AoPS Academy
= 2∠AMP.
, and an odd prime 
not dividing any of them, and an integer 
such that if one considers the infinite sequence 
and looks at the highest power of that divides each of them, these powers are not all zero, and are all at most . Prove that there exists some 
(which may depend on 
) such that whenever divides an element of this sequence, the maximum power of that divides that element is exactly 
.
be a positive integer. Consider 
points on the circumference of a circle. pairs, and line segments are drawn connecting the points in each pair.
} such that for every 2 different number 
and 
in this set at last in one of the permutation comes exactly after
and 
be the altitudes of acute-angled triangle 
. Using a compass and a ruler, construct a point on side 
such that 
.
and be the altitudes of acute-angled triangle . Using a compass and a ruler, construct a point on side such that . such that 
is tangential to . We claim this is our required point. Indeeed note that 
