MathDB

22

Part of 2015 Online Math Open Problems

Problems(2)

2014-2015 Spring OMO #22

Source:

4/14/2015
For a positive integer nn let n#n\# denote the product of all primes less than or equal to nn (or 11 if there are no such primes), and let F(n)F(n) denote the largest integer kk for which k#k\# divides nn. Find the remainder when F(1)+F(2)+F(3)+⋯+F(2015#−1)+F(2015#)F(1) + F(2) +F(3) + \dots + F(2015\#-1) + F(2015\#) is divided by 39999913999991.
Proposed by Evan Chen
Online Math Open
2015-2016 Fall OMO #22

Source:

11/18/2015
Let W=…x−1x0x1x2…W = \ldots x_{-1}x_0x_1x_2 \ldots be an infinite periodic word consisting of only the letters aa and bb. The minimal period of WW is 220162^{2016}. Say that a word UU appears in WW if there are indices k≤ℓk \le \ell such that U=xkxk+1…xℓU = x_kx_{k+1} \ldots x_{\ell}. A word UU is called special if Ua,Ub,aU,bUUa, Ub, aU, bU all appear in WW. (The empty word is considered special) You are given that there are no special words of length greater than 2015.
Let NN be the minimum possible number of special words. Find the remainder when NN is divided by 10001000. Proposed by Yang Liu
Online Math Open