AMC 10 · 2025 · #25

Grade 8 geometry-2d
Archetype Paper Fold-and-Cut Symmetry └ Sub-archetype Composition of Reflections of a Pictured Figure
reflection-unfoldingcoordinate-geometrymodular-arithmetic reflection-unfolding ↑ Prerequisites: coordinate-geometry
📏 Medium solution 💡 4 insights 📊 Diagram
Problem
Square ABCD has side length 4, drawn with A at the bottom left, B at the bottom right, C at the top right, and D at the top left. Point P lies on the left side AD with AP=8/5, and point Q lies on the top side CD with DQ=10/3. A path starts along the segment from P to Q and then bounces off each side with the incoming angle equal to the outgoing angle. If the path hits a vertex it stops there; otherwise it continues forever. At which vertex does the path terminate?

Pick an answer.

(A)
A
(B)
B
(C)
C
(D)
D
(E)
The path continues forever.

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Visualize Spatial Relationships

Chasing a ray as it zig-zags off four walls is hopeless to track by hand. Tool #17 (Visualize Spatial Relationships) supplies the key mental move: instead of bending the ray at each wall, reflect the whole square across that wall and let the ray fly straight into the mirrored copy. The reflected copies tile the plane into a 4×4 grid, and the bounced path becomes one straight line. That is Tool #9 (Solve an Easier Related Problem): a hard bouncing problem turns into an easy straight-line problem. Tool #4 (Introduce a Variable) puts coordinates on the picture so the straight line has an equation, and Tool #1 (Draw a Diagram) anchors which mirrored corner is which.

1STEP 1

Put the square on coordinates

Set A=(0,0), B=(4,0), C=(4,4), D=(0,4). Then AP=8/5 puts P=(0,8/5) and DQ=10/3 puts Q=(10/3,4).

A=(0,0), B=(4,0), C=(4,4), D=(0,4), P=(0,8/5), Q=(10/3,4)
2STEP 2

Write the line through P and Q

Rise 4-8/5=12/5 over run 10/3 gives slope 18/25, and the y-intercept is 8/5, so the ray runs along 25y=18x+40.

y=18/25x+8/5 ⟺ 25y=18x+40
3STEP 3

Unfold the bounces into one straight line

Bouncing is flying straight into mirrored copies tiling the plane; it ends where 25y=18x+40 meets a point with coordinates multiples of 4.

path hits a vertex⇔ x=4a, y=4b for whole numbers a,b
4STEP 4

Find the first grid corner the line reaches

Substituting x=4a, y=4b gives 25b=18a+10, whose smallest positive solution a=5, b=4 makes the first grid corner (20,16).

25b=18a+10 → (a,b)=(5,4) → (x,y)=(20,16)
5STEP 5

Fold the grid corner back to a real vertex

20=4·5 crosses an odd number of strips, so x folds back to 4; 16=4·4 is even, so y folds to 0, giving (4,0), vertex B.

x: 5 odd→ x_real=4, y: 4 even→ y_real=0, (4,0)=B
Answer
B
Check the landing point on the original line: at x=20, y=18/25·20+8/5=360/25+40/25=400/25=16, so (20,16) really is on the ray. Both 20 and 16 are multiples of 4, so it is a genuine grid corner, and it is the first one because a=5 is the smallest positive solution of 25b=18a+10. The parity read-off (x: odd strips → 4; y: even strips → 0) gives (4,0)=B, a real vertex, not choice (E). The path does terminate, ruling out (E), and lands at (B).
💡Key takeaway

To follow a ball bouncing in a mirror-box, don't bend the path — reflect the box and let the ball fly straight, then see which mirrored corner it reaches first.

  • Put the square on coordinates
  • Write the line through P and Q
  • Unfold the bounces into one straight line
  • Find the first grid corner the line reaches
  • Fold the grid corner back to a real vertex

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