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.
be a triangle with 
-mixtilinear incircle 
, and let the tangents from 
and 
to different from 
and 
meet the circumcircle again at 
and 
, and let 
and 
meet 
again at 
and 
. Show that the -excircle of 
is tangent to the circumcircle.
and a point 
outside the circle such that 
. The perpendicular bisector of 
at 
and 
.The segment 
intersects 
at . The perpendicular bisector of
intersects 
at and 
.The rays 
intersect at 
respectively. Prove that the triangles 
and 
are equilateral triangles and the lines , 
are concurrent.
such that for any real numbers 
and 
, the following inequality holds:![\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq (f(x+y))^2
\]](http://latex.artofproblemsolving.com/8/4/3/8437449cd3f8f0da24d17a60b55dcb563e6b5655.png)
of positive integers satsifying 
.
penguin and each one is numbered using numbers 
. penguin 
is called 
if 
where 
is 
prime divisor of (
). what is maximum ratio of 
where 
is number of in antarctica?
+1 w
be real numbers so that 
and 
Show that
+2 w
of positive integers satisfying the following two conditions:
, if 
is in 
, then so are 
and 
; and contains an integer 
such that 
is not divisible by 
.
be a scalene acute triangle with incentre 
and circumcentre 
. Let 
cross at 
. On circle , let and be the mid-arc points of and 
, respectively. Let 
cross 
at 
and let 
cross 
at 
. Prove that the lines 
and 
are concurrent on the external bisector of 
.
+2 w
with integer coefficients, satisfying 
for all non-negative integers .
positive real numbers such that 
. Prove that :
be a non-constant monic polynomial with integer coefficients and let 
be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
.
be an integer, and 
can divide 
let 
be all distinct prime divisors of . Show that 
is an integer. ( where 
is defined as the number of positive integers 
that are relatively prime to .)
be real numbers . Prove that
Where 
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 
be 
.
to 
,
to 
.
