HMMT 2014 #0jl2 — HMMT 2014 · USAMO Guide - Olympiad Mathematics Learning Platform

HMMT 2014 #0jl2

HMMT 2014Hard

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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:

We have two concentric circles $C_{1}$ and $C_{2}$ with radii $1$ and $2$ , respectively. A random chord of $C_{2}$ is chosen. What is the probability that it intersects $C_{1}$ ?

Your answer to this problem must be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are positive integers. If your answer is in this form, your score for this problem will be $\left\lfloor\frac{25 \cdot X}{Y}\right\rfloor$ , where $X$ is the total number of teams who submit the answer $\frac{m}{n}$ (including your own team), and $Y$ is the total number of teams who submit a valid answer. Otherwise, your score is $0$ . (Your answer is not graded based on correctness, whether your fraction is in lowest terms, whether it is at most $1$ , etc.)