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 
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
of functions from the set of real numbers to itself that satisfy ![\[g(f(x+y)) = f(x) + (2x + y)g(y)\]](http://latex.artofproblemsolving.com/0/a/3/0a3bc25299a0bcbd3f3602a0e6881d68fce750c5.png)
for all real numbers 
and 
.
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 
of the partition, the set of difference
be bounded?
of the partition, the set of differencebe bounded?
of the string by 
) to get 
and any 
, the difference between the number of elements of 
in the two sets is 
for some 
.
. Start consecutively choosing from each progression 
two appropriate elements (not already colored) and color them in different colors - one black, the other white. All the remaining non colored naturals may be colored arbitrary. That's it. We just must watch out not to breach the given condition, but we can choose numbers from each progression as big as we want, so it's mot a problem.
.
. Suppose 
has digits in binary. There is a term of the progression 
that begins with a 1 followed by 
zeroes in binary (just take the first term above a large power of two). Consider 
, where is such that the leading 1 in aligns with the leading 1 in , and also consider 
. One of these additions has one carry, and the other has zero. This means their sum of digits in binary differs by one.
and any , the difference between the number of elements of in the two sets is for some .
