AMC 10 · 2014 · #12
Grade 7 geometry-2dcountingPick an answer.
The phrase "integer lengths less than 5" quietly makes the whole problem finite, so the first move is to bound the search (Tool #2, Make a Systematic List): every side sits in {1, 2, 3, 4}, and a sorted triple a ≥ b ≥ c is one of only 20 possibilities. Bounding first matters — listing until examples "run out" proves nothing, while checking a list of known length 20 proves everything. Tool #3 (Eliminate Possibilities) applies the triangle inequality to cut 20 down to the triples that are genuinely triangles. Then Tool #16 (Change Focus) reframes the question: since congruent triangles are also similar, the ban is really a single ban on similarity, so S takes at most one triangle per similarity class and the count asked for is a count of classes. Tool #15 (Organize Information in More Ways) supplies the fingerprint that identifies a class — divide a triple by the gcd of its sides — turning "are these two similar?" into "are these two labels equal?".
Bound the search before listing
Bounding first leaves only 20 candidates.
Counting the candidates first turns an open-ended hunt into a checklist that can actually be finished.
7.SP.C.8Make A Systematic ListKeep only the real triangles
The triangle inequality keeps 13 of them.
Two sticks shorter than the third can never reach across it, so b + c ＞ a is the only test that can fail.
7.G.A.2Eliminate PossibilitiesOne ban, not two
There is really only one ban.
Similarity ignores size, so the 13 triangles are really a smaller number of distinct shapes wearing different sizes.
7.G.A.1Change Focus Count The ComplementFingerprint each shape with the gcd
A common factor fingerprints each shape.
Dividing out the common factor strips away size and leaves the shape's true name.
Dividing out the common factor strips away size and leaves the shape's true name.
▸ Why?
Every number has one prime recipe, so the shared part of three numbers is a single fixed value.
▸ Why?
Triangles of the same shape have their sides in one fixed ratio, so the reduced triple names the shape.
Prove the nine shapes really differ
The surviving shapes are genuinely different.
Scaling a triangle multiplies every side by the same number, so the side ratios stay fixed and act as a name tag for the shape.
7.RP.A.2Organize Information In More WaysMatch the ceiling with a build
So the largest collection is 9, choice (B).
An upper bound alone could be too generous, so showing an actual set of nine locks the answer in.
7.G.A.2Make A Systematic ListCount the whole list of possibilities before checking it, then divide each triangle's sides by their common factor — triangles with the same reduced sides are the same shape, and only different shapes are allowed.
- Bound the search before listing
- Keep only the real triangles
- One ban, not two
- Fingerprint each shape with the gcd
- Prove the nine shapes really differ
- Match the ceiling with a build