HMMT 2013 #0jia

HMMT 2013Hard

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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.

Chinese remainder theoremInverses mod nPrime numbers

Problem

Problem:

Let $m$ be an odd positive integer greater than $1$ . Let $S_{m}$ be the set of all non-negative integers less than $m$ which are of the form $x+y$ , where $x y-1$ is divisible by $m$ . Let $f(m)$ be the number of elements of $S_{m}$ .

a. Prove that $f(m n)=f(m) f(n)$ if $m, n$ are relatively prime odd integers greater than $1$ .

b. Find a closed form for $f\left(p^{k}\right)$ , where $k>0$ is an integer and $p$ is an odd prime.

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