MathDB

4

Part of 2012 Danube Mathematical Competition

Problems(2)

7 elements of set {1,2,3, ...,26}, exist 2 with the same sum

Source: Danube 2012 Junior p4

7/22/2019
Let AA be a subset with seven elements of the set {1,2,3,...,26}\{1,2,3, ...,26\}. Show that there are two distinct elements of AA, having the same sum of their elements.
SumSubsetssetcombinatorics
partition set of n elements in m sets with same sum of elements, criterion

Source: Danube 2012 p4

7/22/2019
Given a positive integer nn, show that the set {1,2,...,n}\{1,2,...,n\} can be partitioned into mm sets, each with the same sum, if and only if m is a divisor of n(n+1)2\frac{n(n + 1)}{2} which does not exceed n+12\frac{n + 1}{2}.
combinatoricsSumSubsetssetdivisor