1998 AMC 12 Problem 29
Attempt Problem 29 of the 1998 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1998 AMC 12 solutions, or check the answer key.
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29.
A point in the plane is called a lattice point if both and are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to
Answer: D
Small Hint:
Three collinear interior lattice points force a fourth, so use a smallest noncollinear lattice triangle
Big Hint:
At a maximal square, opposite sides are pinned by nearby lattice points; compare their separation
Solution:
It suffices to enclose the three noncollinear lattice points In a maximal placement, two opposite sides are pinned by neighboring lattice points; the greatest possible separation is the distance between parallel lines through and Hence the area is at most This is attained by the square bounded by Its area is and exactly the three stated lattice points lie inside. Thus D is correct.
Problem 29 in Other Years
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