and 
in the plane and a subset 
of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair 
of cities, they are connected with a road along the line segment 
if and only if the following condition holds:
distinct from 
and 
,
such[/center]
is directly similar to either 
or 
.
is directly similar to 
if there exists a sequence of rotations, translations, and dilations sending 
to 
,
to 
,
to 
.
and 
be positive integers with 
.
be a polynomial of degree 
such that the polynomial 
divides 
is zero. Prove that 
Y by
Hello everyone, these days I have a burning urge to pick up new math concepts because I think they are important/interesting. But I also feel a constant burnout where I get really tired when I try to solve math problems of these new concepts. I can't and then it gets very demotivating. I don't want to take a break from math because solving problems have become such a natural part of me and also I really want to qualify for usamo next year (my last year I can since it's senior yr). Any suggestions?
