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
2020 CIME I (Problems • Answer Key • Resources) | ||
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Followed by Problem 2 | |
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