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.
. Consider a function 
. A subset 
of 
is said to be invariant if for all 
we have 
. The empty set and are also considered as invariant subsets. By 
we define the number of invariant subsets of for the function 
. such that 
.
there exists a function such that 
.
and 
for 
. Let 
be a positive integer. Suppose that for every positive integer 
there exists a positive integer 
such that 
. Must be a Fibonacci number?
satisfying following conditions.
for all 
.
with 
, equality 
holds.
be an acute triangle with 
and let 
and 
be its orthocenter and circumcenter, respectively. Let 
be the circle 
. The line 
and the circle of radius centered at 
cross at 
and 
, respectively. Prove that , the circle on diameter 
and circle 
are concurrent.
+1 w
such that for all 
be given pairwise coprime positive integers where 
. Let 
be positive integers. We call to be a grandson of if and only if, for all possible piles of stones whose total mass adds up to and consist of stones with masses , it's possible to take some of the stones out from this pile in a way that in the end, we can obtain a new pile of stones with total mass of . Find the greatest possible number that doesn't have any grandsons.
of primes (greater than 5) such that : 
of positive integers such that:
.Prove that: 
has incentre 
and incircle touching 
at . Let 
be the antipode of in the circumcircle of . Point 
is chosen on the internal angle bisector of 
such that 
. Let 
be the midpoint of arc 
, and let 
be the midpoint of 
. Prove that 
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly positive integer factors 
. Assume that no prime larger than 
divides any of the 
's. Show that there must be some perfect cube among the 's.
and 
intersect at points and 
. A line through intersects and at points 
and , respectively. Line 
intersects at point 
, and line 
intersects at point 
. If is the circumcenter of 
, prove that 
.
be such that![\[u_{1} = 1; \quad u_{n+1} = \dfrac{u_{1} + u_{2} +...+u_{n}}{n}+n-1 \quad \forall n \in \mathbb{N^{*}}\]](http://latex.artofproblemsolving.com/1/5/e/15e5bf0cc14794e7028dbfe4a8754f6803e850ea.png)
and 
. Find 
.
denote the set of positive real numbers. Find all functions 
such that for each 
, there is exactly one 
satisfying 
is q QR 
, but as 
, it is equivalently to prove that 
is a QR , which is true , as 
is prime, 
.
is even for all , and 
, we have![\[\left(\frac{n}{4n+1}\right)=\left(\frac{4n+1}{n}\right)\left(\frac{n}{4n+1}\right)=(-1)^{\frac{4n(n-1)}{4}}=(-1)^{n(n-1)}=1.\]](http://latex.artofproblemsolving.com/d/a/7/da7be59fc641f156cc3b2843af8d1fb534f408a2.png)
be an integer such that 
.
.
.
But since 
we have that:
and we are done.
,
.
By FLT, 
.
,