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 
.
+1 w
are natural numbers satisfy that: 
and 
are prime numbers. Prove that 
is a perfect squared number
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 
.
+3 w and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
, the digits in the base-
representation of 
are all greater than .
such that 
is an integer for every positive integer 
be nonnegative real numbers such that 
. Find min
be a prime, and let 
be integers. Show that there exists an integer such that the numbers![\[a_1 + k, a_2 + 2k, \dots, a_p + pk\]](http://latex.artofproblemsolving.com/7/6/2/7624ce20fc9bfb798126ffacdb34110159ef0094.png)
produce at least 
distinct remainders upon division by .
and be positive integers, and let 
be a 
grid of unit squares.
or 
rectangle. A subset 
of grid squares in is domino-tileable if dominoes can be placed to cover every square of exactly once with no domino extending outside of . Note: The empty set is domino tileable. to the upper-right corner of formed by exactly 
edges of the grid squares. and , the number of up-right paths that divide into two domino-tileable subsets.
+2 w
be non-negative real numbers satisfying 
. Find the maximal possible value of
-tuples 
for which this maximal value is achieved.
.
and 
it holds.
, where be non-negative real numbers satisfying .
(the minimum is not attained), as simple as that.
and 
Prove that
Prove that
?