AMC 10 · 2007 · #24
Grade 11 algebraPick an answer.
An equation of the form (something) = (something) has no root count attached to it, but an equation of the form (product) = 0 does: its solutions are exactly the solutions of the factors, and those can be listed. So the whole problem turns on Tool #15 (Organize Information in More Ways) applied once, at the very start — move everything to one side and rewrite the difference sin(nx) - sin x as a product. Because that rewriting is an identity, nothing is gained or lost, and the count becomes the size of a union of two explicit lists. From there Tool #12 (Draw a Venn Diagram) supplies the only remaining idea: the size of a union is the sum of the two sizes minus the size of the overlap, so the entire difficulty of the problem is compressed into one question — when do the two lists share a point? That is the step the problem actually turns on, and it is where a solution based on graphs quietly fails, because the shared point is a place where the curves touch rather than cross. Tool #4 (Introduce a Variable) is what makes the overlap question easy: naming the two half-angles u and v reveals that their difference is x itself, which pins the only possible shared point to x = π/2 before any divisibility argument is needed. Tool #7 (Identify Subproblems) splits the finish into counting each list separately and then summing over n, and Tool #9 (Solve an Easier Related Problem) checks the formula against n = 2 and n = 5, which are small enough to solve by bare hands.
Rewrite the difference as a product
Rewriting the difference as a product splits the equation.
Two sine waves agree exactly when the wave that measures their difference is flat at zero, and the difference of two sines is always a product of one cosine and one sine.
11.F-TF.C.9Organize Information In More WaysTwo lists of solutions
Each factor gives its own family of solutions.
Once the equation reads "product equals zero", every solution is just a place where one of the two factors hits a zero it hits regularly, so the solutions arrive in two evenly spaced families.
11.F-TF.A.2Introduce A VariableCount each list separately
Counting them separately gives a total of n plus one.
Each list is a ruler with evenly spaced marks laid along [0,π], so counting it is just asking how many marks fit before the ruler runs off the end.
Each list of solutions is a ruler with evenly spaced marks laid along the interval, so counting is asking how many marks fit.
▸ Why?
The solutions of one factor come at the same fixed spacing every time, so they form an evenly stepped list.
▸ Why?
Each mark sits at a fixed fraction of the full turn, so the interval holds a definite whole number of them.
The lists can only meet at pi/2
The families can only meet at one point, and only sometimes.
The gap between the two half-angles is the unknown itself, so a shared solution has to sit exactly a quarter turn from the origin — the one spot where both sine curves peak together.
8.EE.C.8Draw A Venn DiagramAssemble the formula for F(n)
That gives a two-case formula.
Every n gives n+1 solutions, and one in every four values of n loses a solution to a double-booking at the midpoint.
9.A-SSE.A.1Solve An Easier Related ProblemAdd it up
Summing with the correction gives 2016532, choice (E).
Sum the clean formula over every n first, then pay back one unit for each of the 501 values of n that lost a solution.
9.A-SSE.A.2Identify SubproblemsTurn "two waves are equal" into "a product is zero", count the two evenly spaced lists of zeros, and subtract the single place they can collide — the midpoint π/2, where both waves peak at once.
- Rewrite the difference as a product
- Two lists of solutions
- Count each list separately
- The lists can only meet at pi/2
- Assemble the formula for F(n)
- Add it up