AMC 10 · 2013 · #11

Grade 8 geometry-3dpattern

Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length

coordinate-geometryspace-diagonal-formulapattern-recognitionspatial-visualization spatial-visualizationpattern-recognition ↑ Prerequisites: coordinate-geometrypythagorean-theorem

📏 Long solution 💡 3 insights

Problem

Two travellers repeat different direction cycles at the same speed. Name their headings at a given separation.

Pick an answer.

(A)

$A\ \text{east, } B\ \text{west}$

(B)

$A\ \text{north, } B\ \text{south}$

(C)

$A\ \text{north, } B\ \text{west}$

(D)

$A\ \text{up, } B\ \text{south}$

(E)

$A\ \text{up, } B\ \text{west}$

How to solve

Strategy Draw a Diagram

Put both flight paths on 3D coordinate axes so every leg becomes a change of one coordinate by one unit. Then, instead of carrying two moving points, carry only the gap vector from B to A. The point of that switch is that the gap vector's three coordinates can be shown to never decrease, which forces the separation to grow steadily and to pass 10 feet exactly once. After that, finding the answer is only a matter of finding which single foot of flight the crossing lands in, and reading off that foot's directions from the two repeating cycles.

1STEP 1

Put the flight on coordinate axes

Coordinates turn each heading into a vector.

north=(0,1,0), east=(1,0,0), up=(0,0,1), south=(0,-1,0), west=(-1,0,0)

2STEP 2

Read off both repeating cycles

Both cycles have a clean closed form.

A(3k)=(k,k,k), B(2k)=(-k,-k,0)

3STEP 3

Follow the gap, not the two bees

The gap is easier to follow than either one.

A north, B south &→ Δ D=(0,2,0) & A north, B west &→ Δ D=(1,1,0) ; A east, B south &→ Δ D=(1,1,0) & A east, B west &→ Δ D=(2,0,0) ; A up, B south &→ Δ D=(0,1,1) & A up, B west &→ Δ D=(1,0,1)

4STEP 4

Show the separation never shrinks

The separation never shrinks.

d(t)² = D₁(t)² + D₂(t)² + D₃(t)², D_i(t) non-negative and non-decreasing → d(t) non-decreasing

5STEP 5

Measure the gap at whole-foot marks

Checking whole steps brackets the moment.

t&=6: & A&=(2,2,2), & B&=(-3,-3,0), & D&=(5,5,2), & d²&=25+25+4=54 ; t&=7: & A&=(2,3,2), & B&=(-3,-4,0), & D&=(5,7,2), & d²&=25+49+4=78 ; t&=8: & A&=(3,3,2), & B&=(-4,-4,0), & D&=(7,7,2), & d²&=49+49+4=102

6STEP 6

Trap the 10-foot moment in one foot

It lands inside a single step.

78 ＜ 100 ＜ 102 ⟹ √(78) ＜ 10 ＜ √(102)

7STEP 7

Name the eighth foot's directions

Remainders name both headings, choice (A).

8 ≡ 2 (mod 3) → A east, 8 ≡ 0 (mod 2) → B west

Answer

A east, B west

Pin the exact instant instead of trusting the trap. Write the time inside the eighth foot as t = 7 + s with 0 ≤ s ≤ 1. Then A = (2+s, 3, 2) and B = (-3-s, -4, 0), so D = (5+2s, 7, 2) and d² = (5+2s)² + 53. Setting d² = 100 gives (5+2s)² = 47, so s = (√(47)-5)/2 ≈ 0.93, which really does lie strictly between 0 and 1. Each wrong choice can also be ruled out by name: A flies north on feet 1, 4, 7, 10, and B flies south on odd feet, so choice (B) means foot 7, where d² = 78 ＜ 100; choice (C) needs A north and B west, which happens on feet 4 and 10, where d² = 26 and 154; choice (D) needs A up and B south, which is foot 9, where d² = 122 ＞ 100; choice (E) needs A up and B west, which is foot 6 or 12, where d² = 54 and 216. Since the separation never shrinks, only foot 8 can straddle 100, so (A) is the only possibility.

💡Key takeaway

Watch the gap between the bees instead of the bees themselves: every foot they fly widens the gap and never narrows it, so they are 10 feet apart exactly once and you only need to find which foot that lands in.

Put the flight on coordinate axes
Read off both repeating cycles
Follow the gap, not the two bees
Show the separation never shrinks
Measure the gap at whole-foot marks
Trap the 10-foot moment in one foot
Name the eighth foot's directions