Difference between revisions of "2022 SSMO Team Round Problems"
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Revision as of 21:43, 31 May 2023
Contents
Problem 1
In triangle , circumcircle
is drawn. Let
be the incenter of
. Let
be the intersection of the
-altitude and
Given that
and
the area of triangle
can be expressed as
for relatively prime positive integers
and
Find
Problem 2
Consider marbles in a line, where the color of each marble is either black or white and is randomly chosen. Define the period of a lineup of 8 marbles to be the length of the smallest lineup of marbles such that if we consider the infinite repeating sequence of marbles formed by repeating that lineup, the original lineup of 8 marbles can be found within that sequence.
A good ordering of these marbles is defined to be an ordering such that the period of the ordering is at most . For example,
is a good ordering because we may consider the lineup
, which has a length equal to
If the probability that the marbles form a good ordering can be expressed as
where
and
are relatively prime positive integers, find
Problem 3
Let be an isosceles trapezoid such that
Let
be a point on
such that
Let the midpoint of
be
such that
intersects
at
and
at
If
and
then
can be expressed as
where
and
are relatively prime positive integers. Find