Vector Ratio

Relevant wiki: Muirhead Inequality

$(a+b)^2+(b+c)^2 +(c+a)^2 = 2(a^2 + b^2 + c^2) + 2(ab + bc + ca)$ .

According to Muirhead's Inequality , the sequence (2,0) majorizes (1,1), and so $a^2 + b^2 \geq 2ab$ . Similarly, $b^2 + c^2 \geq 2bc$ and $c^2 + a^2 \geq 2ca$ .

Thus, $2(a^2 + b^2 + c^2) \geq 2(ab + bc + ca)$ .

Then, $4(a^2 + b^2 + c^2) \geq 2(a^2 + b^2 + c^2) + 2(ab + bc + ca) = (a+b)^2+(b+c)^2 +(c+a)^2$ .

Since $a^2 + b^2 + c^2 > 0$ , then $4 \geq \dfrac{(a+b)^2+(b+c)^2 +(c+a)^2}{a^2 + b^2 + c^2} > 0$ .

Therefore, $2 \geq \dfrac{\sqrt{(a+b)^2+(b+c)^2 +(c+a)^2}}{\sqrt{a^2 + b^2 + c^2}} = \dfrac{OQ}{OP}$ .

As a result, the maximum of this ratio is $\boxed{2}$ which will occur if and only if $a=b=c$ , where the equality holds. In other words, the maximum ratio will be presented along the diagonal of a cubic structure only.