Question 136
Every positive integer is colored by blue or red. Prove that there is a sequence $\{a_n\}$ which has infinite terms, and $a_1 \lt a_2 \lt \dots$ are positive integers, such that $a_1$, $\frac{a_1+a_2}{2}$, $a_2$, $\frac{a_2+a_3}{2}$, $a_3$, $\dots$ is a positive integer sequence with the same color.