BAMO #0hyr — BAMO · USAMO Guide - Olympiad Mathematics Learning Platform

BAMO #0hyr

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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:

In the $O$ - $E$ game, a round starts with player $A$ paying $c$ cents to player $B$ . Then $A$ secretly arranges the numbers $1, 3, 5, 7, 9, 11, 13$ in some order as a sequence $a_{1}, a_{2}, \ldots, a_{7}$ , and $B$ secretly arranges $2, 4, 6, 8, 10, 12, 14$ as a sequence $b_{1}, b_{2}, \ldots, b_{7}$ . Finally, the players show their sequences and $B$ pays $A$ one cent for each $i$ in $X=\{1,2,3,4,5,6,7\}$ such that $a_{i}<b_{i}$ . This finishes the round. What number $c$ would make the game fair? (The game is fair if the total payments by $A$ to $B$ equals the total payments by $B$ to $A$ after all possible distinct rounds are played exactly once.)