2n}$;
[b]iii.[/b] All the roots of f(x) are imaginary numbers with no real part.
Original LaTeX notation
Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions:

[b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2}
x^2 + a_{2n}, a_0 > 0$;
[b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left(
\begin{array}{c}
2n\\
n\end{array} \right) a_0 a_{2n}$;
[b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.

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Source and history

task_id
8fd89143-6146-5231-ae8a-ae4d0c958f19
task_key
test--8fd89143-6146-5231-ae8a-ae4d0c958f19
task_revision_id
1
config / split
/ test

Official source

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