AMC 10 · 2023 · #12
Grade 11 algebraPick an answer.
The operation ⊗ is defined piece by piece, in terms of the real part and the imaginary coefficient of each input. So it says nothing at all while z is still a single letter. Tool #4 (Introduce a Variable) is therefore the entire opening move: write z=a+bi with a and b real, and both sides of the equation turn into something you can actually compute. Tool #15 (Organize Information in More Ways) does the next lift: one equation between complex numbers is secretly two equations between real numbers, and you only see them once each side is sorted into a real part and an imaginary part. Tool #7 (Identify Subproblems) keeps the algebra small, because those two equations do not have to be solved together — the real-part equation hands over b² all by itself, and only then does the imaginary-part equation tie a to b. Tool #3 (Eliminate Possibilities) finishes the second equation: factoring it produces two branches, b=0 and b=2a, and one of them has to be ruled out before you can move on.
Write it with parts showing
The new operation does not mix.
A definition written in terms of components is invisible until you write the components down.
11.N-CN.A.1Introduce A VariableSquare it the ordinary way
Ordinary squaring mixes the parts.
Ordinary multiplication mixes the parts, because i²=-1 drags b² into the real part with a minus sign — exactly the move that ⊗ refuses to make.
11.N-CN.A.2Introduce A VariableSplit one equation into two
It becomes two real equations.
One complex equation is two real equations stacked on top of each other, because the real and imaginary directions are independent.
One complex equation is two real equations stacked on top of each other.
▸ Why?
A complex number is a point with a horizontal and a vertical coordinate, and the two never mix.
▸ Why?
Two complex numbers are equal only when their matching parts are equal, part for part.
What the real part gives
The real part hands over one square.
A term that appears identically on both sides tells you nothing — cancel it and the equation says what it really meant.
9.A-REI.B.3Identify SubproblemsWhat the imaginary part gives
The imaginary part pins the rest.
Factor rather than divide, then let a fact you already own kill off the branch that cannot happen.
9.A-REI.B.4Eliminate PossibilitiesAssemble the modulus
Assembling gives five root two.
The modulus formula only ever wants a² and b², so stopping at the squares was already enough.
11.N-CN.A.3Introduce A VariableWhen a problem invents its own operation, write the number as a+bi and follow the definition literally — then match real parts with real parts and imaginary with imaginary, and the single equation splits into two easy ones.
- Write z with its parts showing
- Square z the ordinary way
- Split one equation into two
- The real part hands over b squared
- The imaginary part pins down a squared
- Assemble the modulus