:![\[
1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3).
\]](http://latex.artofproblemsolving.com/3/1/a/31a0eac3c7ca298c8fa221083dc94e127e705784.png)
the number of positive integer 
satisfying 
is less than or equal to 
where 
is a constant independent of 
on the board. For each procces, the stutent deletes the number x and replaces it with either 
or 
or 
.
on the board?
Y by
From the vertices 
of a triangle 
pass three parallel lines 
respectively, not parallel to the sides of the triangle. Let 
be the midpoints 
respectively. Let 
be the intersection points of the pairs 
and 
and 
and 
, respectively. Prove that lines 
and 
are concurrent.
of a triangle 
pass three parallel lines 
respectively, not parallel to the sides of the triangle. Let 
be the midpoints 
respectively. Let 
be the intersection points of the pairs 
and 
and 
and 
, respectively. Prove that lines 
and 
are concurrent.This post has been edited 7 times. Last edited by parmenides51, Jun 8, 2020, 7:07 AM
