AMC 10 · 2008 · #23

Grade 7 geometry-2dnumber-theory
Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration
simons-favorite-factoring-trickarea-rectangleslinear-diophantine convert-to-algebra ↑ Prerequisites: area-rectangles
📏 Medium solution 💡 3 insights
Problem
A floor is a rectangle measuring a by b feet, with a and b positive integers and b ＞ a. An artist paints a smaller rectangle inside it, its sides parallel to the floor's sides, leaving an unpainted border exactly 1 foot wide all the way around. That border takes up exactly half of the floor's area. Count how many ordered pairs (a, b) make this possible.

Pick an answer.

(A)
1
(B)
2
(C)
3
(D)
4
(E)
5

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Convert to Algebra

The wording hides a clean equation. A quick sketch shows the 1-foot border shaves 2 feet off the width and 2 feet off the length, so the painted rectangle is (a-2) by (b-2). The 'half the area' condition then becomes an equation in a and b; rearranged and factored it turns into a product of two integers equal to 8. Counting the valid pairs is just listing the factor pairs of 8 and keeping the ones with b ＞ a.

1STEP 1

Sketch the border, size the inner rectangle

The 1-foot border eats a foot at both ends of each dimension, so the painted rectangle is (a-2) by (b-2) feet.

painted rectangle = (a-2)×(b-2)
2STEP 2

Turn 'half the area' into an equation

The border is half the floor, so the paint is the other half: (a-2)(b-2) is half of ab, or 2(a-2)(b-2) = ab.

(a-2)(b-2) = 1/2ab → 2(a-2)(b-2) = ab
3STEP 3

Expand, then factor with the +16 trick

Expand and move all to one side: ab - 4a - 4b + 8 = 0. Add 8 so the constant is 16, and it factors as (a-4)(b-4) = 8.

2ab-4a-4b+8 = ab → ab-4a-4b+8 = 0 → ab-4a-4b+16 = 8 → (a-4)(b-4) = 8
4STEP 4

List the factor pairs, keep b ＞ a

With product 8 and a-4 the smaller factor, 1×8 gives (5, 12) and 2×4 gives (6, 8) — just 2 ordered pairs.

(a-4)(b-4)=8: (1,8)→(a,b)=(5,12), (2,4)→(a,b)=(6,8)
Answer
2
Test both floors against the story. For (5, 12): total area is 60, the painted rectangle is 3 by 10 = 30, exactly half, so the border is also 30 - correct. For (6, 8): total area is 48, the painted rectangle is 4 by 6 = 24, exactly half - correct. Could there be more? The number 8 has only the positive factor pairs {1, 8} and {2, 4}; their reverses give b ＜ a and are excluded, and negative pairs give impossible floor sizes. So exactly 2 pairs survive, matching choice (B) and ruling out 1, 3, 4, and 5.
💡Key takeaway

A one-foot border shrinks each side by two; write the half-area rule as an equation, nudge it into a neat product, and count the factor pairs.

  • Sketch the border, size the inner rectangle
  • Turn 'half the area' into an equation
  • Expand, then factor with the +16 trick
  • List the factor pairs, keep b ＞ a

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