AoPS Academy
, a set is said to be -good if the set contains exactly three elements 
and 
. such that 
can be partitioned into several -good sets.
is a prime number. We have a number of cards, each of which has a number written on it such that each of the numbers 
appears at most once and 
exactly once. To design a game, for each pair of cards 
and 
, we want to determine which card wins over the other. The following conditions must be satisfied:
If card wins over card , and card wins over card 
, then card must also win over card .
If card does not win over card , and card does not win over card , then for any card 
, card 
must not win over card 
.
of natural numbers such that for every pair of natural numbers 
and 
, the following inequality holds:![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
and satisfy ![\[ a^2+b^2+c^2 +abc = 4 . \]](http://latex.artofproblemsolving.com/d/3/a/d3ac01295b077425b5eaae556faa4cd678bce32a.png)
Show that ![\[ 0 \le ab + bc + ca - abc \leq 2. \]](http://latex.artofproblemsolving.com/2/0/9/209b3fcb56c01c335bbcb8d5fb781e28e49a0798.png)
be a triangle. Find all points 
on segment 
satisfying the following property: If 
and 
are the intersections of line 
with the common external tangent lines of the circumcircles of triangles 
and 
, then ![\[\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=1.\]](http://latex.artofproblemsolving.com/0/8/0/080a689aea002f1b997294cb6aa48d3334881be7.png)
be positive real numbers. Prove that 
.
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 
.
touches the sides B and of a triangle and intersects its side 
at 
. It is known that the tangent to at is symmetrical to the line with respect to the line 
. What can be the difference 
if 
and 
?
touches the sides B and of a triangle and intersects its side at . It is known that the tangent to at is symmetrical to the line with respect to the line . What can be the difference if and ?
As a hint, think about homothety.