Question 122
Let $n \geq 3$ be a positive integer. On a line, the $n$ points $P_{1}, P_{2}, P_{3}, \ldots, P_{n}$ are marked, in this order and equally spaced. Then each of the points must be colored blue or red so that there are no three points $P_{x}, P_{\frac{x+y}{2}}$ and $P_{y}$ of the same color, with $x+y$ even.
a. Show that for $n=8$ there is a way to color the points $P_{1}, P_{2}, \ldots, P_{8}$ satisfying the given condition.
b. Show that no coloring for $n=9$ satisfies the given condition.
a. Show that for $n=8$ there is a way to color the points $P_{1}, P_{2}, \ldots, P_{8}$ satisfying the given condition.
b. Show that no coloring for $n=9$ satisfies the given condition.