AMC 8 · 2020 · #25

Grade 8 geometry-2dalgebra

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

area-rectanglessystems-of-equationslinear-equations-two-varspatial-visualization convert-to-algebraidentify-subproblems ↑ Prerequisites: systems-of-equationsspatial-visualization

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A big rectangle measuring 3322 wide and 2020 tall is tiled by three squares S₁, S₂, S₃ (side lengths s₁, s₂, s₃) and two rectangles R₁, R₂. From the diagram, S₁ sits in the upper-left, R₁ directly below it, S₂ is the middle square sharing the same vertical column as R₁'s right edge through S₁'s right edge, S₃ sits in the lower-right, and R₂ is above S₃. Find s₂.

Pick an answer.

(A)

651

(B)

655

(C)

656

(D)

662

(E)

666

AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The figure is given but it pays to redraw it and label every segment with s₁, s₂, s₃ — that is Tool #1. Tool #7 (Identify Subproblems) splits the geometry into two independent equations: one from the horizontal direction (total width) and one from the vertical direction (height of the middle column). With two equations linking s₁, s₂, s₃ to the known numbers 3322 and 2020, Tool #13 (Convert to Algebra) lets us add or subtract the equations to eliminate s₁ and s₃ in a single move and read off s₂ — no need to solve for all three side lengths.

1STEP 1

Redraw and label: the three squares' widths sit side by side across the whole top edge, S₁ then S₂ then S₃.

top edge = s₁ + s₂ + s₃

2STEP 2

The labeled top edge equals the given width 3322 — that is the first equation.

s₁ + s₂ + s₃ = 3322 (Eq. 1)

3STEP 3

Each side column is 2020 tall, so reading the middle gap gives s₁ - s₂ + s₃ = 2020 — the second equation.

s₂ = (2020 - s₁) - s₃ ⇔ s₁ - s₂ + s₃ = 2020 (Eq. 2)

4STEP 4

Subtract Eq. 2 from Eq. 1 — s₁ and s₃ cancel, s₂ doubles: 2 s₂ = 1302.

(s₁ + s₂ + s₃) - (s₁ - s₂ + s₃) = 3322 - 2020 → 2 s₂ = 1302

5STEP 5

Divide by 2: s₂ = 651, matching choice (A).

s₂ =

\frac{1302}{2}

= 651 → (A)

Answer

651

The answer must lie between 0 and min(3322, 2020) = 2020, and 651 fits comfortably. As a sanity check, since s₁ + s₂ + s₃ = 3322 and s₁ + s₃ - s₂ = 2020, adding the two equations gives 2(s₁ + s₃) = 5342, so s₁ + s₃ = 2671. That leaves s₂ = 3322 - 2671 = 651. Matches. The figure also implies s₁, s₃ ＞ s₂ (the middle square is the smallest), and any split of 2671 into two values each above 651 is consistent (e.g. s₁ ≈ 1369, s₃ ≈ 1302).

💡Key takeaway

This AMC 8 problem only needs Grade 8 simultaneous equations you already know — two pictures of the rectangle's edges become two equations, and subtracting them cancels the squares you don't care about!

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