AMC 10 · 2003 · #7

Grade 7 geometry-2d

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

polygon-inequalitysystematic-enumeration bound-inequality-then-enumeratesystematic-enumeration ↑ Prerequisites: polygon-inequality

📏 Medium solution 💡 2 insights

Problem

Count the triangles whose three side lengths are whole numbers and add up to 7, where two triangles that are the same shape and size (congruent) count only once.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Make a Systematic List

The question asks "how many" whole-number triangles fit a fixed perimeter, which is a finite counting task — tool #2 (Make a Systematic List) is built for exactly this, and writing sides in order a ≤ b ≤ c makes the list complete with no repeats. Tool #14 (Extreme Principle) tames the search first: the single longest side c is the tightest constraint, and the triangle inequality caps how big it can be, which pins c to one value before any listing starts. Tool #3 (Eliminate Possibilities) does the final filtering: after listing the integer side combinations, throw out any that fail the triangle inequality, and count what survives.

1STEP 1

Bound the longest side

Order the sides so the longest is last; the triangle rule then reads 7 minus c greater than c.

a+b ＞ c → 7-c ＞ c

2STEP 2

Pin the longest side to a single value

Solving with the fact that c is the longest pins it to c = 3, leaving the others to add to 4.

7-c ＞ c → c ＜ 3.5, c ≥ 7/3 → c=3, a+b=4

3STEP 3

List the side combinations and keep the real triangles

The splits give (1,3,3) and (2,2,3), both genuine triangles, so the count is 2, choice (B).

(1,3,3): 1+3 ＞ 3 ✓ (2,2,3): 2+2 ＞ 3 ✓ → 2 triangles (B)

Answer

Sanity-check by ignoring the triangle rule and just listing integer sides a ≤ b ≤ c with a+b+c=7: (1,1,5),(1,2,4),(1,3,3),(2,2,3) — four combinations. The first two fail the triangle inequality (1+1 ＜ 5 and 1+2 ＜ 4: too flat to close up), leaving exactly the two found, (1,3,3) and (2,2,3). This confirms 2 and shows why the answer is not the larger 4 (choice D).

💡Key takeaway

For integer-sided triangles with a fixed perimeter, first bound the longest side with the triangle inequality, then list the few remaining side combinations that fit.

Bound the longest side
Pin the longest side to a single value
List the side combinations and keep the real triangles

Bound the longest side

Pin the longest side to a single value

List the side combinations and keep the real triangles

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