HMMT 2013 #0jg5

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.

CirclesAngle chasingTriangles

Problem

Problem:

Let $\omega_{1}$ and $\omega_{2}$ be circles with centers $O_{1}$ and $O_{2}$ , respectively, and radii $r_{1}$ and $r_{2}$ , respectively. Suppose that $O_{2}$ is on $\omega_{1}$ . Let $A$ be one of the intersections of $\omega_{1}$ and $\omega_{2}$ , and $B$ be one of the two intersections of line $O_{1} O_{2}$ with $\omega_{2}$ . If $A B = O_{1} A$ , find all possible values of $\frac{r_{1}}{r_{2}}$ .

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