\problemdef{Combinatorics}{1}{
	\statement{
		A real estate investor asked a developer to paint 2017 houses so that at least 1000 are green and 1000 are red. What is the maximum number of colors the developer can use?
	}

	\answer{19}
}

\problemdef{Combinatorics}{2}{
	\statement{
		How many ways are there to cut one $3 \times 5$ rectangle into five $1 \times 3$ rectangles?
	}

	\answer{4}
}

\problemdef{Combinatorics}{3}{
	\statement{
		How many different integral solutions $(x, y, z)$ are there to $x+y+z = 20$?
	}

	\answer{$C_{19}^2$}
}


\problemdef{Combinatorics}{4}{
	\statement{
		Consider an uncolored $8 \times 8$ board. How many ways are there to paint the squares black or white so that we end up with exactly 31 black squares, none of which share an edge?
	}

	\answer{68}
}

\problemdef{Combinatorics}{5}{
	\statement{
		Let $A$ be the set of four-digit integers in which the first digit is equal to the sum of the other three. \par
		Let $B$ be the set of four-digit integers in which the last digit is equal to the sum of the other three. \par
		Which set is larger, and by how many elements?
	}

	\answer{$A$, by 54.}
}