[/quote]
,
satisfying
for all integers 
.
,
,
means 
applied 
,
means 
applied 
satisfying 
.
such that 
(the number of positive divisors of 
be integers and 
be a prime for which![\[ x^2-3xy+p^2y^2=12p \]](http://latex.artofproblemsolving.com/2/4/c/24cb51eb55144a9a1e0e489d115993496a052e79.png)
.
satisfying 
for all positive integers 
.
be the midpoint of 
.
and 
the circumcircles of the triangles 
and 
,
be the crossing point of the tangent to 
with the tangent to 
.
is tangent to 
denote the feet of the altitudes from 
,
.
and 
be the circles through 
,
such that![\[ f(x+f(y))=y+f(x+1),\]](http://latex.artofproblemsolving.com/b/a/3/ba35828951778c0d35f50b5b557796bd2cf9e338.png)
for all 
.
where degrees are used.
and 
Prove that
Let 
Prove that
Let 
and 
Prove that
Let 
Prove that
Y by
Let 
be a quadrilateral with parallel sides 
and 
, 
and 
are the midpoints of its sides 
and 
, respectively. The straight line 
bisects the segment connecting the centers of the circumcircles of triangles 
and 
. Prove that is a parallelogram.
be a quadrilateral with parallel sides 
and 
, 
and 
are the midpoints of its sides 
and 
, respectively. The straight line 
bisects the segment connecting the centers of the circumcircles of triangles 
and 
. Prove that is a parallelogram.This post has been edited 1 time. Last edited by parmenides51, Sep 27, 2024, 4:12 PM
