2011 AMC 12B Problems/Problem 11
Problem
A frog located at , with both
and
integers, makes successive jumps of length
and always lands on points with integer coordinates. Suppose that the frog starts at
and ends at
. What is the smallest possible number of jumps the frog makes?
Solution
Since the frog always jumps in length and lands on a lattice point, the sum of its coordinates must change either by
(by jumping parallel to the x- or y-axis), or by
or
(based off the 3-4-5 right triangle).
Because either ,
, or
is always the change of the sum of the coordinates, the sum of the coordinates will always change from odd to even or vice versa. Thus, it is impossible for the frog to go from
to
in an even number of moves. Therefore, the frog cannot reach
in two moves.
However, a path is possible in 3 moves: from to
to
to
.
Thus, the answer is
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2011 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
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