Matematické Fórum

stuart clark · 09. 11. 2011 19:15

Find largest positive value of $n$ in $n.\left(\frac{abc}{ab+bc+ca}\right)\leq (a+b)^2+(a+b+4c)^2$

where $a\;,b\;,c\in\mathbb{R}$

laszky · 09. 03. 2018 17:03

Such $n$ does not exist.

We prove it by contradiction: assume that such largest $n$ exists and choose $a=b=c=\frac{n}{k}$ for some $k>0$ .

Then the inequality changes into

$n \cdot \frac{\left(\frac{n}{k}\right)^3}{3\left(\frac{n}{k}\right)^2} \leq 4\left(\frac{n}{k}\right)^2 + 36\left(\frac{n}{k}\right)^2$ ,

which can be rewritten as

$\frac{n^2}{3k}\leq \frac{40n^2}{k^2}$ .

This inequality holds only for $k\leq120$ . For $k>120$ we obtain a contradiction.

Matematické Fórum

#1 09. 11. 2011 19:15

largest value of n

#2 09. 03. 2018 17:03

Re: largest value of n

Zápatí