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.
such that for all 
we have:
be defined by![\[
x_1 = 2,\quad x_{n+1} = x_n + \frac{n}{x_n},\quad \text{for all } n \geq 1.
\]](http://latex.artofproblemsolving.com/f/2/8/f28193d8ea3fa6da8bbdb51f9a7e2e9d8a7d3c9f.png)
Prove that the sequences 
and 
have finite limits, and find those limits.
of natural numbers such that for every pair of natural numbers 
and 
, the following inequality holds:![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
+1 w
be reals such that 
and 
Find the value of 
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
and 
be prime numbers such that 
where 
Find all possible values of 
cups labeled 
, where the 
-th cup has capacity liters. In total, there are liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
aand 
Prove that
Let 
aand Prove that
Let 
aand 
Prove that 
and 
and the radius of the inscribed circle is 
. If and are natural numbers and 
, find the value of 
and .
be a point lying on the Neuberg cubic of 
and let 
be the circumcenter of 
respectively. Prove that the reflection of 
in 
respectively are concurrent.
be a point lying on the Neuberg cubic of and let be the circumcenter of respectively. Prove that the reflection of in respectively are concurrent.
.![$$a+b+c\geq 3\sqrt[4]{21}$$](http://latex.artofproblemsolving.com/6/9/f/69f4b29e2eb96f1baedde3ea7e352cc360ceb963.png)
Prove that
Where 
such that : 
and 
