AoPS Academy
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 
, there exists 
such[/center]
is directly similar to either 
or 
.[/center]
is directly similar to 
if there exists a sequence of rotations, translations, and dilations sending 
to 
, 
to 
, and 
to 
.
on the circle 
and write 
. Then 
. Also 
hence 
. It follows that we want 
for all 
hence 
.
we have 
and with 
we find 
. Hence all pairs with 
.
.
defined on 
by![\[f(x, y) = x^4 + y^4 - 2(x - y)^2.\]](http://latex.artofproblemsolving.com/1/d/9/1d9e85722971f67e3ba2dee0e12d119ceabc671a.png)
(and determine them) such that![\[\forall (x, y) \in \mathbb{R}^2, f(x, y) \geq \alpha \| (x, y) \|^2 + \beta,\]](http://latex.artofproblemsolving.com/e/b/3/eb384916db4e019c7ba11f52fadf973f515891e2.png)
where 
denotes the Euclidean norm.
