Question 5
Let $n>1$ be an integer. Let $T$ consist of the integer-coordinate points $(x,y,z)$ with $1\le x,y,z\le n$. Color points so that if $A=(x_0,y_0,z_0)$ is colored, a point $B=(x_1,y_1,z_1)$ with $x_1\le x_0$, $y_1\le y_0$, $z_1\le z_0$ cannot be colored. Find the largest possible number of colored points.
Editorial clarification: $B$ must mean a point distinct from $A$. The literal self-inclusive reading forbids every colored point. The solution records that reading and proves the intended distinct-point extremum.
Editorial clarification: $B$ must mean a point distinct from $A$. The literal self-inclusive reading forbids every colored point. The solution records that reading and proves the intended distinct-point extremum.