This problem is in the plane. Given a point $Q$ and $n$ distinct points $P_1, P_2, P_n$. For each $1\leq k \leq n$ $|QP_k|$ is different. Furthermore for each $1\leq k\leq n$ consider the circle with center $P_k$ and radius $P_kQ$. Suppose that neither of the circles contain more than two points (these two points are $Q$ and $P_k$), i.e. there is no other point inside or on the circle. Find the maximum value of $n$.
I found it in a book, and I have no idea to solve it. How comes graph theory here?? Help guys please!
The maximal $n$ is 5.
Clearly $Q, P_i$ and $P_j$ cannot be colinear. Let $P_1, P_2, \cdots, P_n$ surrounds $Q$ counterclockwise. Then for two neighboring points $P_i$ and $P_{i+1}$, we must have $\overline{P_iP_{i+1}} > \max\{\overline{P_iQ}, \overline{P_{i+1}Q}\}$, so $\angle P_iQP_{i+1}$ is the largest angle in $\triangle P_iQP_{i+1}$, hence $\angle P_iQP_{i+1} > 60^\circ$.