# Sayan Condition

Probability Level 2

Determine the least positive integer $n \geq 3$ for which the following condition holds: No matter how the elements of the set of the first $n$ positive integers, i.e. $\{1, 2, \ldots n \}$, 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.

×