[/quote]
,
be integers and 
be a prime for which![\[ x^2-3xy+p^2y^2=12p \]](http://latex.artofproblemsolving.com/2/4/c/24cb51eb55144a9a1e0e489d115993496a052e79.png)
.
from the nonnegative integers to the integers satisfying 
and ![\[ f(mn) = f(m) + f(n) + f(m)f(n) \]](http://latex.artofproblemsolving.com/e/f/f/eff10a32cffaec742403ebfc46ff09bc8ef7b7c5.png)
for all nonnegative integers 
and 
.
satisfying the condition: 
with 
,
by 
,
,
be the reflected point of 
wrt 
.
and 
meet at 
.
.
.
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
,
is such that the quadrilateral 
is cyclic and the circumcircle of 
is tangent to the line 
.
is the second common point of the line 
and the circumcircle of 
.
of positive integers such that 
is divisible by 
.
and circumcircle 
.
intersects the circumcircle of triangle 
.
intersects circumcircle of triangle 
again at 
and intersects circumcircle of triangle 
.
be the intersection of 
.
lies on 
.
and 
do not depend on the choices of 
.
,
at 
,
to 
, respectively. Rays 
intersect 
at 
,
touches incircle at 
.
for all pairs of positive integers 
and 
.
for all Canadian functions 
.
on 
respectively. Circumcircle 
of triangle 
intersects 
.
intersects 
at 
cuts 
respectively. Construct parallelogram 
.
are collinear.
-
and 
such that 
.
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
