AMC 10 · 2006 · #19

Grade 8 arithmetic
Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search
sequences-arithmeticangle-sum-trianglesystematic-enumeration bound-inequality-then-enumerate ↑ Prerequisites: angle-sum-trianglesequences-arithmetic
📏 Medium solution 💡 3 insights
Problem
A triangle's three angle measures are distinct positive whole numbers of degrees and they form an arithmetic progression, meaning each angle is a fixed step larger than the one before. Two triangles with the same three angles are similar and count as one shape. How many non-similar triangles are possible?

Pick an answer.

(A)
0
(B)
1
(C)
59
(D)
89
(E)
178

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The angles are tied together by two rules — they form an arithmetic progression and they sum to 180° — so the natural first move is to name them with a variable (Tool #4). Writing the three angles as 60-d, 60, 60+d makes the common difference d the single thing that decides the whole triangle. Then the only question is: which values of d are allowed? That is a boundary question, so the Extreme Principle (Tool #14) pins down the smallest and largest legal d from the 'positive integer' and 'distinct' rules. Finally, because each allowed d gives one differently-angled (non-similar) triangle, counting the whole-number values of d in that range is a clean systematic count (Tool #2).

1STEP 1

Name the three angles

Call the smallest angle a and the common step d, with d a positive whole number; the three angles are then a, a+d, a+2d.

angles = a, a+d, a+2d (d ＞ 0)
2STEP 2

Use the triangle angle-sum rule

Adding them gives a+(a+d)+(a+2d) = 3(a+d), and a triangle's angles sum to 180°, so 3(a+d) = 180.

a+(a+d)+(a+2d) = 3(a+d) = 180
3STEP 3

The middle angle is fixed at 60

Dividing by 3 gives a+d = 60, so the middle angle is always 60° and the three angles are 60-d, 60, 60+d.

a+d = 60 → angles = 60-d, 60, 60+d
4STEP 4

Find the allowed values of d

The smallest angle 60-d must be at least 1° and distinctness needs d at least 1, so the whole-number step d runs over 1 to 59.

60-d ≥ 1 → d ≤ 59, d ≥ 1 → 1 ≤ d ≤ 59
5STEP 5

Count the triangles

Different steps give different angle sets, and -d only relabels the same set, so the count is 59 triangles.

d ∈ {1,2,…,59} → 59 triangles → (C) 59
Answer
59
Spot-check the endpoints. With d=1 the angles are 59,60,61 — three distinct positive integers summing to 180, valid. With d=59 they are 1,60,119 — still valid. With d=60 the smallest angle would be 0°, which is not a real angle, so 59 is correctly the last one. There is exactly one triangle per step size, so the total is 59, matching choice (C). The tempting wrong answers check out as traps: 89 would come from mistakenly allowing the middle angle to vary, and 178 from double-counting d and -d as separate triangles.
💡Key takeaway

Because three angles in arithmetic progression must average 60°, the middle angle is always 60; the step size can be any whole number from 1 to 59, giving 59 different triangles.

  • Name the three angles
  • Use the triangle angle-sum rule
  • The middle angle is fixed at 60
  • Find the allowed values of d
  • Count the triangles

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