Difference between revisions of "2013 USAJMO Problems/Problem 2"

(Created page with "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 ce...")
 
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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:
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Each cell of an <math> m\times n </math> 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 . (ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to .
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(i) The difference between any two adjacent numbers is either 0 or <math> 1 </math>.  
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(ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to <math>0</math> .
 
   
 
   
Determine the number of distinct gardens in terms of and .
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Determine the number of distinct gardens in terms of <math>m</math> and <math>n</math> .

Revision as of 19:26, 11 May 2013

Each cell of an $m\times n$ 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 0 or $1$. (ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to $0$ .

Determine the number of distinct gardens in terms of $m$ and $n$ .