China Western Mathematical Olympiad

It is well-known that there are positive real numbers $\beta ,\gamma$ such that $$\overrightarrow{OA}+\beta \overrightarrow{OB}+\gamma \overrightarrow{OC}=\vec{0}.$$ So for positive integer $k$, we have $$k\overrightarrow{OA}+k\beta \overrightarrow{OB}+k\gamma \overrightarrow{OC}=\vec{0}.$$ Let $m(k)=[k\beta ]$, $n(k)=[k\gamma ]$, where $[x]$ is the biggest integer which is less than or equal to $x$, and $\{x\}=x-[x]$.

Assume $T$ is an integer larger than $\max \left\{\frac{1}{\beta },\frac{1}{\gamma }\right\}$. Then the sequences $\{m(kT)\mid k=1,2,\dots \}$ and $\{n(kT)\mid k=1,2,\dots \}$ are increasing, and $$\begin{array}{rl}& |kT\overrightarrow{OA}+m(kT)\overrightarrow{OB}+n(kT)\overrightarrow{OC}|\\ & =|-\{kT\beta \}\overrightarrow{OB}-\{kT\gamma \}\overrightarrow{OC}|\\ & \le |\overrightarrow{OB}|\cdot \{kT\beta \}+|\overrightarrow{OC}|\cdot \{kT\gamma \}\\ & \le |\overrightarrow{OB}|+|\overrightarrow{OC}|.\end{array}$$ This shows there exists infinitely many vectors such that $$kT\overrightarrow{OA}+m(kT)\overrightarrow{OB}+n(kT)\overrightarrow{OC},$$ whose endpoint is in a circle $O$ of radius $|\overrightarrow{OB}|+|\overrightarrow{OC}|$. So there are two of the vectors, such that the distance between the endpoints of the two vectors is less than $\frac{1}{2007}$, this means there exists two integers $k_1<k_2$, such that $$\begin{array}{rl}& |(k_{2}T\overrightarrow{OA}+m(k_{2}T)\overrightarrow{OB}+n(k_{2}T)\overrightarrow{OC})\\ & -(k_{1}T\overrightarrow{OA}+m(k_{1}T)\overrightarrow{OB}+n(k_{1}T)\overrightarrow{OC})|\\ & <\frac{1}{2007}.\end{array}$$ So, if we let $p=(k_2-k_1)T$, $q=m(k_{2}T)-m(k_{1}T)$, $r=n(k_{2}T)-n(k_{1}T)$, then $p,q,r$ are integers, and $$|p\overrightarrow{OA}+q\overrightarrow{OB}+r\overrightarrow{OC}|<\frac{1}{2007}.$$