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.
points in a plane, we will make the lines that passes by any two points in this plane and we obtain exactly 
distinct lines, prove that least points are collinears!!
points in the plane, any from them not collinear, exist 
distinct lines which pass through points.
, exist at most 
distinct lines. points. collinear points exists a single line which passes through them, instead distinct lines in the case of non-collinear points. collinear points decreases with the number of distinct lines.
sets of collinear points and none with at least collinear points, the number of distinct lines is 
, hence an even number and this number can not be , contradiction. points contains a set of at least collinear points.
