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}
}


\problemdef{Misc}{6}{
	\statement{
		All faces of the cube are painted in different colors (each face is painted with the same color). If you look at this cube from one side, then you can see the blue, white and yellow faces, on the other side you can see the black, blue and red faces, and on the third side you can see the green, black and white faces. Which face is opposite to the white one?
	}

	\answer{Red}
}

\problemdef{Misc}{7}{
	\statement{
		A huge military band performed in a field. First, the musicians lined up in a square. Then they rearranged themselves into a rectangle, and the number of ranks increased by 5. How many musicians were in the orchestra?
	}

	\answer{400}
}