\problemdef{Algebra}{1}{
	\statement{
		Evaluate
		$
			\bigl( 1 - \frac{1}{4} \bigr)
			\bigl( 1 - \frac{1}{9} \bigr)
			\bigl( 1 - \frac{1}{16} \bigr)
			~ ... ~
			\bigl( 1 - \frac{1}{255} \bigr)
		$
	}
	\answer{$\frac{8}{15}$}
}



\problemdef{Algebra}{2}{
	\statement{
		$(a + b)(a + b - 1) = ab$ and $a^2 + b^2 = 3$. \par
		Find $a^3 + b^3$.
	}
	\answer{3}
}



\problemdef{Algebra}{3}{
	\statement{
		Simplify $(2^{62} + 1)/(2^{31} + 2^{16} + 1)$
	}
	\answer{$2^{31} - 2^{16} + 1$}
}



\problemdef{Algebra}{4}{
	\statement{
		$x, y, z > 0$ and $xyz = 1$. \par
		Also, $x + 1/z = 5$ and $y + 1/x = 29$.

		Find $z + 1/y$.
	}
	\answer{$z + 1/y = 1/4$}
}



\problemdef{Algebra}{5}{
	\statement{
		Factor $x^8 + x^4 + 1$ into four quadratics.
	}
	\answer{$(x^2 - \sqrt{3} x + 1)~(x^2 + \sqrt{3} x + 1)~(x^2 - x + 1)~(x^2 + x + 1)$}
}

\problemdef{Algebra}{6}{
	\statement{
		Sophia bought a Greyhound ticket, but then her plans changed and she sold it back for $\$24$. The percent of the ticket's cost that she lost in the sale is equal to the dollar value of her initial ticket. How much did she buy it for? List all options.
	}

	\answer{$\$40$ or $\$60$}
}

\problemdef{Algebra}{7}{
	\statement{
		How do you cut a cake into 6 pieces so that it can be distributed equally to both three guests and four guests?
	}

	\answer{$\frac{3/12} + \frac{3}{4}$ or $\frac{2}{12} + \frac{2}{6} + \frac{2}{4}$}
}

\problemdef{Algebra}{8}{
	\statement{
		On the first day the grocery store sold $\frac{1}{2}$ of all the geese and half a goose, on the second --- $\frac{1}{3}$ of the remainder and another $\frac{1}{3}$ of the goose, on the third --- $\frac{1}{4}$ of the new remnant and another 3/4 of the goose, on the fourth --- $\frac{1}{5}$ of the remainder and another $\frac{1}{5}$ of the goose. On the fifth day, the store sold the remaining 19 geese. How many geese were there in the store?
	}

	\answer{101}
}