AMC 10 · 2023 · #6

Grade 11 algebrageometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

logarithm-propertiescoordinate-geometrysymmetric-polynomialsquadratic-equations convert-to-algebraidentify-subproblems ↑ Prerequisites: logarithm-propertiescoordinate-geometry

📏 Medium solution 💡 3 insights

Problem

Two points sit on the base-two logarithm curve, and the midpoint of the segment joining them is given. Find the positive difference between their x-coordinates.

Pick an answer.

(A)

$2\sqrt{11}$

(B)

$4\sqrt{3}$

(C)

(D)

$4\sqrt{5}$

(E)

How to solve

Strategy Introduce a Variable

The two points look like four unknowns, but they are not. Tool #4 (Introduce a Variable) cuts that in half immediately: a point on y = log₂ x is decided by its x-coordinate alone, so naming x_A and x_B names everything. Tool #13 (Convert to Algebra) turns the one geometric fact, "the midpoint is (6,2)", into two ordinary equations — one from averaging the x-coordinates, one from averaging the y-coordinates. The second of those is a sum of logarithms, and the log product rule converts it into a statement about x_A x_B. That leaves a sum and a product, which is where tool #15 (Organize Information in More Ways) earns its place: instead of solving for x_A and x_B separately, read the difference straight off the identity (x_A - x_B)² = (x_A + x_B)² - 4x_A x_B. Tool #7 (Identify Subproblems) handles the small leftover job of simplifying the radical at the end.

1STEP 1

Let the curve supply the y-coordinates

The curve supplies the y-coordinates.

A = (x_A, log₂ x_A), B = (x_B, log₂ x_B)

2STEP 2

Split the midpoint into two averages

Split the midpoint into two averages.

(x_A + x_B)/2 = 6 ⟹ x_A + x_B = 12; (log₂ x_A + log₂ x_B)/2 = 2 ⟹ log₂ x_A + log₂ x_B = 4

3STEP 3

Turn the log sum into a product

The log sum becomes the log of a product.

log₂ x_A + log₂ x_B = log₂(x_A x_B) = 4 ⟹ x_A x_B = 2⁴ = 16

4STEP 4

Read the difference off sum and product

Sum and product give the difference.

(x_A - x_B)² = (x_A + x_B)² - 4x_A x_B = 12² - 4 · 16 = 144 - 64 = 80

5STEP 5

Take the positive square root

Take the positive square root.

|x_A - x_B| = √(80) = √(16 · 5) = 4√(5)

6STEP 6

Check the two points exist

Both points really exist, so the answer is four root five.

t² - 12t + 16 = 0 ⟹ t = 6 ± 2√(5); (6 + 2√(5)) - (6 - 2√(5)) = 4√(5) → (D)

Answer

4√(5)

Test the recovered points directly. With x_A = 6 + 2√(5) ≈ 10.4721 and x_B = 6 - 2√(5) ≈ 1.5279, the x-average is (10.4721 + 1.5279)/2 = 6. The heights are log₂ 10.4721 ≈ 3.3885 and log₂ 1.5279 ≈ 0.6115, whose average is (3.3885 + 0.6115)/2 = 2.0000 — exactly the required midpoint (6,2). The difference is ≈ 8.9443, and 4√(5) ≈ 8.9443, so choice (D) is confirmed. A second sanity check comes from the shape of the curve: y = log₂ x bends downward, so the midpoint of any chord must sit strictly below the curve. At x = 6 the curve is at log₂ 6 ≈ 2.585, and the given midpoint height 2 is indeed below it — the problem is consistent. The two answers near the true value are the tempting ones: 9 = √(81) is what a rounded √(80) looks like, and 8 = √(64) is what you get by mistakenly rooting the subtracted term 4 · 16 = 64 instead of the difference 80. The smaller choices 2√(11) = √(44) and 4√(3) = √(48) come from a wrong product, so the step worth double-checking is log₂ x_A + log₂ x_B = 4 → x_A x_B = 2⁴.

💡Key takeaway

Adding two logs multiplies the numbers inside, so the midpoint's height becomes a product — and once you know a sum and a product, the difference falls out of (a-b)² = (a+b)² - 4ab.

Let the curve supply the y-coordinates
Split the midpoint into two averages
Turn the log sum into a product
Read the difference off sum and product
Take the positive square root
Check the two points exist