Vis at
[tex]{\left( {a + \frac{1}{b} - 1} \right)} \left( {b + \frac{1}{c} - 1} \right) +{\left( {b + \frac{1}{c} - 1} \right)} \left( {c + \frac{1}{a} - 1} \right)+{\left( {c + \frac{1}{a} - 1} \right)} \left( {a + \frac{1}{b} - 1} \right) \ge 3[/tex]
Eller
[tex]\sum\limits_{cyc}^{a,b,c} {\left( {a + \frac{1}{b} - 1} \right)} \left( {b + \frac{1}{c} - 1} \right) \ge 3[/tex]
[tex]a,b,c > 0[/tex]
Nok en ulikhet
Moderatorer: Vektormannen, espen180, Aleks855, Solar Plexsus, Gustav, Nebuchadnezzar, Janhaa
Artig oppgave, lett å rote seg bort her... 
[tex]\sum_{cyc}^{a,b,c} (a+\frac{1}{b}-1)(b+\frac{1}{c}-1) \geq \sum_{cyc}^{a,b,c} (2 \sqrt{\frac{a}{b}}-1)(2 \sqrt{\frac{b}{c}}-1)=3[/tex]
Her brukte jeg bare at [tex]x+\frac{1}{y} \geq 2\sqrt{\frac{x}{y}}[/tex] som er AM-GM
EDIT: Sorry folks, rota meg visst bort (ble litt for seint
):
[tex]\sum_{cyc}^{a,b,c} (2 \sqrt{\frac{a}{b}}-1)(2 \sqrt{\frac{b}{c}}-1) \neq 3[/tex]
[tex]\sum_{cyc}^{a,b,c} (a+\frac{1}{b}-1)(b+\frac{1}{c}-1) \geq \sum_{cyc}^{a,b,c} (2 \sqrt{\frac{a}{b}}-1)(2 \sqrt{\frac{b}{c}}-1)=3[/tex]
Her brukte jeg bare at [tex]x+\frac{1}{y} \geq 2\sqrt{\frac{x}{y}}[/tex] som er AM-GM
EDIT: Sorry folks, rota meg visst bort (ble litt for seint
):[tex]\sum_{cyc}^{a,b,c} (2 \sqrt{\frac{a}{b}}-1)(2 \sqrt{\frac{b}{c}}-1) \neq 3[/tex]
Sist redigert av Zivert den 02/09-2008 00:05, redigert 2 ganger totalt.