AMC 10 · 2020 · #17

Grade 11 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Lattice or Non-Aligned Polygon Area

logarithm-propertiesarea-trapezoidpolynomial-substitutionquadratic-equationscoordinate-geometry convert-to-algebraidentify-subproblemspolynomial-substitution ↑ Prerequisites: logarithm-propertiesarea-trapezoid

📏 Long solution 💡 4 insights

Problem

Four points sit on the natural logarithm curve. Their x-coordinates are four consecutive positive integers. Joining the four points in order makes a quadrilateral whose area is the natural logarithm of ninety-one ninetieths. Find the x-coordinate of the leftmost point.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Introduce a Variable

Tool #4 (Introduce a Variable): call the leftmost x-coordinate n and write all four vertices in terms of n, so the area becomes one formula instead of five separate computations. Tool #1 (Diagram): a quick sketch shows the four points sit on a downward-bending curve, so the shape is bounded above by three short chords and below by one long chord. Tool #7 (Subproblems): that picture turns the area into trapezoids under the chords, which subtract cleanly and make almost every logarithm cancel. Tool #3 (Eliminate): once the leftover expression is compared with ln91/90, testing the five choices confirms which n works and rules out the rest.

1STEP 1

Name the leftmost coordinate

Write all four with one letter.

A=(n,ln n), B=(n+1,ln(n+1)), C=(n+2,ln(n+2)), D=(n+3,ln(n+3))

2STEP 2

Sketch the shape

The curve is concave, so the shape behaves.

top boundary: A→ B→ C→ D; bottom boundary: A→ D

3STEP 3

Split into trapezoids

Write the area as a difference of trapezoids.

[quad] = ((ln n+ln(n+1))/2+(ln(n+1)+ln(n+2))/2+(ln(n+2)+ln(n+3))/2)-3·(ln n+ln(n+3))/2

4STEP 4

Cancel and combine the logs

Most cancels, leaving one logarithm.

[quad]=ln(n+1)+ln(n+2)-ln n-ln(n+3)=ln(n+1)(n+2)/(n(n+3))

5STEP 5

Strip off the logarithm

Stripping the log leaves a fraction equation.

(n+1)(n+2)/(n(n+3))=91/90

6STEP 6

Spot the repeated block

The same block appears on both sides.

(m+2)/m=91/90 → 90m+180=91m → m=180

7STEP 7

Solve the quadratic

Solve the quadratic.

n²+3n-180=0 → (n+15)(n-12)=0 → n=12

8STEP 8

Check the choices

The positive root is 12.

n=12: 182/180=91/90 → (D)

Answer

The area ln91/90≈ 0.011 is tiny, which fits: far out along y=ln x the curve is almost straight, so four nearby points nearly line up and enclose very little area. The formula ln(n+1)(n+2)/(n(n+3)) agrees — it shrinks toward 0 as n grows, and it is always positive because (n+1)(n+2)=n(n+3)+2 exceeds n(n+3). Since the area is strictly decreasing in n, the answer is unique, and n=12 reproduces 182/180=91/90 exactly.

💡Key takeaway

Points on a curve are easier to handle when the area is built from trapezoids: almost every logarithm cancels, what is left is ln(n+1)(n+2)/(n(n+3)), and matching it to ln91/90 gives n²+3n=180, so n=12 — answer (D).

Name the leftmost x-coordinate
Sketch the shape on the curve
Split the area into trapezoids
Cancel and combine the logs
Strip off the logarithm
Spot the repeated block
Solve the quadratic
Check every answer choice