handouts/Problems/problems/misc.tex

\problemdef{Misc}{1}{
	\difficulty{1}

	\statement{
		A carpenter cut a chessboard into $1 \times 1$ squares in 70 minutes. \par
		How long will it take him to cut it into $2 \times 2$ squares?
	}

	\answer{30}
}


\problemdef{Misc}{2}{
	\difficulty{2}


	\statement{
		There are two kinds of books on a shelf: those on permissible magic and those on black magic. Two books on permissible magic cannot be set between exactly three other books, and two books on black magic may not stand next to each other. \par
		What is the maximal amount of books that may be placed on the shelf?
	}

	\answer{8}
}


\problemdef{Misc}{3}{
	\difficulty{3}

	\statement{
		The numbers $1 ... 9$ are arranged in a $3 \times 3$ grid. \par
		The sum of each row and column is then computed. \par
		What is the maximum number of consecutive integers one may find in the set of these sums?
	}

	\answer{5}
}


\problemdef{Misc}{4}{
	\difficulty{4}

	\statement{
		16 rugby teams participate in a regional championship. Each pair of teams plays against each other twice. The 8 teams with the most wins will proceed to the national championship. If there is a tie in this ranking, the tied teams will draw lots. \par
		Assume a rugby game can never tie. What is the minimum number of wins a team needs to guarantee a spot in the nationals?
	}

	\answer{23}
}




\problemdef{Misc}{5}{
	\difficulty{5}

	\statement{
		Five boxes are filled with pastries. We know that box C contains a third of the pastries in E, and that B contains two times more than C and E combined. A contains half the number of pastries in E, and a tenth of those in D. Box B contains four times more pastries than D. \par
		What is the minimal possible positive number of pastries in all the boxes put together?
	}

	\answer{310}
}