MathDB
Two bounds to find

Source: ISL 2020 A1

July 20, 2021
algebraIMO ShortlistInequalityIMO Shortlist 2020

Problem Statement

Version 1. 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 . Version 2. 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 .
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