USAMO 1973 #4August 19, 2009

Posted by lumixedia in algebra, Problem-solving.
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A fairly straightforward algebra problem. Could appear on a modern AMC-12, though the decoy answers would have to be carefully written.

USAMO 1973 #4. Determine all the roots, real or complex, of the system of simultaneous equations

$\displaystyle x+y+z=3$

$\displaystyle x^2+y^2+z^2=3$

$\displaystyle x^3+y^3+z^3=3$

USAMO 1973 #3August 17, 2009

Posted by lumixedia in combinatorics, Problem-solving.
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USAMO 1973 #3. Three distinct vertices are chosen at random from the vertices of a given regular polygon of ${(2n+1)}$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points? (more…)

USAMO 1973 #2August 11, 2009

Posted by lumixedia in Problem-solving.
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USAMO 1973 #2. Let ${\{X_n\}}$ and ${\{Y_n\}}$ denote two sequences of integers defined as follows:

$\displaystyle X_0=1,\hspace{0.1cm}X_1=1,\hspace{0.1cm}X_{n+1}=X_n+2X_{n-1}\hspace{0.1cm}(n=1,2,3,...)$

$\displaystyle Y_0=1,\hspace{0.1cm}Y_1=7,\hspace{0.1cm}Y_{n+1}=2Y_n+3Y_{n-1}\hspace{0.1cm}(n=1,2,3,...)$

Thus, the first few terms of the sequence are:

$\displaystyle X:\hspace{0.1cm}1,1,3,5,11,21,...$

$\displaystyle Y:\hspace{0.1cm}1,7,17,55,161,487,...$

Prove that, except for “1”, there is no term which occurs in both sequences. (more…)

USAMO 1973 #1August 7, 2009

Posted by lumixedia in Problem-solving.
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USAMO 1973 #1. Two points, ${P}$ and ${Q}$, lie in the interior of a regular tetrahedron ${ABCD}$. Prove that angle ${PAQ<60^{\circ}}$. (more…)