:![\[
1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3).
\]](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 Adventure10, Mango247
assume ABCD a convex quadrilatral. P and Q are on BC and DC respectively such that angle BAP= angle DAQ .prove that [ADQ]=[ABP]
([ABC] means its area ) iff the line which crosses through the orthocenters of these traingles , is perpendicular to AC.
([ABC] means its area ) iff the line which crosses through the orthocenters of these traingles , is perpendicular to AC.
