HMMT February #0l5v — HMMT February · USAMO Guide - Olympiad Mathematics Learning Platform

HMMT February #0l5v

HMMT FebruaryHard

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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: Let $f$ be a function from nonnegative integers to nonnegative integers such that $f(0) = 0$ and

f(m) = f\left(\left\lfloor \frac{m}{2} \right\rfloor\right) + \left\lceil \frac{m}{2} \right\rceil^2

for all positive integers $m$ . Compute

\frac{f(1)}{1 \cdot 2} + \frac{f(2)}{2 \cdot 3} + \frac{f(3)}{3 \cdot 4} + \dots + \frac{f(31)}{31 \cdot 32}.

(Here, $\lfloor z\rfloor$ is the greatest integer less than or equal to $z$ , and $\lceil z\rceil$ is the least positive integer greater than or equal to $z$ .)