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.
and 
be arbitrary points on the sides 
and 
of triangle 
, respectively. The circumcircle of 
meets for the second time the circumcircle of 
at point 
. Line 
meets line 
at point 
, while line 
meets line 
at point 
. Prove that lines 
and 
pass through the same point.
with circumcenter 
. 
and 
intersect the altitude from 
to at points 
and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
and 
Find the value of 
and 
such that 
and 
is a prime number
such that 
for all 
+1 w
be an even integer and let 
be real numbers satisfying 
.
be a positive integer. Ana and Banana play a game. Banana thinks of a function 
and a prime number 
. He tells Ana that 
is nonconstant, 
, and 
for all integers 
. Ana's goal is to determine the value of . She writes down integers 
. After seeing this list, Banana writes down 
in order. Ana wins if she can determine the value of from this information. Find the smallest value of for which Ana has a winning strategy.
Let 
be positive reals such that 
.Prove the inequality 
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
be a set of 
points in the plane, no three of which are collinear. Initially these points are connected with segments so that each point in is the endpoint of exactly two segments. Then, at each step, one may choose two segments 
and 
sharing a common interior point and replace them by the segments and 
if none of them is present at this moment. Prove that it is impossible to perform 
or more such moves.
is the center of the inscribed circle and holds 
. Prove that 
.
be a primes ,
be positive integers such that 
and 
.
be a triangle with 
and 
. Let be the point on AB such that 
and let be the point on such that 
. Let 
intersect 
at 
. Prove that 
(extended) is perpendicular to .
the feet of the perpendicular from on . I'm going to show that 
concur.
, 
, 
, 
, 
, 
we get 
, 
, 
, 
, 
, 
.
we can simplify to 
concur
Prove that
