AMC 10 · 2017 · #23
Grade 8 geometry-2dPick an answer.
AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Counting good triangles directly is hard because the bad cases hide everywhere. Tool #16 (Count the Complement) flips it: count every way to pick 3 of the 25 points, then subtract only the bad picks — the ones where the 3 points lie on a single line and the area is zero. Tool #7 (Identify Subproblems) sorts those flat triples by the kind of line they sit on (flat, upright, and slanted), and Tool #2 (Make a Systematic List) counts each kind without missing or double-counting any line.
Count every triple of points
25 points, and a triangle uses any 3 of them in any order, so C(25, 3) = 2300 possible picks — but some are not real triangles yet.
Count the whole pile first; it is easier to throw out the few bad picks than to hunt for the good ones one by one.
7.SP.C.8Count The ComplementSpot the bad picks: collinear triples
Three points make no triangle exactly when they are collinear — zero area — so subtract every collinear triple from the 2300.
A triangle collapses to a flat line exactly when its three corners share one slope, so collinear means zero area.
8.EE.B.6Identify SubproblemsCount lines that hold 5 points
5 rows, 5 columns, and 2 main diagonals — 12 lines of 5 points — each give C(5, 3) = 10 collinear triples, for 120.
A line through 5 points hides C(5, 3) flat triples, so the long lines are the biggest source of zero-area picks.
7.SP.C.8Make A Systematic ListCount the shorter diagonals (slope ± 1)
The shorter slope ± 1 diagonals — four with 4 points and four with 3 points — add 4·C(4, 3) + 4·C(3, 3) = 20 more.
Off the main diagonal the slanted lines get shorter, so they hold fewer points and fewer flat triples.
8.EE.B.6Make A Systematic ListCount the steeper and shallower lines
Lines with slope 2, -2, 1/2, or -1/2 fit only 3 points, in 3 spots each — 4× 3 lines give 12·C(3, 3) = 12 more.
A steeper step uses up the grid fast, so each slanted slope squeezes in only a handful of 3-point lines.
8.EE.B.6Make A Systematic ListAdd the bad picks and subtract
Add the flat triples: 120 + 20 + 12 = 152 collinear. Subtract from 2300: 2300 - 152 = 2148, which is answer (B).
Every pick is either a triangle or a flat line, so the triangles are just the whole pile minus the flat ones.
4.NBT.B.4Count The ComplementCount every way to pick 3 points, then take away the picks that land on one straight line — what is left are the real triangles.
- Count every triple of points
- Spot the bad picks: collinear triples
- Count lines that hold 5 points
- Count the shorter diagonals (slope ± 1)
- Count the steeper and shallower lines
- Add the bad picks and subtract
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