Vector Ratio

Algebra Level 3

In the xyzxyz-plane above, let O=(0,0,0)O=(0,0,0) be the point of origin, P=(a,b,c)P=(a,b,c), and Q=(a+b,b+c,c+a)Q=(a+b, b+c, c+a), where a,b,ca,b,c are non-zero real numbers.

What is the maximum value of the ratio OQOP?\frac{OQ}{OP}?

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