HMMT February 2020 #0kdl

HMMT February 2020Hard

From module Generating Functions (Intro)

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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.

Generating functionsCatalan numbers, partitionsAlgebraic properties of binomial coefficients

Problem

Problem: Anastasia is taking a walk in the plane, starting from $(1,0)$ . Each second, if she is at $(x, y)$ , she moves to one of the points $(x-1, y)$ , $(x+1, y)$ , $(x, y-1)$ , and $(x, y+1)$ , each with $\frac{1}{4}$ probability. She stops as soon as she hits a point of the form $(k, k)$ . What is the probability that $k$ is divisible by $3$ when she stops?

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