Harvard-MIT Mathematics Tournament #0iad

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: A cell of a 2-configuration of a set $A$ is a nonempty subset $C$ of $A$ such that i. for any two distinct elements $a, b$ of $C$ , there exists a sequence $c_{0}, c_{1}, \ldots, c_{n}$ of elements of $A$ with $c_{0}=a, c_{n}=b$ , and such that $\left\{c_{0}, c_{1}\right\},\left\{c_{1}, c_{2}\right\}, \ldots,\left\{c_{n-1}, c_{n}\right\}$ are all elements of the 2-configuration, and ii. if $a$ is an element of $C$ and $b$ is an element of $A$ but not of $C$ , there does NOT exist a sequence $c_{0}, c_{1}, \ldots, c_{n}$ of elements of $A$ with $c_{0}=a, c_{n}=b$ , and such that $\left\{c_{0}, c_{1}\right\},\left\{c_{1}, c_{2}\right\}, \ldots,\left\{c_{n-1}, c_{n}\right\}$ are all elements of the 2-configuration. Also, we define a 2-configuration of $A$ to be barren if there is no subset $\left\{a_{0}, a_{1}, \ldots, a_{n}\right\}$ of $A$ , with $n \geq 2$ , such that $\left\{a_{0}, a_{1}\right\},\left\{a_{1}, a_{2}\right\}, \ldots,\left\{a_{n-1}, a_{n}\right\}$ and $\left\{a_{n}, a_{0}\right\}$ are all elements of the 2-configuration. Show that, given any 2-configuration of a set $A$ , every element of $A$ belongs to exactly one cell.