Question 101
Let $n$ be an integer greater than $12$. Points $P_{1}, P_{2}, \ldots, P_{n}, Q$ in the plane are distinct. Prove that for some $i$, at least $n / 6 - 1$ of the distances
$$
P_{1} P_{i}, P_{2} P_{i}, \ldots, P_{i-1} P_{i}, P_{i+1} P_{i}, \ldots, P_{n} P_{i}
$$
are less than $P_{i} Q$.
$$
P_{1} P_{i}, P_{2} P_{i}, \ldots, P_{i-1} P_{i}, P_{i+1} P_{i}, \ldots, P_{n} P_{i}
$$
are less than $P_{i} Q$.