\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 $a^2 + b^2 = 3$. 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{$(x^2 - \sqrt{3} x + 1)~(x^2 + \sqrt{3} x + 1)~(x^2 - x + 1)~(x^2 + x + 1)$}
}