AoPS Academy
be a triangle, 
its circumcircle and 
the center of . Let 
be a point on the segment 
. We denote by 
the second intersection point of the circumcircles of triangles 
and 
. Prove that the line 
and the tangent to at point 
intersect on the circumcircle of triangle 
.
, 
, 
be positive real numbers such that 
. Prove that:![\[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\]](http://latex.artofproblemsolving.com/d/1/8/d18152c5635ee2f2059a860bf2fa362fe525ebb3.png)
+1 w
satisfying the following two conditions:
such that, for all indices 
, 
.
and for any index , the number![\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]](http://latex.artofproblemsolving.com/5/4/a/54ac17063104439114f3c7b07de8fd27450ea1eb.png)
is a multiple of .
and 
, there exists a positive perfect square with sum of digits equal to 
.
, where the number of 
s is 
(one can also just cite 2003 n4 for 
) 
satisfy 
is the smallest positive perfect square with sum of digits equal to for all positive integers 
. It's clear that doesn't divide , that 
for all , and since is a perfect square, 
.
for some positive integer 
with 
. The values 
are exactly 
, which are clearly all distinct perfect squares.
denote the sum of digits function. Does there exist 
such that![\[
n, s(n), \dots, s^{2024}(n)
\]](http://latex.artofproblemsolving.com/c/3/3/c33803a77912ed4dde5a5c0ae05934496f04707f.png)
are all distinct perfect squares?
,
such that 
.
times.![\[y=\sum_{i=0}^{x-1} 10^{2^i} > x.\]](http://latex.artofproblemsolving.com/c/6/0/c60c3b85bac8e4a3ccaaa66830f9b0b496895f38.png)
Observe that
Here, any term of the form 
with 
uniquely determines 
and cannot be a power of 
.
,
:![\[81, 9801, 998001, 99980001, 9999800001 \dots\]](http://latex.artofproblemsolving.com/9/8/2/9829753bc3cf7feda72593b745c8cf1e3316cfe2.png)
This finishes.