[/quote]
,
and 
Prove that 
be real numbers such that 
.
Prove that 
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
or 
Soo... can u guys help? thx!
positive integers. Product of any one of them with sum of remaining numbers increased by 
is divisible with sum of all 
be positive reals such that 
.
Given 
non-zero reals 
such that their sum is 
.
such that 
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
*Note: Whenever I have an equation, AoPS gives the error "New users are not allowed to post images in the Community." To circumvent this I've removed all dollar signs from the equations. Sorry for making this unreadable! If anyone has a better solution, please notify me!
Let ABC be a triangle with |AB|\neq|BC| and the circle \omega, passing through the points A and C, intersect sides AB and BC again at points D and E respectively. The tangents to \omega at the points A and E intersect at X. Prove that AC, DE and BX are concurrent.
I stumbled into this result whilst playing around with the nine-point circle. I wasn't able to find this on the internet. If anybody finds it, please reply and give me advice on how to search for such geometry problems! I don't know how to prove this. I can't figure out how to use the tangency.
Let ABC be a triangle with |AB|\neq|BC| and the circle \omega, passing through the points A and C, intersect sides AB and BC again at points D and E respectively. The tangents to \omega at the points A and E intersect at X. Prove that AC, DE and BX are concurrent.
I stumbled into this result whilst playing around with the nine-point circle. I wasn't able to find this on the internet. If anybody finds it, please reply and give me advice on how to search for such geometry problems! I don't know how to prove this. I can't figure out how to use the tangency.
This post has been edited 1 time. Last edited by Gimbrint, Saturday at 10:26 PM
Reason: Added a correction
Reason: Added a correction
and the circle 
,
and 
,
and 
be cyclic, 
,
.
and 
.
collinear. Now this is a special case of Pascal's theorem (https://en.wikipedia.org/wiki/Pascal%27s_theorem), where the "hexagon" in question is really a quadrilateral, two of whose sides are zero hence why two of the lines in question are tangents.
