AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
with 
odd, which satisfy the equation:
such that: 
be an odd prime number. Prove that there exist sequence such that: 
such that :
denotes the number of distinct prime factors dividing 
with incircle 
, let 
be the incenter and 
be the point where touches 
. Let 
be the point on 
with 
and 
be the orthocenter of 
, so that 
on 
also satisfies 
. Prove that 
, the reflection of over the midpoint of 
, lies on .
for which all positive divisors of can be put into the cells of a rectangular table under the following constraints:
be the Fermat point of a 
. Prove that the Euler line of the triangles 
, 
, 
are concurrent and the point of concurrence is 
, the centroid of .
such that
that is not isosceles at 
, let 
be its incircle, which is tangent to 
at 
, respectively. The lines 
and 
intersect the line passing through and parallel to 
at 
and 
, respectively. The lines passing through 
and perpendicular to 
intersect 
and 
at 
and 
, respectively.
are collinear and that 
are concurrent.
be a prime number. Prove that there exists an integer 
with 
such that neither 
nor 
is divisible by 
.
and 
lie on the lateral sides 
and 
respectively of an isosceles triangle in such a way that 
. Let 
be the reflection of about 
. Prove that the line 
passes through the incenter of .
be bisector of 
. Take 
reflection of 
about 
and: 
, which proves that 
are concurrent, and 
being angle bisectors, we are done.
be a triangle with incenter 
two points of 
s.t. 
is the intersection of the parallel to through 
and the parallel to 
through 
.
are collinear.
be the infinity point of 
; 
be the infinity point of 
; Let 
be the infinity point of 
we get are collinear .
thus 
besides
moves linearly on , then also and move linearly. So it suffices to consider the two extremal cases 
and 
. and move on are parallel to each other (which fortunately is true in this case). Otherwise, all such lines are not guaranteed to pass through a fixed point.
be nonnegative real numbers such that 
.
and 
Prove that
and 
,
and the condition on the angles shows that 
is the angular bisector of 
;
.
,
,
,
,
,
.