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 inscribed in circle 
with center 
. Let 
be the orthocenter of triangle and let 
be the midpoint of 
. Tangent of at 
intersects the perpendicular bisector of 
at 
. Tangent of at 
intersects the perpendicular bisector of 
at 
. Prove that 
and 
are perpendicular.
+2 w
be a sequence of positive integers with 
, 
and![\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]](http://latex.artofproblemsolving.com/1/b/f/1bf1a88979e7d2b33530de979d66a4e81032953d.png)
for all 
. Show that if 
is a prime less than 
for some positive integer 
, then there exists 
such that 
.
+1 w
in 
colors, there exists an arithmetic progression of size . How can we derive Van der Waerden's theorem for 
from this?
be the set of positive integers. Find all functions 
, such that for all positive integers 
and prime numbers :
of positive integers for which there exist positive integers 
and 
such that
holds for all integers 
(Note that 
denotes the greatest common divisor of integers 
and 
)
be a strictly decreasing function
then for all 
is given which 
.
is the interior and the incircle of meets 
at 
. 
and 
meet at 
. Let 
be a tangent from D to the circumcircle of 
, and define 
and 
on 
and 
, respectively.
meet at one point.
is an infinite sequence of integers satisfying the following two conditions:
divides 
for 
such that 
for all 
such that 
for all .
denote the set of positive integers. Consider a function 
. For any 
we write 
. Suppose that 
has the following two properties:, then 
;
is finite.
is periodic. be a triangle with circumcircle 
and incentre . A line 
intersects the lines 
, 
, and 
at points 
, , and , respectively, distinct from the points 
, , , and . The perpendicular bisectors , 
, and 
of the segments , , and 
, respectively determine a triangle 
. Show that the circumcircle of the triangle is tangent to .
so that there exists a unique 
satisfying 
and![\[f(x+g(y))+f(-x-g(-y))=g(x+f(y))+g(-x-f(-y))\]](http://latex.artofproblemsolving.com/d/e/0/de0ea33c3a679ff02762be39e9013b72974e4287.png)
for all 
.
be the assertion 
and 
both are solutions and so we need 
even.
and so 
is odd and we can write 
for some odd 
becomes then 
,
is even and 
such that 
, there exists infinitely many solutions, for example :
exist, for example even periodic functions)
such that , then implies 
and unique solution 
not periodic was enough, but it is not : It exists some non periodic even functions such that such that )
and 
Prove that
