Question

Aratati daca a^2/b^2+b^2/c^2+c^2/a^2=a/c+b/a+c/b, atunci a=b=c

Answer 1

$Avem~de~aratat~ca~daca~ \Sigma \frac{a^2}{b^2} =\Sigma \frac{a}{c} ~atunci~a=b=c. \\ -------------------------- \\ Folosim~identitatea~\Sigma~ x^2 \geq\Sigma~x,pentru~orice~x,y,z~reale. \\ \Sigma \frac{a^2}{b^2} =\Sigma( \frac{a}{b})^2 \geq \Sigma \frac{a}{b}\cdot \frac{b}{c} =\Sigma \frac{a}{c}. \\ Cu~egalitate~pentru~a=b=c.$

Aceste tipuri de ecuatii (care aduc a inegalitati) se rezolva evident intai demonstrand cazul de inegalitate iar apoi rezumandu-ne la cel de egalitate.