HMMT 二月 2014 · 冲刺赛 · 第 36 题
HMMT February 2014 — Guts Round — Problem 36
题目详情
- [ 25 ] We have two concentric circles C and C with radii 1 and 2, respectively. A random chord of C 1 2 2 is chosen. What is the probability that it intersects C ? 1 m Your answer to this problem must be expressed in the form , where m and n are positive integers. If n 25 · X your answer is in this form, your score for this problem will be b c , where X is the total number of Y m teams who submit the answer (including your own team), and Y is the total number of teams who n 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.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT 2014, 22 FEBRUARY 2014 — GUTS ROUND Organization Team Team ID#
解析
- [ 25 ] We have two concentric circles C and C with radii 1 and 2, respectively. A random chord of C 1 2 2 is chosen. What is the probability that it intersects C ? 1 m Your answer to this problem must be expressed in the form , where m and n are positive integers. If n 25 · X your answer is in this form, your score for this problem will be b c , where X is the total number of Y m teams who submit the answer (including your own team), and Y is the total number of teams who n 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.) Answer: N/A The question given at the beginning of the problem statement is a famous problem in probability theory widely known as Bertrand’s paradox. Depending on the interpretation of the phrase “random chord,” there are at least three different possible answers to this question: • If the random chord is chosen by choosing two (uniform) random endpoints on circle C and 2 taking the chord joining them, the answer to the question is 1 / 3. • If the random chord is chosen by choosing a (uniformly) random point P the interior of C (other 2 than the center) and taking the chord with midpoint P , the answer to the question becomes 1 / 4. • If the random chord is chosen by choosing a (uniformly) random diameter d of C , choosing a point P on d , and taking the chord passing through P and perpendicular to d , the answer to the question becomes 1 / 2. (This is also the answer resulting from taking a uniformly random horizontal chord of C .) 2 You can read more about Bertrand’s paradox online at http://en.wikipedia.org/wiki/Bertrand_ paradox_(probability) . We expect that many of the valid submissions to this problem will be equal to 1 / 2 , 1 / 3, or 1 / 4. However, your score on this problem is not based on correctness, but is rather proportional to the number of teams who wrote the same answer as you! Thus, this becomes a problem of finding what is known in game theory as the “focal point,” or “Schelling point.” You can read more about focal points at http://en.wikipedia.org/wiki/Focal_point_(game_theory) or in economist Thomas Schelling’s book The Strategy Of Conflict .