ELMO SL 2014 C4

Let $r$ and $b$ be positive integers. The game of Monis, a variant of Tetris, consists of a single column of red and blue blocks. If two blocks of the same color ever touch each other, they both vanish immediately. A red block falls onto the top of the column exactly once every $r$ years, while a blue block falls exactly once every $b$ years.

(a) Suppose that $r$ and $b$ are odd, and moreover the cycles are offset in such a way that no two blocks ever fall at exactly the same time. Consider a period of $rb$ years in which the column is initially empty. Determine, in terms of $r$ and $b$ , the number of blocks in the column at the end.

(b) Now suppose $r$ and $b$ are relatively prime and $r+b$ is odd. At time $t=0$ , the column is initially empty. Suppose a red block falls at times $t = r, 2r, \dots, (b-1)r$ years, while a blue block falls at times $t = b, 2b, \dots, (r-1)b$ years. Prove that at time $t=rb$ , the number of blocks in the column is $\left\lvert 1+2(r-1)(b+r)-8S \right\rvert$ , where $$S = \left\lfloor \frac{2r}{r+b} \right\rfloor + \left\lfloor \frac{4r}{r+b} \right\rfloor + ... + \left\lfloor \frac{(r+b-1)r}{r+b} \right\rfloor .$$ Proposed by Sammy Luo