\( \newcommand{\combin}[2]{{}^{#1}C_{#2} } \newcommand{\cmod}[3]{#1 \equiv #2\left(\bmod {}{#3}\right)} \newcommand{\mdc}[2]{\left( {#1},{#2}\right)} \newcommand{\mmc}[2]{\left[ {#1},{#2}\right]} \newcommand{\cis}{\mathop{\rm cis}} \newcommand{\sen}{\mathop{\rm sen}} \newcommand{\tg}{\mathop{\rm tg}} \newcommand{\cotg}{\mathop{\rm cotg}} \newcommand{\cosec}{\mathop{\rm cosec}} \newcommand{\cotgh}{\mathop{\rm cotgh}} \newcommand{\cosech}{\mathop{\rm cosech}} \newcommand{\sech}{\mathop{\rm sech}} \newcommand{\sh}{\mathop{\rm sh}} \newcommand{\ch}{\mathop{\rm ch}} \newcommand{\th}{\mathop{\rm th}} \newcommand{\senEL}[1]{\mathop{\rm sen}^{#1}} \newcommand{\tgEL}[1]{\mathop{\rm tg}^{#1}} \newcommand{\cotgEL}[1]{\mathop{\rm cotg}^{#1}} \newcommand{\cosecEL}{\mathop{\rm cosec}^{#1}} \newcommand{\shEL}[1]{\mathop{\rm sh^{#1}}} \newcommand{\chEL}[1]{\mathop{\rm ch^{#1}}} \newcommand{\thEL}[1]{\mathop{\rm th^{#1}}} \newcommand{\cotghEL}[1]{\mathop{\rm cotgh^{#1}}} \newcommand{\cosechEL}[1]{\mathop{\rm cosech^{#1}}} \newcommand{\sechEL}[1]{\mathop{\rm sech^{#1}}} \newcommand{\senq}{\senEL{2}} \newcommand{\tgq}{\tgEL{2}} \newcommand{\cotgq}{\cotgEL{2}} \newcommand{\cosecq}{\cosecEL{2}} \newcommand{\cotghq}{\cotghEL{2}} \newcommand{\cosechq}{\cosechEL{2}} \newcommand{\sechq}{\sechEL{2}} \newcommand{\shq}{\shEL{2}} \newcommand{\chq}{\chEL{2}} \newcommand{\arctg}{\mathop{\rm arctg}} \newcommand{\arcsen}{\mathop{\rm arcsen}} \newcommand{\argsh}{\mathop{\rm argsh}} \newcommand{\argch}{\mathop{\rm argch}} \newcommand{\vect}[1]{\overrightarrow{#1}} \newcommand{\tr}[1]{ \textnormal{Tr}\left({#1}\right)} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\H}{\mathbb{H}} \newcommand{\vect}[1]{\overrightarrow{#1}} \newcommand{\Mod}[1]{\ (\mathrm{mod}\ #1)} \)

05/07/2017

Um inteiro às fatias


Problema: Seja $a_n=2-\displaystyle\frac{1}{n^2+\sqrt{n^4+\frac{1}{4}}}$, $n=1,2,...$ .
Mostre que $\sqrt{a_1}+\sqrt{a_2}+...+\sqrt{a_{119}}$ é um inteiro.

Problema proposto por Américo Tavares no facebook, no dia 4 de Julho de 2017.

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