Miscellaneous Problems

Problem 20. How many numbers are there in the set {2, 4, 6, . . . , 2006}, that is, in the set of all strictly positive natural numbers between 2 and 2006? What is (2 + 4 +···+ 2006) − (1 + 3 +···+ 2005), that is, what is the difference between the sum of all the odd positive integers up to 2005 and the sum of all the even positive integers up to (and including) 2006?

Problem 19. Bobby was playing on the elevator of a very tall building. Starting from the floor where he was, he went five floors up, four down, three up, four up, and two down. If he is now in the 30th floor, what was the original floor from where he started?

Problem 18. How many addition signs should be put between digits of the number 987654321 and where should we put them to get a total of 99?

Problem 17. You have a seven-inch gold bar, that is already segmented into seven equal pieces. You are allowed to make two cuts to it. How can you pay an employee that demands to be paid one gold piece daily for the seven days that he works for you?

Problem 16. How many different sums can be made when two non-necessarily distinct numbers from the set {1, 3, 4, 5, 7} are taken?