AMC 10 · 2005 · #17

Grade 6 geometry-2d
Archetype Pair-Sum Invariant └ Sub-archetype Universe Total Minus Known
sequences-arithmeticdouble-counting pattern-recognitionidentify-subproblems ↑ Prerequisites: sequences-arithmeticmean-median-mode-range
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
The numbers 3, 5, 6, 7, 9 are placed one at each of the five points A, B, C, D, E of a five-pointed star. Each of the five edges AB, BC, CD, DE, EA gets the sum of its two endpoint numbers, and those five sums, in some order, form an arithmetic sequence. Find the middle term of that sequence.

Pick an answer.

(A)
9
(B)
10
(C)
11
(D)
12
(E)
13

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Change Focus / Count the Complement

Placing 3, 5, 6, 7, 9 correctly is a maze of 120 orderings, and chasing the right one is slow. Tool #16 (Change Focus) sidesteps it: instead of finding the arrangement, add up all five edge sums at once. Tool #5 (Look for a Pattern) supplies the key structural fact — every point touches exactly two edges, so that grand total counts each number twice. Tool #4 (Introduce a Variable) names the middle term m and turns the arithmetic-sequence fact into a single equation.

1STEP 1

Stop arranging, start totaling

Hunting for the right arrangement is slow. Change focus: ask what all five edge sums add up to together.

S₁ + S₂ + S₃ + S₄ + S₅ = ?
2STEP 2

Each point is counted twice

Every point lies on two edges, so each number counts twice — the grand total is 2(3+5+6+7+9), or 60.

S₁ + S₂ + S₃ + S₄ + S₅ = 2(3+5+6+7+9) = 2 · 30 = 60
3STEP 3

Middle term equals the average

With five terms, the middle term is the average, so the total is five times it. Call the middle term m: 5m = 60.

5m = S₁ + S₂ + S₃ + S₄ + S₅ = 60
4STEP 4

Solve for the middle term

Divide 5m = 60 by 5: the middle term is m = 12, choice (D).

5m = 60 → m = 60/5 = 12 → (D)
Answer
12
The middle term must be the average of the five edge sums, and that average never depends on how the numbers are arranged — it is always 60/5 = 12. Check that a real arrangement exists so the problem is well-posed: place the numbers around the star in the cyclic order 9, 5, 6, 7, 3. The edge sums are 9+5=14, 5+6=11, 6+7=13, 7+3=10, 3+9=12, i.e. {10,11,12,13,14} — an arithmetic sequence with common difference 1 and middle term 12. Also, 12 is one of the answer choices and sits right in the middle of the list, both reassuring signs.
💡Key takeaway

When the exact arrangement is hard to pin down, add everything up at once — the total often hands you the answer.

  • Stop arranging, start totaling
  • Each point is counted twice
  • Middle term equals the average
  • Solve for the middle term

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