USAMO 1972 #5August 4, 2009

Posted by lumixedia in Problem-solving.
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USAMO 1972 #5. A given convex pentagon ${ABCDE}$ has the property that the area of each of the five triangles ${ABC}$, ${BCD}$, ${CDE}$, ${DEA}$, ${EAB}$ is unity. Show that every non-congruent pentagon with the above property has the same area, and that, furthermore, there are an infinite number of such non-congruent pentagons. (more…)

USAMO 1972 #4July 26, 2009

Posted by lumixedia in Problem-solving.
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USAMO 1972 #4. Let ${R}$ denote a non-negative rational number. Determine a fixed set of integers ${a}$, ${b}$, ${c}$, ${d}$, ${e}$, ${f}$ such that, for every choice of ${R}$,

$\displaystyle |\frac{aR^2+bR+c}{dR^2+eR+f}-\sqrt[3]{2}|<|R-\sqrt[3]{2}|.$ (more…)

USAMO 1972 #2, #3July 21, 2009

Posted by lumixedia in Problem-solving.
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I think I might as well just start going through the USAMOs in chronological/numerical order.

USAMO 1972 #2. A given tetrahedron ${ABCD}$ is isosceles, that is ${AB=CD}$, ${AC=BD}$, ${AD=BC}$. Show that the faces of the tetrahedron are acute-angled triangles. (more…)

USAMO 1972 #1July 18, 2009

Posted by lumixedia in Problem-solving.
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USAMO 1972 # 1. The symbols ${(a,b,...,g)}$ and ${[a,b,...,g]}$ denote the greatest common divisor and the least common multiple, respectively, of the positive integers ${a,b,...,g}$. For example, ${(3,6,18)=3}$ and ${[6,15]=30}$. Prove that
$\displaystyle \frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}.$