2011 AMC 12B Problems/Problem 11

Problem

A frog located at $(x,y)$ , with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$ . What is the smallest possible number of jumps the frog makes?

$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

Solution

Since the frog always jumps in length $5$ and lands on a lattice point, the sum of its coordinates must change either by $5$ (by jumping parallel to the x- or y-axis), or by $3$ or $1$ (by making a move similar to a knight on a chessboard, with two units along one axis and one unit along the other.)

Because either $1$ , $3$ , or $5$ 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 $(0,0)$ to $(1,0)$ in an even number of moves. Therefore, the frog cannot reach $(1,0)$ in two moves.

However, a path is possible in 3 moves: from $(0,0)$ to $(5,0)$ to $(3,1)$ to $(1,0)$ .

Thus, the answer is

\[= \boxed{3\  \(\textbf{(B)}}\] (Error compiling LaTeX. Unknown error_msg)

2011 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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2011 AMC 12B Problems/Problem 11

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