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 military band was performing 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}
}


\problemdef{Misc}{8}{
	\statement{
		Out of five coins, three coins are real and two are fake. Both fake coins weigh the same, but each weighs less than a real coin. What is the least number of times you have to use the balance scale to guarantee that you can find at least one real coin?
	}

	\answer{1}
}


\problemdef{Misc}{9}{
	\statement{
		Out of 81 coins one is fake and 80 are genuine. A fake coin weighs less than a genuine coin. What is the least number of times you can use a balance scale to find the fake coin?
	}

	\answer{4}
}


\problemdef{Misc}{10}{
	\statement{
		The numbers $1, 2, 3, ..., 99, 100$ are written on the board. It is allowed to erase any two numbers a and b and write the number $a + b - 1$ instead. What number can remain on the board after 99 such operations?
	}

	\answer{4951}
}


\problemdef{Misc}{11}{
	\statement{
		Numbers from 1 to 2021 are written on the board. With one operation it is allowed to erase two numbers and write their positive difference instead. At some point there will be only one number left on the board. What is the minimal possible value of this number?
	}

	\answer{1}
}


\problemdef{Misc}{12}{
	\statement{
		On the island of knights, knaves and spies, you come across three people. One wears blue, one wears red, and one wears green. You know that one is a knight, one is a knave, and one is a spy. ``Who is the spy?'' you ask. \par

		The man wearing blue says, ``That man in red is the spy.'' \par
		The man wearing red says, ``No, the man in green is the spy.'' \par
		The man wearing green says, ``No, the man in red is in fact the spy. \par

		Knights always tell the truth, knaves always lie, and spies do whatever they want. Who is the spy? Who is the knight and who is the knave?
	}

	\answer{Blue is knave, Red is knight, Green is spy.}
}

\problemdef{Misc}{13}{
	% Induction
	\statement{
		Find a formula for $1 + 2 + ... + n$.
	}

	\answer{$\frac{n(n+1)}{2}$}
}

\problemdef{Misc}{14}{
	% Induction
	\statement{
		Find a formula for $(1 \times 2) + (2 \times 3) +... + (n \times (n + 1))$.
	}

	\answer{$\frac{n(n+1)(n+2)}{3}$}
}

\problemdef{Misc}{15}{
	% Induction
	\statement{
		$x + \frac{1}{x} = 10$. Find $x^2 + \frac{1}{x^2}$, $x^3 + \frac{1}{x^3}$, and $x^4 + \frac{1}{x^4}$
	}

	\answer{98, 970, and 9602}
}


\problemdef{Misc}{16}{
	% Induction
	\statement{
		Let $F_n$ be the $n$th Fibonacci number. $F_0 = 0, F_1 = 1$. \par
		How many of $F_0, F_1, ..., F_{2021}$ are even?
	}

	\answer{674}
}

\problemdef{Misc}{17}{
	% Induction
	\statement{
		How many ways are there to split a $2 \times 10$ into $1 \times 2$ or $2 \times 1$ dominoes?
	}

	\answer{89}
}

\problemdef{Misc}{18}{
	\statement{
		Split the shape below into six equal parts. \par
		Each part must have the same shape, but they may be rotated.

		\begin{center}
			\begin{tikzpicture}[scale = 0.4]
				\foreach \x in {0, ..., 7} {
				\foreach \y in {0, ..., 7} {
					\draw[draw=black] (\x,\y) rectangle +(1,1);
				}}

				\draw[draw=black] (0,8) rectangle +(1,1);
				\draw[draw=black] (1,8) rectangle +(1,1);
				\draw[draw=black] (-1,1) rectangle +(1,1);
				\draw[draw=black] (-1,0) rectangle +(1,1);
				\draw[draw=black] (6,-1) rectangle +(1,1);
				\draw[draw=black] (7,-1) rectangle +(1,1);
				\draw[draw=black] (8,6) rectangle +(1,1);
				\draw[draw=black] (8,7) rectangle +(1,1);
			\end{tikzpicture}
		\end{center}
	}

	\solution{
		\begin{center}
		\begin{tikzpicture}[scale = 0.4]
			\fill[yellow] (-1,0) rectangle +(4,2);
			\fill[yellow] (0,1) rectangle +(1,4);
			\fill[yellow] (0,4) rectangle +(2,1);

			\fill[red] (6,-1) rectangle +(2,4);
			\fill[red] (3,0) rectangle +(4,1);
			\fill[red] (3,0) rectangle +(1,2);

			\fill[blue] (4,1) rectangle +(2,4);
			\fill[blue] (1,2) rectangle +(4,1);
			\fill[blue] (1,2) rectangle +(1,2);

			\fill[cyan] (2,3) rectangle +(2,4);
			\fill[cyan] (3,5) rectangle +(4,1);
			\fill[cyan] (6,4) rectangle +(1,2);

			\fill[green] (0,5) rectangle +(2,4);
			\fill[green] (1,7) rectangle +(4,1);
			\fill[green] (4,6) rectangle +(1,2);

			\fill[gray] (5,6) rectangle +(4,2);
			\fill[gray] (7,3) rectangle +(1,4);
			\fill[gray] (6,3) rectangle +(2,1);

			\foreach \x in {0, ..., 7} {
			\foreach \y in {0, ..., 7} {
				\draw[draw=black] (\x,\y) rectangle +(1,1);
			}}

			\draw[draw=black] (0,8) rectangle +(1,1);
			\draw[draw=black] (1,8) rectangle +(1,1);
			\draw[draw=black] (-1,1) rectangle +(1,1);
			\draw[draw=black] (-1,0) rectangle +(1,1);
			\draw[draw=black] (6,-1) rectangle +(1,1);
			\draw[draw=black] (7,-1) rectangle +(1,1);
			\draw[draw=black] (8,6) rectangle +(1,1);
			\draw[draw=black] (8,7) rectangle +(1,1);
		\end{tikzpicture}
		\end{center}
	}
}