AMC 10 · 2023 · #25
Grade 12 algebraPick an answer.
The number 2023 is there to intimidate, not to matter. Nothing about the problem depends on that particular value, so the first move is to replace it with 3, then 5, then 7, and watch what the tangent addition formula produces. Those small cases show the shape is real and hand over a candidate pattern for the top coefficient. A pattern from three cases is a guess, though, so the second move supplies the mechanism: multiplying an angle by n is the same as raising a complex number to the nth power, and once both sides are divided by cosⁿ x the whole trig identity collapses into (1 + itan x)ⁿ. The Binomial Theorem then names every coefficient at once — they are binomial coefficients wearing alternating signs — and the top one is the easiest of the lot, since C(2023, 2023) = 1. The last piece of the plan is a separate subproblem the small cases cannot settle: showing no second integer list exists, which is what lets the derived coefficients be called the a_i.
Name the tangent
Give the tangent a short name.
Naming tan x turns a trig identity into a fraction of two polynomials, and a polynomial's leading coefficient is a single visible number.
9.A-SSE.A.1Introduce A VariableTry a small multiple
Ask the same question at three times.
A giant multiplier hides the structure, so shrink it until the structure is something you can write down in one line.
11.F-TF.C.9Solve An Easier Related ProblemTwo more cases
Two more cases fix the pattern.
Pascal's triangle showing up uninvited is a signal that something is being raised to a power behind the scenes.
9.A-SSE.A.2Look For A PatternMultiplying an angle is a power
Multiplying an angle is a complex power.
Dividing by cosⁿ x is what erases the cosines, because a ratio does not care about a factor both parts share.
12.N-CN.B.5Convert To AlgebraThe binomial theorem names them
The binomial theorem gives every coefficient.
Powers of i cycle through i, -1, -i, 1, so a single expansion splits itself into an odd half and an even half with alternating signs.
Powers of the imaginary unit cycle through four values, so one expansion splits into an odd half and an even half.
▸ Why?
A complex number is a point with a direction, and multiplying by the imaginary unit turns it a quarter.
▸ Why?
Four quarter turns make one full turn, which brings every value back exactly where it started.
Check the list is unique
Numerator and denominator are coprime, so it is unique.
Two fractions that agree everywhere must be the same fraction once the top and bottom share no factor and the scale is pinned down.
11.A-APR.C.4Identify SubproblemsRead off the top coefficient
The last coefficient is negative one.
The last coefficient is the easiest one, because the last entry of any Pascal row is just 1 and only its sign is in question.
9.A-SSE.A.1Eliminate PossibilitiesMultiplying an angle by n is the same as raising 1 + itan x to the nth power, so every coefficient in a multiple-angle formula is just a binomial coefficient with a sign on it.
- Give the tangent a short name
- Try the same question on tan 3x
- Two more cases fix the pattern
- Multiplying an angle is raising a power
- The Binomial Theorem names every coefficient
- Check that no second list exists
- Read off the top coefficient