2. Each cell of an 
board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it satisfies the following two conditions:
board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it satisfies the following two conditions:
(i) The difference between any two adjacent numbers is either 
or 
.
or .
(ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to .
.
Determine the number of distinct gardens in terms of 
and 
.
and .
3. In triangle 
, points 
lie on sides 
respectively. Let 
, 
, 
denote the circumcircles of triangles 
, 
, 
, respectively. Given the fact that segment 
intersects , , again at 
respectively, prove that 
.
, points lie on sides respectively. Let , , denote the circumcircles of triangles , , , respectively. Given the fact that segment intersects , , again at respectively, prove that .
USA Junior Mathematical Olympiad 2013
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