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Omni-MATH / [i]Version 1[/i]. Let n be a positive integer, and set N=2^(n). Determine the smallest real number a_(n) such that, for all real x, root[N]((x^(2N)+1)/(2))≤a_(n)(x−1)²+x. [i]Versio…

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problem

[i]Version 1[/i]. Let nn be a positive integer, and set N=2nN=2^{n}. Determine the smallest real number ana_{n} such that, for all real xx, x2N+12N≤an(x−1)2+x. \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . [i]Version 2[/i]. For every positive integer NN, determine the smallest real number bNb_{N} such that, for all real xx, x2N+12N≤bN(x−1)2+x. \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
Plain-text mathematical notation (without MathML)
[i]Version 1[/i]. Let n be a positive integer, and set N=2^(n). Determine the smallest real number a_(n) such that, for all real x,
root[N]((x^(2N)+1)/(2))≤a_(n)(x−1)²+x.
[i]Version 2[/i]. For every positive integer N, determine the smallest real number b_(N) such that, for all real x,
root[N]((x^(2N)+1)/(2))≤b_(N)(x−1)²+x.
Original LaTeX notation
[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
[i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]

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Source and history

task_id
489e9e7d-f17c-5c4e-a1f1-f582cdd7511f
task_key
test--489e9e7d-f17c-5c4e-a1f1-f582cdd7511f
task_revision_id
1
config / split
/ test

Official source

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