Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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...")
 
Line 1: Line 1:
−
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:
+
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 .
+
(i) The difference between any two adjacent numbers is either 0 or <math> 1 </math>.  
 +
(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 .
+
Determine the number of distinct gardens in terms of <math>m</math> and <math>n</math> .

Revision as of 18:26, 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 $n$ .

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2013_USAJMO_Problems/Problem_2&oldid=52725"