<s>[INST] Find the value of $\frac{a+b+c}{d+e+f}$ Given real numbers $a, b, c, d, e, f$, such that:
$a^2 + b^2 + c^2 = 25$
$d^2 + e^2 + f^2 = 36$
$ad + be + cf = 30$
What is the value of $\frac{a+b+c}{d+e+f}$?
I've tried combining equations in several ways but haven't gotten very far. Any hints would be appreciated.
 [/INST] By C-S $$(a^2+b^2+c^2)(d^2+e^2+f^2)\geq(ad+be+cf)^2,$$
where the equality occurs for $(a,b,c)||(d,e,f)$, which we got.
Let $(a,b,c)=k(d,e,f).$ 
Thus, $5=|k|6$ and $|k|=\frac{5}{6}$ and
$$\frac{a+b+c}{d+e+f}=\frac{5}{6}$$ or
 $$\frac{a+b+c}{d+e+f}=-\frac{5}{6}$$
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