2022 SSMO Accuracy Round Problems/Problem 3

Problem

Let $A=(6,3,2), B=(2,-9,-6),$ and $O=(0,0,0).$ Suppose that $D$ is a point in space such that $OD$ bisects $\angle{AOB}$ and $O,D,A,B$ are coplanar. In addition, $\angle{DAO}=90^{\circ}.$ If $DO$ can be expressed as $\frac{a\sqrt{b}}{c}$ , where $a$ and $c$ are relatively prime positive integers and $b$ is squarefree, find $a+b+c.$

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2022 SSMO Accuracy Round Problems/Problem 3

Problem

Solution