?
be nonnegative real numbers such that 
.
be the orthocenter of acute triangle 
,
be the foot of the altitude from 
to 
,
be the reflection of 
.
intersects line 
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 
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 
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 
.
.
be a polynomial of degree 
such that the polynomial 
divides 
is zero. Prove that 
Y by
Let 
be a positive integer. We say that a positive integer 
is -good if 
divides for some integer 
. We say a positive integer is -bad if it is not -good.
(a) Is it true that for every positive integer
there exist consecutive -bad positive integers?
(b) Is it true that for every positive integer there exist consecutive -good positive integers?
be a positive integer. We say that a positive integer 
is -good if 
divides for some integer 
. We say a positive integer is -bad if it is not -good.(a) Is it true that for every positive integer
there exist consecutive -bad positive integers?(b) Is it true that for every positive integer
there exist consecutive -good positive integers?
has to divide some integer when 
.
,
,
,
,
and 
.
,
be 
that is 
for each 
.
are all 
