and 
and 
Prove that
Where 
with 
.
and 
are on 
and 
respectively such that 
.
and 
meet at 
.
is the incenter of 
and 
is the orthocenter of 
.
and 
are circles with diameter 
and 
,
and 
.
.
and 
are externally tangent at a point 
be a line tangent to 
and 
.
and 
be diameters in 
and 
lies on 
and 
are tangent to 
and 
lies on 
and 
are tangent to 
.
,
times the radius of 
to 
.
intersects line 
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 
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 
is a point not on the line and that 
and 
Find the area of 
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 
.
Y by
Players 
and 
play a game with a stack of 
coins. Beginning with , each player takes turns to choose a prime 
and remove 
coins from the stack. The player who takes the last coin wins. Prove that there are infinitely many integers such that has a winning strategy.
and 
play a game with a stack of 
coins. Beginning with , each player takes turns to choose a prime 
and remove 
coins from the stack. The player who takes the last coin wins. Prove that there are infinitely many integers such that has a winning strategy.
so that hte problem has a sense
rocks.
,
is composite.
