[/quote]
,
be a triangle with incentre 
.
tangent to 
at 
again at 
.
tangent to 
again at 
.
is tangent to the incircle of 
is a prime number
defined by:![\[
T_0 = 0,\quad T_1 = T_2 = 1,
\]](http://latex.artofproblemsolving.com/a/8/a/a8a2935b678607acac4180530e4b485673c3ced6.png)
and![\[
T_{n+3} = T_{n+2} + T_{n+1} + T_n \quad \text{for all } n \geq 0.
\]](http://latex.artofproblemsolving.com/8/b/0/8b044e593142ee6b2b75f28575c9b9eaa1ec7b4c.png)
Prove that, for every positive integer 
,![\[
T_{3n} = a_0 T_0 + a_1 T_1 + a_2 T_2 + \cdots + a_{2n} T_{2n},
\]](http://latex.artofproblemsolving.com/f/b/c/fbcbfd6edb17f4c7d7e30d2f6abc75718183acd6.png)
where, for each 
,
is the coefficient of 
in the expansion of the polynomial 
.
and 
on 
and 
are on the same side of 
be midpoint of segment 
be midpoint of segment 
be intersection of 
with 
and 
,
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 
.
and 
be difined as:
Which is larger, 
or 
?
be a square with diagonals 
,
touch sides 
and 
at 
and 
respectively. Let 
and 
denote the circumcircles of triangle 
and 
respectively.
and 
intersect at 
and 
and 
intersect at 
and 
and 
and 
Prove that 
and 
Show that 
and 
are concurrent.
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
such that, for all positive integers 
and 
,
,
.
,
.
intersects 
.
is orthocenter of triangle 
.
coincide
,
.
lies on segment 
such that 
.
and 
be the circumcenter and incenter of triangle 
,
and 
be the circumcenter and incenter of triangle 
,
and 
intersect at point 
.
.
the number of positive integer 
satisfying 
is less than or equal to 
where 
is a constant independent of 
Y by
Define act <Castle> as following:
For rectangle with dimensions i * j, doing <Castle> means to change its dimensions to (i+p) * (j+q) where p,q is a natural number smaller than 3.
Define 1*1 rectangle as "C0" rectangle, and define "Cn" ("n" is a natural number) as a rectangle that can be created with "n" <Castle>s.
Plus, there is a constraint for "Cn" rectangle. The constraint is that "Cn" rectangle's area must be bigger than n^2 and be same or smaller than (n+1)^2. (n^2 < Area =< (n+1)^2)
Let all "C20" rectangle's area's sum be A, and let all "C20" rectangles perimeter's sum be B.
What is A-B?
For rectangle with dimensions i * j, doing <Castle> means to change its dimensions to (i+p) * (j+q) where p,q is a natural number smaller than 3.
Define 1*1 rectangle as "C0" rectangle, and define "Cn" ("n" is a natural number) as a rectangle that can be created with "n" <Castle>s.
Plus, there is a constraint for "Cn" rectangle. The constraint is that "Cn" rectangle's area must be bigger than n^2 and be same or smaller than (n+1)^2. (n^2 < Area =< (n+1)^2)
Let all "C20" rectangle's area's sum be A, and let all "C20" rectangles perimeter's sum be B.
What is A-B?
This post has been edited 1 time. Last edited by jaydenkaka, Apr 23, 2025, 2:23 PM
