Difference between revisions of "2020 CIME I Problems/Problem 1"
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Revision as of 19:54, 30 August 2020
Problem 1
A knight begins on the point 
in the coordinate plane. From any point 
the knight moves to either 
or 
. Find the number of ways the knight can reach the point 
.
Solution
Let 
denote a move of 
units north and 
unit east, and let 
denote a move of unit north and units east. To get to the point using only these moves, say 
moves in direction and 
moves in direction , we must have 
because both the and -coordinates have increased by 
since the knight started. Solving this system of equations gives us 
. This means we need the knight to make 
moves, 
of which are headed in direction , and the remaining are headed in direction . Any combination of these moves work, so the answer is 
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