HMMT November #0jej
HMMT NovemberHard
From module Telescoping Sums and Products
Community difficultyNo ratings yetBe the first to rate this problem.
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.
Recurrence relationsTelescoping series
Problem
Problem:
Gary plays the following game with a fair -sided die whose faces are labeled with the positive integers between and , inclusive: if , he stops; otherwise he rolls the die, and starts over with a -sided die, where is the number his -sided die lands on. (In particular, if he gets , he will stop rolling the die.) If he starts out with a -sided die, what is the expected number of rolls he makes?