HMMT 十一月 2010 · 冲刺赛 · 第 36 题
HMMT November 2010 — Guts Round — Problem 36
题目详情
- [ 25 ] Paul Erd˝ os was one of the most prolific mathematicians of all time and was renowned for his many collaborations. The Erd˝ os number of a mathematician is defined as follows. Erd˝ os has an Erd˝ os number of 0, a mathematician who has coauthored a paper with Erd˝ os has an Erd˝ os number of 1, a mathematician who has not coauthored a paper with Erd˝ os, but has coauthored a paper with a mathematician with Erd˝ os number 1 has an Erd˝ os number of 2, etc. If no such chain exists between Erd˝ os and another mathematician, that mathematician has an Erd˝ os number of infinity. Of the mathematicians with a finite Erd˝ os number (including those who are no longer alive), what is their average Erd˝ os number according to the Erd˝ os Number Project? If the correct answer is X and you write down A , your team will receive max (25 − b 100 | X − A |c , 0) points where b x c is the largest integer less than or equal to x . Page 3 of 3
解析
- [ 25 ] Paul Erd˝ os was one of the most prolific mathematicians of all time and was renowned for his many collaborations. The Erd˝ os number of a mathematician is defined as follows. Erd˝ os has an Erd˝ os number of 0, a mathematician who has coauthored a paper with Erd˝ os has an Erd˝ os number of 1, a mathematician who has not coauthored a paper with Erd˝ os, but has coauthored a paper with a mathematician with Erd˝ os number 1 has an Erd˝ os number of 2, etc. If no such chain exists between Erd˝ os and another mathematician, that mathematician has an Erd˝ os number of infinity. Of the mathematicians with a finite Erd˝ os number (including those who are no longer alive), what is their average Erd˝ os number according to the Erd˝ os Number Project? If the correct answer is X and you write down A , your team will receive max (25 − b 100 | X − A |c , 0) points where b x c is the largest integer less than or equal to x . Answer: 4.65 We’ll suppose that each mathematician collaborates with approximately 20 people (except for Erd˝ os himself, of course). Furthermore, if a mathematician has Erd˝ os number k , then 1 we’d expect him to be the cause of approximately of his collaborators’ Erd˝ os numbers. This is k 2 because as we get to higher Erd˝ os numbers, it is more likely that a collaborator has a lower Erd˝ os number already. Therefore, we’d expect about 10 times as many people to have an Erd˝ os number of 2 than with an Erd˝ os number of 1, then a ratio of 5, 2 . 5, 1 . 25, and so on. This tells us that more mathematicians have an Erd˝ os number of 5 than any other number, then 4, then 6, and so on. If we use this approximation, we have a ratio of mathematicians with Erd˝ os number 1, 2, and so on of about 1 : 10 : 50 : 125 : 156 : 97 : 30 : 4 : 0 . 3, which gives an average Erd˝ os number of 4 . 8. This is close to the actual value of 4.65. Guts Round