AoPS Academy
+2 w
[/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.
be an acute triangle with 
, and let 
be the projection from 
onto 
. Let 
be a point on the extension of 
past such that 
. Let 
be on the perpendicular bisector of 
such that and 
are on the same side of and![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of across 
and 
be 
and 
, respectively. Let 
and 
such that 
. Let 
and 
intersect the circumcircles of 
and 
at 
and 
, respectively. Let 
and 
intersect 
and 
at 
and 
. Let 
intersect at 
. Prove that 
.
+1 w with incircle 
. Let 
be the point on such that the circumcircle of 
is tangent to and let the -excircle be tangent to at 
. Prove that the tangent from to and the tangent from to (distinct from ) meet on the line parallel to and passing through .
by 
, 
for every positive integer 
. Prove that 
leave distinct remainders when divided by 
.
of distinct positive integers such that there exists a positive integer 
for which the number of divisors of 
and of 
are equal.
and 
be two sequences of integers such that 
and 
for all 
. Suppose that 
divides 
for an infinity of integers . Prove that there exist an integer 
such that 
for all
are coprime. Consider the sequence 
given by 
. Suppose that there exist positive integers 
such that 
is even and 
is an integer. Prove that the numerator in its simplest form of 
is an odd integer greater than 
.
be an integer and let 
be distinct subsets of 
. Show that there exists 
such that the n subsets 
are also disjoint.
satisfies 
. If 
, find the maximum possible value of 
.
such that for all irrational numbers 
and 
,
gives point lies on bisector of 
. Line 
intersect circumcircle 
of triangle at 
. Circle 
passes through , touch 
at and intersect segment at 
and 
at 
., , 
lies at one line.
![\[\angle AQM=\angle ACM=\angle AXM-\angle XAC=\angle PXM+\angle AXP-\angle PAX=\angle AXP=\angle AQP\]](http://latex.artofproblemsolving.com/3/7/5/3751cf50fcea9e05a9641a419d8780b3fe5a6c43.png)
so 
is a straight line.
the second intersection point of 
and . Because 
is cyclic, we have that 
, and also 
because is tangent to so 
and therefore 
. Now 
so , , and lie on one line.
gives point lies on bisector of . Line intersect circumcircle of triangle at . Circle passes through , touch at and intersect segment at and at ., , lies at one line.
be the internal bisector of angle 
in 
. Circle passes through 
and is tangent to side 
. Let meet 
at . Take an arbitrary point on line ; and let 
meet line at and again at 
. Prove that 
and are collinear. with constant velocity on line . Note that 
and 
are projective mappings; hence 
is also projective. To see that this map coincides with perspectivity about , we need to check the claim for three positions of . Take 
and see that each of these works, so we are done! 
and 
.
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
s
where 
and 
be 
.
to 
,
to 
.
