Question

exista x,y si z pozitive astfel incat { {$x+ \frac{1}{x}$} + {$y+ \frac{1}{y}$} + {$z+ \frac{1}{z}$} } = $\frac{1}{3}$?

Answer 1

$Aceasta~este~o~problema~de~existenta.~Pentru~a~o~demonstra \\ \\ este~suficient~sa~gasim~un~exemplu. \\ \\ Sa~luam,~de~exemplu,~x=y=z=a. \\ \\ Obtinem~astfel:~ \{ \{a+ \frac{1}{a}\}+ \{a+ \frac{1}{a} \}+ \{ a+\frac{1}{a} \} \} = \frac{1}{3}. \\ \\ Echivalent~cu~ \{3 \{{a+ \frac{1}{a} \} \} = \frac{1}{3}.$

$Observam~ca~ \{3 \cdot \{ \frac{1}{9}\} \}= \frac{1}{3},~deci~putem~analiza~cazul \\ \\ \{a+\frac{1}{a}\}=9. \\ \\ Un~exemplu~de~solutie~pentru~a~ar~fi~a=9. \\ \\ Deci~exista~numere~pozitive~x,y~si~z~astfel~incat~relatia~sa~fie~ \\ \\ indeplinita. \\ \\ Observatie:~acestea~nu~sunt~unicele~valori!$