1998 AMC 12 Problem 29

Question

A point

(x,y)

in the plane is called a lattice point if both

x

and

y

are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to

4.0

4.2

4.5

5.0

5.6

Three collinear interior lattice points force a fourth, so use a smallest noncollinear lattice triangle

At a maximal square, opposite sides are pinned by nearby lattice points; compare their separation

It suffices to enclose the three noncollinear lattice points

(0,0),(0,1),(1,0).

In a maximal placement, two opposite sides are pinned by neighboring lattice points; the greatest possible separation is the distance

\sqrt5

between parallel lines through

(1,1)

and

(0,-1).

Hence the area is at most

5.

This is attained by the square bounded by

\begin{aligned} y&=2x+2,\\ y&=2x-3,\\ 2y+x&=3,\\ 2y+x&=-2. \end{aligned}

Its area is

5

and exactly the three stated lattice points lie inside. Thus D is correct.

Problem 29 in other years