Question

Determine the least positive integer n for which the following condition holds: No matter how the elements of the set of the first n positive integers, i.e. {1,2,…n}, are colored in red or blue, there are (not necessarily distinct) integers x,y,z, and w in a set of the same color such that x+y+z=w.

Details and assumptions:
The phrase not necessarily distinct means that the integers can be repeated. For example, if 1,2,4 are all colored red, then we have 1+1+2=4 which would satisfy the condition.

Answer 1

345

Answer 2

wrong