[/quote]
,
be a scalene triangle with circumcircle 
.
be the midpoint of 
.
is selected in the line segment 
.
and 
intersect 
and 
,
and 
intersect (a second time) the circumcircles to triangles 
and 
,
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.)
such that 
divides 
for all positive integers 
and 
with 
.
,
such that 
is a perfect square? (Here 
denotes the integer part of the real number 
?
,
be the intouch triangle with 
,
,
,
be the intersecttion of the perpendicular from 
with 
,
.
and 
the set 
can be partitioned into 
Y by
many older / past links inside aops have gone dead / become broken
such as the 1st link from here (this one )
and older such as this one from here [this one uses the url http://www.mathlinks.ro]
I have seen it in many links before year 2010
is their any way to make those old links work / stay alive?
such as the 1st link from here (this one )
and older such as this one from here [this one uses the url http://www.mathlinks.ro]
I have seen it in many links before year 2010
is their any way to make those old links work / stay alive?
This post has been edited 8 times. Last edited by parmenides51, Nov 6, 2022, 10:55 AM
