Question

Aratati ca daca a, b, c apartin Q astfel incat 1/a+1/b=1/c, atunci √(a^2+b^2+c^2 ) apartine Q.

Answer 1

[tex] \frac{1}{a}+\frac{1}{b}=\frac{a+b}{ab}=\frac{S}{P}=\frac{1}{c}\Rightarrow S=\frac{P}{c}\\ \sqrt{a^2+b^2+c^2}=\sqrt{(a+b)^2-2ab+c^2}=\sqrt{\frac{P^2}{c^2}-2P+c^2}=\sqrt{(\frac{P}{c}-c)^2}\\ =|\frac{P}{c}-c|\in \mathbb{Q}[/tex]