AMC 10 · 2024 · #23
Grade 12 algebraPick an answer.
Each individual tangent here is an ugly nested radical, so any plan that computes them one at a time is doomed. Two observations rescue the problem, and they fit together. First, four numbers make six pairwise products, and the sum only asks for four of them — so the natural move is to take all six and subtract the two that were skipped, which turns a lopsided expression into a fully symmetric one. Second, the four angles are exactly the odd multiples of π/16 below π/2, which is precisely the condition cos 8θ = 0. That condition converts into a single polynomial equation in tanθ, so the four squared tangents become the four roots of one quartic. Symmetric functions of roots are exactly what a polynomial's coefficients already are, so the symmetric sum can be read off a coefficient without ever computing a single tangent.
Name the four numbers
The target uses four of the six pairs.
Four things make six pairs, so a sum of only four products is the full pair-sum with two pieces cut out.
9.A-SSE.A.1Introduce A VariableAsk for all six, subtract the strays
The target is all six minus two.
Add the missing terms back and subtract them again, and a lopsided sum turns into a symmetric one.
9.A-SSE.A.2Change Focus Count The ComplementThe skipped pairs are complementary
Being complementary, each stray pair is 1.
Two angles that add to a right angle have tangents that multiply to 1, so those products collapse to nothing.
Two angles that add to a right angle have tangents that multiply to one, so those products collapse.
▸ Why?
A tangent is the far leg over the near one, and the two angles trade those legs between them.
▸ Why?
A number multiplied by its reciprocal gives one, so the pair contributes nothing to the total.
Turn eight-fold angles into a polynomial
De Moivre hands us a quartic.
Multiplying an angle by 8 is the same as raising a complex number to the eighth power, which is how a trig condition becomes a polynomial.
12.N-CN.B.5Convert To AlgebraConfirm all four roots are accounted for
The four values are exactly the quartic's roots.
Four distinct numbers that all satisfy a degree-four equation must be its complete list of roots.
11.A-APR.B.3Make A Systematic ListRead the answer off a coefficient
Seventy minus two is 68.
Vieta lets a coefficient answer a question about the roots, so a symmetric sum is free even when every individual root is hideous.
11.A-APR.C.4Convert To AlgebraWhen a sum leaves a couple of terms out, put them all back in and subtract the strays — the complete symmetric sum is the one a polynomial will hand you for free.
- Name the four numbers
- Ask for all six, subtract the strays
- The skipped pairs are complementary
- Turn eight-fold angles into a polynomial
- Confirm all four roots are accounted for
- Read the answer off a coefficient