1. Let be a non-increasing sequence so that . Prove that .
And this problem is for the younger kids. (So give them a chance to solve it).
2. Given a positive integer define a -partition to be a sum of positive integers which sum to . For example, . The following are -partitions. and and . Notice that are considered distinct*. Say you a given a specific . And given a specific value of , can you find the total number of -partitions of this integer, with a formula?** Now try to see how many partitions (again not counting order) exist for a given integer (the answer is really supprising).
Hint: Review your Combinatorics formula for this one.
*)This is done to simplify this problem. When these are not considered distinct this forms a problem from number theory called the "partition problem". It is a very complicated problem.
**)The problem is not whether you can. But rather how you can. If it was the later the answer would be "yes" turning this problem into a worthless problem.