MathDB

29

Part of 2013 Online Math Open Problems

Problems(2)

2012-2013 Winter OMO #29

Source:

1/16/2013
Let ϕ(n)\phi(n) denote the number of positive integers less than or equal to nn that are relatively prime to nn, and let d(n)d(n) denote the number of positive integer divisors of nn. For example, ϕ(6)=2\phi(6) = 2 and d(6)=4d(6) = 4. Find the sum of all odd integers n≤5000n \le 5000 such that n∣ϕ(n)d(n)n \mid \phi(n) d(n).
Alex Zhu
Online Math Opennumber theoryrelatively prime
2013-2014 Fall OMO #29

Source:

10/30/2013
Kevin has 255255 cookies, each labeled with a unique nonempty subset of {1,2,3,4,5,6,7,8}\{1,2,3,4,5,6,7,8\}. Each day, he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie, and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the cookie labeled with {1,2}\{1,2\}, he eats that cookie as well as the cookies with {1}\{1\} and {2}\{2\}). The expected value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in the form mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm + n.
Proposed by Ray Li
Online Math Openprobabilityexpected value