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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.
times as many students as Lara’s high school. The two high schools have a total of
students. How many students does Portia’s high school have?
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 .
that satisfy the following condition: For every integer k such that 
there always exists a positive integer 
that is divisible by n and 
. 
: sum of elements of
and 
. Prove that
Equality holds when 
or 
or 
is drawn externally on side 
of 
. Point 
is taken on 
. Find 
if 
.
for all 
.
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