BMO 2017

Let the colors be $d_1$, $d_2$, $d_3$, $\dots$, $d_n$. Look at the coordinates $$(k,0+(n+1)abr),(k,ab+(n+1)abr),(k,2ab+(n+1)abr),\dots ,(k,nab+(n+1)abr)$$ for integers $k$ and $r$. By the pigeonhole principle there are two points of the same color. For every pair $(k,r)$ we say the color $d_i$ is $(k,r)$-good if at least two coordinates $$(k,0+(n+1)abr),(k,ab+(n+1)abr),(k,2ab+(n+1)abr),\dots ,(k,nab+(n+1)abr)$$ are colored by color $d_i$. Fixing $r$ and taking $k=0,ab,2ab,\dots ,n^{2}ab$ get that some color, say $d_1$, was $(k,r)$-good for at least $n+1$.

Among the $n+1$ pairs $(x,y)$ there exists two which share the same $x$ coordinate. We call such quadruple $r$-great. In every $r$-great quadruple there are two triangles whose vertices are all the same color and whose two sides are divisible by $ab$. Taking $$r=0,1,2,\dots ,n\left(c\left(\left(\genfrac{}{}{0ex}{}{n+1}{3}\right)\left(\genfrac{}{}{0ex}{}{n^2+1}{3}\right)+1\right)+1\right)+1$$ we get that there is one color which is in a $r$-great quadruple for at least $$c\left(\left(\genfrac{}{}{0ex}{}{n+1}{3}\right)\left(\genfrac{}{}{0ex}{}{n^2+1}{3}\right)+1\right)+1$$ different values of $r$. Let this color be $d_1$. Since there are less than $\left(\genfrac{}{}{0ex}{}{n+1}{3}\right)\left(\genfrac{}{}{0ex}{}{n^2+1}{3}\right)$ possible triangles in any $r$-great quadruple (among $c\left(\left(\genfrac{}{}{0ex}{}{n+1}{3}\right)\left(\genfrac{}{}{0ex}{}{n^2+1}{3}\right)+1\right)+1$ $r$-great quadruples with the color $d_1$) we get that there are $c+1$ triangles which are the same and the same color $d_1$ and with two sides divisible by $ab$. This concludes the problem.