Let $S=\{(i,j) \vert i,j=1,2,\ldots ,100\}$ be a set consisting of points on the coordinate plane. Each element of $S$ is colored one of four given colors. A subset $T$ of $S$ is called colorful if $T$ consists of exactly $4$ points with distinct colors, which are the vertices of a rectangle whose sides are parallel to the coordinate axes. Find the maximum possible number of colorful subsets $S$ can have, among all legitimate coloring patters.
Problem
Source: 2019 CWMI P3
Tags: rectangle, combinatorics
13.08.2019 14:23
So, should $|T|=4$ or should it contain $4$ points with the desired property?
13.08.2019 16:32
15.08.2019 02:21
answer is 9375000
27.08.2019 02:05
CantonMathGuy wrote:
Hi, I know that your solution seems to be correct. But I don’t really understand how you get that expression for the number of rainbow Ls. Btw, $24 * 25^2 =15000$
25.11.2019 21:53
Solution please
27.11.2019 19:40
CantonMathGuy wrote:
My interpretation/guess is that they want $|T|=4$ because $T$ "consists of exactly $4$ points..." If they want $|T| \geq 4$ then better wording would have been $T$ "contains..." Also, if they want $ |T| \geq 4$ then the word "exactly" would be useless and confusing.
04.02.2023 11:29
Similar Problem: All-Russian 2000 Grade 11 P8
04.02.2023 22:11
Yay nice problem