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.
that is not isosceles at 
, let 
be its incircle, which is tangent to 
at 
, respectively. The lines 
and 
intersect the line passing through and parallel to 
at 
and 
, respectively. The lines passing through 
and perpendicular to 
intersect 
and 
at 
and 
, respectively.
are collinear and that 
are concurrent.
positive integer numbers such that the least common multiple (LCM) of all these numbers is not a perfect square. Mykhailo consecutively hides one of these numbers and writes down the LCM of the remaining 
numbers that are not hidden. What is the maximum number of the written numbers that can be perfect squares?
is a prime number and 
are positive integers. Prove that there exist infinitely 
such that 
are divisible by 
, let 
if is not squarefree and 
if is a product of 
primes, and let 
be the sum of the divisors of . Prove that for all we have![\[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \]](http://latex.artofproblemsolving.com/8/e/b/8eb88de7cd7ea8e2853dbc80cae24bdc6b77f9be.png)
be a sequance of natural integers. Proof that, any number in the sequance can be written only as or 
for any 
and not necessarily distinct primes 
for 
. And and 
are same in some case.
has no integer solutions
and 
play a game with a stack of coins. Beginning with , each player takes turns to choose a prime and remove 
coins from the stack. The player who takes the last coin wins. Prove that there are infinitely many integers such that has a winning strategy.
)
)
). I would also highly recommend taking Helen Yang's classes(almost all NJ ppl have gone through her, with the exceptions of dotted and advait).
satisfying: 
.
![\[2^{xy}f(xy-1)+2^{x+y+1}f(x)f(y)=4xy-2,\forall x,y\in\mathbb{R}\]](http://latex.artofproblemsolving.com/d/1/d/d1de7196fd3ba60732c14fa2b156ec880b4829d0.png)
