In this note, I have a selection of questions for you to prove. It is for all levels to try. The theme is combinatorics. Good luck.
1) is a subset of with the property that no two elements of differ by . What is the maximum number of elements that can have?
2) Brilli the Ant and Brian Till play a game by taking turns removing some stones from a pile. The number of stones removed at each turn must be a divisor of the number of stones in the pile at the start of that turn, and no player is ever allowed to take all the stones in the pile. The person who cannot move is the loser.
If they start with 2008 stones and Brilli goes first, who has the winning strategy?
3) How many sequences of 30 terms are there, such that the following three conditions are satisfied:
no sequence has three consecutive terms equal to each other
every term of the sequence is equal to or
the sum of all terms of each sequence is greater than 8?
4) Let , where is a positive integer greater than or equal to . A subset is full if it has the following three properties:
has at least elements
The elements of form an arithmetic progression
The arithmetic progression cannot be extended in either direction using elements of .
How many full subsets of are there?
5) On an chess board we place an Xon the bottom left corner. All other squares are blank. Any row or column of squares is called a playable set if either
it contains exactly two squares with an X in each, one of which must be the middle square of the playable set, or
it contains exactly two blank squares, one of which must be the middle square of the playable set.
At any stage it is permitted to choose any playable set and simultaneously change every X into a blank, and every blank into a square with an X. After a finite number of such changes, Brilli has reached a configuration with exactly one square containing an X. Which square or squares could it be?
6) Let be a positive integer and let be an integer such that . Consider all the element subsets of the set . For each such element subset, consider its smallest member. Prove that the average of these smallest numbers equals .