Harvard-MIT Mathematics Tournament #0irl

Harvard-MIT Mathematics TournamentHard

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Problem

Problem: Let $\mathcal{C}$ be the hyperbola $y^{2}-x^{2}=1$ . Given a point $P_{0}$ on the $x$ -axis, we construct a sequence of points $\left(P_{n}\right)$ on the $x$ -axis in the following manner: let $\ell_{n}$ be the line with slope 1 passing through $P_{n}$ , then $P_{n+1}$ is the orthogonal projection of the point of intersection of $\ell_{n}$ and $\mathcal{C}$ onto the $x$ -axis. (If $P_{n}=0$ , then the sequence simply terminates.) Let $N$ be the number of starting positions $P_{0}$ on the $x$ -axis such that $P_{0}=P_{2008}$ . Determine the remainder of $N$ when divided by 2008.