[/quote]
,
that verify the following equation for all 
:
is a prime number
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
such that 
.
be a quadrilateral inscribed in a circle 
,
such that 
and 
are tangent to 
intersects 
and the line 
at 
.
be the second point of intersection between 
and 
,
be a triangle with incenter 
,
and 
at 
.
and 
the reflections of 
with 
,
be the midpoint of 
and 
are perpendicular.
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
be the circle with diameter 
be the intersection of 
and 
be the intersection of 
be the lines through 
respectively. Let 
and 
.
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
is constant.
be tangent to the side 
.
on 
on 
lies on the circumcircle of triangle 
.
and 
and 
are on the same side of 
be midpoint of segment 
with 
and 
be positive real integer such that 
Prove that
such that 
and 
be a positive integer, and let its positive divisors be![\[
d_1 < d_2 < \cdots < d_k.
\]](http://latex.artofproblemsolving.com/6/3/e/63ec82aba0d6c4eece89af6a1247a89498616281.png)
Define 
to be the number of ordered pairs 
with 
such that 
.
.![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](http://latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers.
.
meets the side 
and 
are 
and 
and 
.
is a rectangle if and only if 
.
Y by Adventure10
Let 
be the orthocenter of the intouch triangle of 
. Let the reflection of on the perpendicular bisectors of 
be 
. Let 
, be the midpoints of 
. Let 
be the midpoints of and 
be the excenters of . Prove that 
are concurrent at the radical center of the nine-point circles of triangles 
.
Sagnik Majumder, IMOTC-2016, Senior Batch
be the orthocenter of the intouch triangle of 
. Let the reflection of on the perpendicular bisectors of 
be 
. Let 
, be the midpoints of 
. Let 
be the midpoints of and 
be the excenters of . Prove that 
are concurrent at the radical center of the nine-point circles of triangles 
.Sagnik Majumder, IMOTC-2016, Senior Batch
This post has been edited 2 times. Last edited by Ankoganit, May 21, 2016, 3:13 AM
Reason: Thanks WizardMath for correction
Reason: Thanks WizardMath for correction
is parallel to the isogonal of AP and DQ is also parallel to it.
.But let's do that. 
