[/quote]
,
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
positive integers. Product of any one of them with sum of remaining numbers increased by 
is divisible with sum of all 
be a triangle with 
,
be the foot of the altitude from 
.
be a point in the interior of the segment 
.
be the point on the segment 
such that 
.
be the point on the segment 
such that 
.
be the point of intersection of 
and 
.
.
.
.
is selected in the line segment 
.
and 
intersect 
,
and 
intersect (a second time) the circumcircles to triangles 
,
passes through a fixed point 
distinct from 
.
,
,
,
for all 
,
be two positive sequences defined by 
and![\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \]](http://latex.artofproblemsolving.com/2/0/1/2012dc6ff3a41478547cea4aa9b6bccbe17a3623.png)
for all 
.
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
with 
,
and 
for all positive integers 
where 
is a prime.(Don't get mad at me,I've used the search function and did not see a correct and complete solution anywhere.)
Y by TheOneYouWant, Tawan, skys, Adventure10, Mango247
Let the nervous waiting commence!
No talking about about how to solve the problems.
I've done ~4, and fully locked down 2.
How is everyone else doing?
No talking about about how to solve the problems.
I've done ~4, and fully locked down 2.
How is everyone else doing?
This post has been edited 1 time. Last edited by xPythagoras, Feb 15, 2016, 10:10 PM
Reason: No one seemed to like the other 'rules'
Reason: No one seemed to like the other 'rules'
skills
