and 
for 
.
.
for which 
is divisible by 
.
is divisible by 
satisfy,
Where 
}
be a fixed square and 
a variable point on segment 
.
is constructed such that 
is on segment 
and 
is on segment 
.
be the intersection point of lines 
and 
.
be a function taking in positive integers and outputting nonnegative integers, defined as follows:
is the number of positive integers 
with 
such that the equation 
has an integer solution 
.
such tha
Adit Aggarwal, India.
Y by Mango247, Mango247, Mango247
Let 
be a triangle inscribed in circle 
with its altitudes 
intersect at orthocenter 
(
). Let 
be the midpoint of 
be the orthogonal projection of on 
. 
intersects 
at 
. Let 
be the intersection of tangent of 
which passes through 
with 
be the reflection of through . Prove that 
.
be a triangle inscribed in circle 
with its altitudes 
intersect at orthocenter 
(
). Let 
be the midpoint of 
be the orthogonal projection of on 
. 
intersects 
at 
. Let 
be the intersection of tangent of 
which passes through 
with 
be the reflection of through . Prove that 
.
intersect 
clearly 
.
,
,
centered at 
is the intersection of 
,
.
,
the midpoint of 
,
,
,
and 
are both cyclic and hence 
is cyclic too. Now 
and 
,
is midpoint of 
.
and 
