USA IMO #0ibk

USA IMOHard

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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.

Inclusion-exclusionGenerating functionsEnumeration with symmetryRoots of unityComplex numbers

Problem

Let $n$ be a positive integer. Consider sequences $a_0, a_1, \dots, a_n$ such that $a_i \in \{1, 2, \dots, n\}$ for each $i$ and $a_n = a_0$ .

a. Call such a sequence good if for all $i = 1, 2, \dots, n$ , $a_i - a_{i-1} \not\equiv i \pmod n$ . Suppose that $n$ is odd. Find the number of good sequences.

b. Call such a sequence great if for all $i = 1, 2, \dots, n$ , $a_i - a_{i-1} \not\equiv i, 2i \pmod n$ . Suppose that $n$ is an odd prime. Find the number of great sequences.

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