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 
such that 
and 
is a prime number
and 
Find the value of 
such that 
for all 
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 an acute triangle with incentre 
and 
. Let lines 
and 
intersect the circumcircle of at 
and 
, respectively. Consider points 
and 
such that 
and 
are parallelograms (with 
, and 
). Let 
be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.
. Then 
is cyclic and 
.
. Since 
and 
we have that 
is the diameter of the circumscribed circle of 
. Then 
.
is cyclic, which implies that, 
Also, 
, Now since, 
Prove that
be a square and 
points on sides 
respectively such that 
.
where 
are points of intersection of 
.
it's said 
?
and 
are cyclic. angle bash some more. and then prove 
is cyclic, upon noticing 
is diameter. 
is centre hence 
oops sorry. i thought ur soln had an error. but it was actually a typo. sorry 
