AoPS Academy
, define 
and 
, meaning 
is the least common multiple of the numbers 
. Determine whether or not there exists a constant 
such that
for every positive integer .
and 
be positive integers with 
. There are tokens placed initially in the leftmost cell of a horizontal row consisting of cells. A move consists of shifting any token 
, 
, 
, 
, 
, or 
positions to the right. Andrés and Beto alternate turns, starting with Andrés. The winner is the player who moves a token into the rightmost cell. Determine who has a winning strategy in terms of and .
be an acute triangle with circumcenter 
. Draw altitude 
, with 
on side 
. The parallel line to 
passing through intersects line 
at point 
. Prove that point and the midpoints of sides 
and are collinear.
has a radius 
satisfying 
.
, so 
is obtuse. Let 
be the antipode of 
with respect to 
. Then Pythagorean Theorem yields![\[
AB^2 + BD^2 = AC^2 + CD^2 = 4R^2,
\]](http://latex.artofproblemsolving.com/4/a/b/4abba8f2b7719d12b5731ea2f037edeecf9ff82e.png)
so substituting into the given yields![\[
BC^2 + CD^2 + DB^2 = 9R^2.
\]](http://latex.artofproblemsolving.com/e/c/3/ec3eb7244fa86a73c5a5932f8d11f41639ac08ce.png)
But it's known that, in any triangle with orthocenter 
and circumcenter , that 
. Hence the orthocenter and circumcenter of 
coincide, implying is equilateral. All in all, 
and 
. yields only one degree of freedom heavily implies this is the equality case of an inequality.
