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King's Maths School

Challenge 414: A Sum-set

Another challenging challenge from Y13 student Adem!

Adem recently sent me an email claiming the following:

"Take a set of n+1 positive integers which sum to 2n. It is always possible to form a subset of those integers with sum n."

For instance, {1, 1, 1, 2, 3} is a set of 5 integers which sum to 8. I can form the subset {1, 3} which has a sum of 4.

Is Adem right? Can you prove his claim, or find a counterexample?

Submit your solution

Please do send in your solution to this problem to weeklymaths@kcl.ac.uk You can scan or photograph your written work, or type your solutions. If this is your first weekly maths challenge solution, please include your year group and the name of the school you attend. We'll be happy to provide feedback on your solution, assuming that you are in year 11 or below. If you are older than this, we hope you enjoy trying the problems and reviewing your solutions against those we publish on the website.