52 lines
831 B
TeX
Executable File
52 lines
831 B
TeX
Executable File
\problemdef{Algebra}{1}{
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\statement{
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Evaluate
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$
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\bigl( 1 - \frac{1}{4} \bigr)
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\bigl( 1 - \frac{1}{9} \bigr)
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\bigl( 1 - \frac{1}{16} \bigr)
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~ ... ~
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\bigl( 1 - \frac{1}{255} \bigr)
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$
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}
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\answer{$\frac{8}{15}$}
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}
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\problemdef{Algebra}{2}{
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\statement{
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$(a + b)(a + b - 1) = ab$ and $a^2 + b^2 = 3$. Find $a^3 + b^3$.
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}
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\answer{3}
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}
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\problemdef{Algebra}{3}{
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\statement{
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Simplify $(2^{62} + 1)/(2^{31} + 2^{16} + 1)$
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}
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\answer{$2^{31} - 2^{16} + 1$}
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}
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\problemdef{Algebra}{4}{
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\statement{
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$x, y, z > 0$ and $xyz = 1$. \par
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Also, $x + 1/z = 5$ and $y + 1/x = 29$.
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Find $z + 1/y$.
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}
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\answer{$z + 1/y = 1/4$}
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}
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\problemdef{Algebra}{5}{
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\statement{
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Factor $x^8 + x^4 + 1$ into four quadratics.
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}
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\answer{$(x^2 - \sqrt{3} x + 1)~(x^2 + \sqrt{3} x + 1)~(x^2 - x + 1)~(x^2 + x + 1)$}
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} |