AMC 10 · 2021 · #22
Grade 11 algebraPick an answer.
Do not try to find the polynomial as a whole. Vieta's relations say each coefficient is one symmetric combination of the roots, so abc splits into three separate, smaller questions: the sum of the three cosines, the sum of their three pairwise products, and their product. Each of those three is a short trigonometry exercise once the angles are renamed as θ, 2θ, 4θ, because that renaming exposes a doubling pattern that sine identities collapse.
Split the product into three parts
Write the coefficients as symmetric functions.
A monic cubic's coefficients are nothing but the sum, pair-sum, and product of its roots, so you never have to know the roots individually.
A monic cubic's coefficients are nothing but the sum, the pair sum and the product of its roots.
▸ Why?
The coefficients record exactly those symmetric quantities and nothing else about the roots.
▸ Why?
A cubic vanishes only where one of its linear factors vanishes, so the roots really do rebuild it.
Rename as a doubling ladder
The angles form a doubling ladder.
Renaming the angles as θ, 2θ, 4θ turns a random-looking trio into a doubling ladder, which is exactly the shape sine identities know how to climb.
11.F-TF.A.2Introduce A VariableFind the sum of the roots
Multiplying by a sine makes it telescope.
Multiplying a run of cosines by the right sine turns each one into a difference of sines, and a chain of differences telescopes down to just its two ends.
11.F-TF.C.9Look For A PatternFind the pairwise sum
Turning products into sums reproduces the earlier value.
This set of angles is closed under adding and subtracting, so a product of two of these cosines always folds back into the same three cosines.
11.F-TF.C.9Organize Information In More WaysFind the product of the roots
The sine double-angle formula collapses the product.
Each doubling halves the front coefficient, and after three doublings the angle has travelled a full circle and landed back where it started.
11.F-TF.A.2Look For A PatternMultiply the three coefficients
Multiplying gives one thirty-second.
Two negative factors make a positive product, so the answer had to be one of the positive choices.
9.A-SSE.A.1Identify SubproblemsYou never need the roots themselves: a monic cubic's coefficients are just the roots' sum, pair-sum, and product, and multiplying a chain of cosines by the right sine makes each of those three collapse to a simple fraction.
- Vieta splits abc into three parts
- Rename the angles as a doubling ladder
- Sum of the roots is -1/2
- Pairwise sum is also -1/2
- Product of the roots is 1/8
- Multiply the three coefficients