# Notice on POW 2019-12

POW 2019-12 is still open and anyone who first submits a correct solution will get the full credit.

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# Solution: 2019-13 Property R

Let $$A_{a, b} = \{ (x, y) \in \mathbb{Z}^2 : 1 \leq x \leq a, 1 \leq y \leq b \}$$. Consider the following property, which we call Property R:

“If each of the points in $$A$$ is colored red, blue, or yellow, then there is a rectangle whose sides are parallel to the axes and vertices have the same color.”

Find the maximum of $$|A_{a, b}|$$ such that $$A_{a, b}$$ has Property R but $$A_{a-1, b}$$ and $$A_{a, b-1}$$ do not.

The best solution was submitted by 하석민 (수리과학과 2017학번). Congratulations!

Here is his solution of problem 2019-13.