AMC 10 · 2013 · #24
Grade 11 geometry-2dPick an answer.
Tool #16 (Count the Complement) is primary for a countable reason: the three chosen lengths come from only six values, so there are C(8, 3) = 56 possible length patterns, of which 46 work and 10 fail. Listing ten failures is honest work; listing forty-six successes is a bookkeeping accident waiting to happen. Tool #14 (Extreme Principle) does the structural cutting twice — once to note that only the longest of the three sides can break the inequality, and once to note that any pair of lengths already past the diameter can never fail, which kills most of the case work before it starts. Tool #1 (Draw a Diagram) supplies the fact that makes six values possible at all: a chord of a circle depends only on how many steps it spans. Tool #2 (Make a Systematic List) then sweeps the surviving pairs in a fixed order and attaches the right multiplicity to each pattern. The one place none of these tools helps is the crux: two of the failures are exact equalities, not near misses, and only algebra decides those.
Six lengths, fixed by the step
The step size gives only six lengths.
Rotate the 12-gon onto itself and any two vertices the same number of steps apart land on another such pair, so only the step can matter.
Turning the polygon onto itself carries any two vertices to another pair the same number of steps apart, so only the step matters.
▸ Why?
A rotation moves the figure without stretching, so the length of a chord is carried along unchanged.
▸ Why?
After a full lap the vertices return to their start, so the step count is all that ever distinguishes a pair.
Write the six lengths exactly
Each length is written out exactly.
Half a chord, a radius, and the perpendicular from the centre form a right triangle, so every chord is 2Rsin(half its central angle).
10.G-SRT.C.8Introduce A VariableCount the segments in each class
Counting per class gives the sample space.
Each class is a ring of twelve segments except the diameters, where walking around the circle traces every one of them twice.
11.S-CP.B.9Make A Systematic ListOnly the longest side can fail
Only the longest side can fail the test.
The longest side plus anything already outreaches a shorter side, so the only live question is whether the two short ones together outreach the long one.
7.G.A.2Extreme PrincipleTwo exact ties, not near misses
Two exact ties appear, not near misses.
Two sines add up into a single cosine, and at these particular angles that cosine is exactly another chord of the same 12-gon.
11.F-TF.C.9Introduce A VariablePairs with no shortest chord
Pairs without the shortest chord always pass.
Once both short sides reach the hexagon and square chords their sum has already passed the diameter, and nothing in the figure is longer than a diameter.
9.A-SSE.A.2Extreme PrincipleThresholds for the pairs starting at a₁
The remaining thresholds are checked one by one.
Each short pair has exactly one threshold, the first length it can no longer reach, and every longer length is out of reach too.
9.A-SSE.A.2Make A Systematic ListWeight the ten failing patterns
Ten failing patterns weigh 10080 in total.
A pattern like {1,1,6} is not one outcome but a whole block of them, so each pattern must carry the weight of how many ways it can actually be picked.
7.SP.C.8Make A Systematic ListTake the complement and divide
The complement gives 223/286, choice (E).
Ten failing patterns are far easier to pin down than forty-six working ones, so count what breaks and keep the rest.
11.S-CP.B.9Change Focus Count The ComplementSort the three lengths first, because only the two short ones against the long one can fail — and settle the close calls with exact radicals, since in a regular 12-gon two of those sums land dead on another length and a flat triangle has no area.
- Six lengths, fixed by the step
- Write the six lengths exactly
- Count the segments in each class
- Only the longest side can fail
- Two exact ties, not near misses
- Pairs with no shortest chord
- Thresholds for the pairs starting at a₁
- Weight the ten failing patterns
- Take the complement and divide