2011 AMC 12B Problem 11

Question

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?

2

3

4

5

6

One jump is impossible since the distance is

1,

not

5

A midpoint of two jumps would lie on

x=\tfrac12,

which has no lattice points; try three jumps

One jump cannot work, since

(0,0)

and

(1,0)

are only

1

apart. Two jumps also fail: the intermediate point would be at distance

5

from both, forcing it onto the perpendicular bisector

x=\dfrac12,

which contains no lattice points.

Three jumps suffice, for example

(0,0)\to(3,4)\to(6,0)\to(1,0),

where each step has length

5.

Thus, the correct answer is B.

Problem 11 in other years