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and 
for which:
with 
. Points 
and 
are on 
and 
respectively such that 
. Lines 
and 
meet at 
. 
is the incenter of 
and 
is the orthocenter of 
. 
and 
are circles with diameter 
and 
, respectively, intersecting each other at points 
and 
. Prove that and lie on 
.
and 
are externally tangent at a point 
. Let 
be a line tangent to at 
and at 
. Let 
and 
be diameters in and respectively. Suppose points 
and 
lies on such that 
and 
are tangent to , and points 
and 
lies on such that 
and 
are tangent to ., , , lie on a circle 
., , , determine a quadrilateral with an incircle 
, and its radius is 
times the radius of . be the orthocenter of acute triangle , let be the foot of the altitude from 
to , and let be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points and . Prove that is the midpoint of .
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 
.
and lie in a straight line in that order. Suppose that 
is a point not on the line and that 
and 
Find the area of 
be a positive integer, and let 
be nonnegative integers such that 
Prove that![\[
\sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}.
\]](http://latex.artofproblemsolving.com/c/4/9/c49d0ccfef69402d05dc44d721172498c605cf7e.png)
Note: 
for all nonnegative integers 
.
and 
Prove that
Where 
be nonnegative real numbers such that 
Prove that
Let 
Prove that
<===>
<===>
<===>
<===>
Very very good.
