,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
of non-negative real numbers such that the following three conditions are all satisfied:
;
;
of 
[/list]
is a permutation of 
,
.
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?
having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer 
of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation 
.
Y by Adventure10, Mango247, and 1 other user
Let 
be a diameter of a circle; let 
and 
be the tangents at 
and 
, respectively; let 
be any point other than on ; and let 
be arcs on the circle determined by two lines through . Prove that the lines 
and 
determine a segment on equal in length to that of the segment on determined by 
and 
be a diameter of a circle; let 
and 
be the tangents at 
and 
, respectively; let 
be any point other than on ; and let 
be arcs on the circle determined by two lines through . Prove that the lines 
and 
determine a segment on equal in length to that of the segment on determined by 
and 
)
to be the same as that of 
,
)
,
,
,
be the intersections of lines 
with 
be the other tangent from 
.
is harmonic, the perspectivity at 
,
is the point at infinity on 
is the midpoint of segment 
.
,
and the result follows. 
