AMC 10 · 2021 · #19
Grade 10 countinggeometry-2dPick an answer.
Four polygons on one circle is a picture nobody can count by eye, so tool #7 (Identify Subproblems) does the first cut: every crossing involves exactly two sides from two different polygons, so the whole count splits cleanly into the six two-polygon pictures. That turns the task into tool #9 (Solve an Easier Related Problem) — settle once and for all how many crossings one m-gon and one n-gon make, then reuse the result six times. For that single pair, tool #1 (Draw a Diagram) supplies the only fact needed: two chords of a circle meet inside exactly when their four endpoints alternate around the rim. Tool #16 (Change Focus) decides which polygon to count from. Counting along the sides of the bigger polygon is uneven — some of its sides get hit, some do not — but counting along the sides of the smaller polygon is perfectly even, and that evenness is what makes the pair count a clean formula. Tool #2 (Make a Systematic List) then runs the six pairs without missing or repeating one, and tool #15 (Organize Information in More Ways) adds them up by regrouping: instead of six separate pair totals, ask how many times each polygon is the smaller one.
Split into six two-polygon pictures
Split into six two-polygon pictures.
Sides of one polygon only ever meet on the circle, so every point inside the circle is a meeting of two different polygons and the count breaks apart pair by pair.
10.G-CO.A.1Identify SubproblemsOne pair only: compare the arcs
For one pair, compare the arcs.
Fewer sides means longer arcs, so the arc cut off by the smaller polygon is always wide enough to swallow at least one corner of the larger one.
10.G-C.B.5Solve An Easier Related ProblemWhen do two chords cross inside?
Decide when two chords cross inside.
Two chords cross only if each one separates the other's endpoints, and on a circle that separation is just the endpoints taking turns as you walk around.
Two chords cross inside only when their endpoints alternate around the circle.
▸ Why?
Alternating endpoints means each chord separates the other's ends, which is exactly what a crossing is.
▸ Why?
Each crossing belongs to exactly one pair of chords, so counting pairs counts crossings once each.
Every side of the smaller polygon is hit twice
Every side of the smaller one is cut twice.
Walking around the larger polygon you must enter the arc once and leave it once, and those two moments are the two crossings.
10.G-C.A.2Change Focus Count The ComplementList the six pairs
List all six pairs.
Only the smaller member of each pair matters, so the six-row table is really just a list of which polygon is outnumbered.
7.SP.C.8Make A Systematic ListRegroup and add
Regrouping and adding gives 68.
Each polygon pays 2 crossings per side for every polygon that has more sides than it, so the tally is really a count of who is outnumbered by whom.
6.EE.A.3Organize Information In More WaysTwo chords of a circle cross inside exactly when their endpoints take turns around the rim, so in any pair of inscribed regular polygons the one with fewer sides sets the count: each of its sides is crossed exactly twice.
- Split into six two-polygon pictures
- One pair only: compare the arcs
- When do two chords cross inside?
- Every side of the smaller polygon is hit twice
- List the six pairs
- Regroup and add