,
by 
is a power of 
such that if any 
elements are removed from the set 
,
from each of the integers at two neighboring vertices and adding 
to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount 
index this yr (118.5 on amc10+ 9 on AIME) i’m in 7th rn btw first year comp math also. I will grind so hard until like 30 hrs/week. I’m ok at proofs.
is selected inside acute 
so that 
and 
.
is chosen on ray 
so that 
.
be the midpoint of 
.
is tangent to the circumcircle of triangle 
.
that satisfy![\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]](http://latex.artofproblemsolving.com/f/b/b/fbb693770cd90d8a144026608074122e54da5802.png)
for all 
.
be positive integers. Let 
be the set of integer points 
with 
and 
.
rectangles is called happy if each point in 
a rotation of 
counterclockwise around the origin;
a rotation of 
a reflection across the 
-
a reflection across the 
-
Y by Adventure10
Square 
in the coordinate plane has vertices at the points 
and 
Consider the following four transformations:
Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying
and then 
would send the vertex 
at 
to 
and would send the vertex 
at 
to itself. How many sequences of 
transformations chosen from 
will send all of the labeled vertices back to their original positions? (For example, 
is one sequence of 
transformations that will send the vertices back to their original positions.)
in the coordinate plane has vertices at the points 
and 
Consider the following four transformations:
a rotation of 
counterclockwise around the origin;
a rotation of clockwise around the origin;
a reflection across the 
-axis; and
a reflection across the 
-axis.
Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying
and then 
would send the vertex 
at 
to 
and would send the vertex 
at 
to itself. How many sequences of 
transformations chosen from 
will send all of the labeled vertices back to their original positions? (For example, 
is one sequence of 
transformations that will send the vertices back to their original positions.)
This post has been edited 2 times. Last edited by AOPS12142015, Feb 6, 2020, 5:57 PM
Reason: source
Reason: source
.
bc 
could be switched.
ways to do this.
for 
ways, 
ways, 
ways, 
.
moves; and then by parity there is exactly one 20th move that will return it to the start, so there are 
ways for A to return to its starting position. Similarly, there are 
by multiplying (or we can skip the last part of the solution by noting that C is the only answer that is a perfect square).
in the following generating function:
We end up getting 
