MathDB

1

Part of 2013 239 Open Mathematical Olympiad

Problems(2)

Double permutations

Source: 239 2013 S1

8/7/2020
Consider all permutations of natural numbers from 11 to 100100. A permutation is called \emph{double} when it has the following property: If you write this permutation twice in a row, then delete 100100 numbers from them you get the remaining numbers 1,2,3,…,1001, 2, 3, \ldots , 100 in order. How many \emph{double} permutations are there?
Divisors mod 2013

Source: 239 2013 J1

8/7/2020
Among the divisors of a natural number nn, we have numbers such that when they are devided by 20132013, give us remainders 1001,1002,…,20121001, 1002, \ldots, 2012. Prove that among the divisors of the number n2n^2, there exist numbers such that when they are divided by 20132013, give us reminders 1,2,3,…,20121, 2, 3, \ldots, 2012.