AMC 10 · 2011 · #19

Grade 8 number-theory

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

perfect-squaresdifference-of-squaresfactors convert-to-algebracasework ↑ Prerequisites: perfect-squares

📏 Long solution 💡 3 insights

Problem

In 1991 the population of a town was a perfect square. Ten years later, in 2001, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, after another increase of 150 people, the population is a perfect square again. Which choice is closest to the percent growth of the town's population over this twenty-year period?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The clues describe three unknown populations linked by fixed jumps of 150. I name the perfect squares with letters, turn each clue into an equation, and get a difference of two squares equal to a small number. A difference of squares factors, so I can list the few factor pairs and test which one also satisfies the 2011 condition.

1STEP 1

Name the perfect squares

Let 1991 be n², so 2001 is n² + 150 = m² + 9 and 2011 is n² + 300 = k².

1991: n² 2001: n²+150=m²+9 2011: n²+300=k²

2STEP 2

Turn the 2001 clue into an equation

Collect the numbers in n² + 150 = m² + 9: m² - n² = 141, so the two squares differ by that much.

n² + 150 = m² + 9 → m² - n² = 141

3STEP 3

Factor the difference of squares

Since (m-n)(m+n) = 141 and 141 = 3 × 47, the only factor pairs are 1 × 141 and 3 × 47.

(m-n)(m+n)=141=1×141=3×47

4STEP 4

Solve for n in each case

Subtracting m-n=a from m+n=b gives 2n=b-a: the pair 1 × 141 gives n = 70, and 3 × 47 gives n = 22.

1×141: n=(141-1)/2=70 3×47: n=(47-3)/2=22

5STEP 5

Keep the case that fits 2011

70² + 300 = 5200 is not a square, so n = 70 is out; n = 22 gives 484 in 1991 and 784 = 28² in 2011.

70²+300=5200 (not square) 22²+300=784=28² (square)

6STEP 6

Compute the percent growth

Growth compares the rise to the start: 300/484 ≈ 0.620, about 62 percent, which is choice (E).

(784-484)/484=300/484≈0.620=62%

Answer

Check all three clues with n=22: 1991 is 484=22² (square), 2001 is 634 = 625 + 9 = 25² + 9 (9 more than a square), 2011 is 784=28² (square). All fit. The growth 300/484 is a bit under two-thirds, and 62% is the only choice near that, so (E) is sound.

💡Key takeaway

When two perfect squares differ by a small number, factor that number as (m-n)(m+n) to find the squares, then measure growth against where you started.

Name the perfect squares
Turn the 2001 clue into an equation
Factor the difference of squares
Solve for n in each case
Keep the case that fits 2011
Compute the percent growth