MathDB

3

Part of 2008 May Olympiad

Problems(2)

n such that 1010... 101 is prime

Source: XIV May Olympiad (Olimpiada de Mayo) 2008 L2 P3

9/19/2022
In numbers 1010...1011010... 101 Ones and zeros alternate, if there are nn ones, there are n−1n -1 zeros (n≥2n \ge 2 ).Determine the values of nn for which the number 1010...1011010... 101, which has nn ones, is prime.
number theoryprime
1-2008 written on blackboard, replace two numbers with their difference

Source: XIV May Olympiad (Olimpiada de Mayo) 2008 L1 P3

9/22/2022
On a blackboard are written all the integers from 11 to 20082008 inclusive. Two numbers are deleted and their difference is written. For example, if you erase 55 and 241241, you write 236236. This continues, erasing two numbers and writing their difference, until only one number remains. Determine if the number left at the end can be 20082008. What about 20072007? In each case, if the answer is affirmative, indicate a sequence with that final number, and if it is negative, explain why.
combinatoricsgame