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OlympiadBench / 1872 / Let n be a positive integer, and set N=2^(n). Determine the smallest real…

Problem

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

question

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

Plain-text mathematical notation (without MathML)

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

Original LaTeX notation

Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,

$$
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x
$$

answer type

Expression

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 1872
task_id: acdf9280-b473-5b32-9e55-405f8e8cb855
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--1872
task_revision_id: 3
config / split: OE_TO_maths_en_COMP / train

Official source

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