THE 68th NMO SELECTION TESTS FOR THE JUNIOR BALKAN MATHEMATICAL OLYMPIAD

For $n=12$ there exists a coloring of the numbers from $\{2,3,\dots ,12\}$ with two colors such that the equation $x+y=z$ has no monochrome solution: we color the numbers from $A=\{2,3,4,11,12\}$ with one color, and the elements of $B=\{5,6,7,8,9,10\}$ with the second color. For $n<12$ we color with the first color the numbers from $\{2,3,\dots ,n\}\cap A$ and with the second one the elements of $\{2,3,\dots ,n\}\cap B$. Again, the equation $x+y=z$ has no monochrome solution. It follows that $n\ge 13$. We prove that, for any coloring with two colors of the elements of the set $\{2,3,\dots ,13\}$, the equation $x+y=z$ has monochrome solutions. Assume the contrary: there exists a coloring such that there is no monochrome solution to the equation. Case 1: If $2$, $3$, $4$ have color 1, then $5=2+3$, $6=2+4$, $7=3+4$ need to have color 2, hence $11=5+6$, $12=5+7$, $13=6+7$ need to be of color 1. But then $2+11=13$, and the equation has a monochrome solution (of color 1).

Case 2: If $2$ and $3$ are of color 1 while $4$ has color 2, then $5=2+3$ has color 2, and $9=4+5$ has color 1. But $3+6=9$, and numbers $3$ and $9$ have color 1, hence $6$ needs to be of color 2. Similarly, as $2+7=9$ and $2$, $9$ have color 1, it follows that $7$ has color 2. If $11$ has color 1, then $2+9=11$ leads to a monochrome solution. If $11$ has color 2, then $5+6=11$ is a monochrome solution.

Case 3: If $2$ and $4$ have color 1 while $3$ has color 2, then $6$ has color 2, $9$ has color 1. As $4$ and $9$ have color 1, we need $5$ to have color 2. Similarly, $2$ and $9$ are of color 1, therefore $7$ has color 2, $8=3+5$ has color 1. But $12=4+8=5+7$, hence we have a monochrome solution (of color 1 or color 2, depending on the color of $12$).

Case 4: $2$ has color 1, $3$ and $4$ have color 2. Then $7$ has color 1. But $2$ and $7$ having color 1 means that $5$ needs to be of color 2. This leads to $9=2+7=4+5$ and, again, regardless on the color of $9$, the equation has a monochrome solution.