USAMO 1973 #4 August 19, 2009Posted by lumixedia in algebra, Problem-solving.
Tags: algebra, contest math, olympiad math, USAMO, USAMO 1973
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
USAMO 1973 #3 August 17, 2009Posted by lumixedia in combinatorics, Problem-solving.
Tags: combinatorics, contest math, olympiad math, USAMO, USAMO 1973
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USAMO 1973 #3. Three distinct vertices are chosen at random from the vertices of a given regular polygon of 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 #2 August 11, 2009Posted by lumixedia in Problem-solving.
Tags: algebra, contest math, number theory, olympiad math, USAMO, USAMO 1973
USAMO 1973 #2. Let and denote two sequences of integers defined as follows:
Thus, the first few terms of the sequence are:
Prove that, except for “1”, there is no term which occurs in both sequences. (more…)