AMC 10 · 2021 · #18
Grade 11 algebraPick an answer.
Absolute-value bars block algebra, so Tool #15 (Organize Information in More Ways) rewrites each |w|² as the product ww — the same quantity, now expandable. Once that is done, every expansion produces only two combinations, z + z and zz, because the equation cannot distinguish z from its conjugate. Tool #4 (Introduce a Variable) names those two combinations s and p; both are real, so Tool #13 (Convert to Algebra) turns a statement about a complex number into one ordinary real equation in s and p. Completing the square collapses that equation to a sum of two squares equal to zero, and Tool #14 (Extreme Principle) finishes it: a square is never smaller than zero, so a zero total forces each square to be zero and pins s and p exactly.
Trade moduli for conjugate products
Rewrite each as a conjugate product.
A modulus hides a square root, but pairing a number with its conjugate replaces it by an ordinary product you can expand.
11.N-CN.A.3Organize Information In More WaysName the two real quantities
Both the sum and the product are real.
The equation cannot tell z apart from z, so it can only depend on their sum and their product.
9.A-CED.A.2Introduce A VariableExpand each term
Write every term with those two.
Both expansions collapse onto s and p alone, which is exactly the payoff of naming them.
11.N-CN.A.2Introduce A VariableAssemble one real equation
Every imaginary part disappears.
Once every piece is written through s and p, a question about complex numbers becomes plain two-variable real algebra.
9.A-SSE.A.2Convert To AlgebraComplete the square twice
Two perfect squares appear.
Completing the square regroups a scattered equation into squared distances, and here the total distance turns out to be zero.
9.A-SSE.B.3Organize Information In More WaysA zero sum forces both
The zero sum forces each to vanish.
Zero is the smallest value a square can take, so a zero total leaves no room for either square to be positive.
Zero is the smallest value a square can take, so a zero total leaves no room for either square to be positive.
▸ Why?
If either square were positive the total would be positive, since the other can never pull it back down.
▸ Why?
A total of zero means nothing is left over, so neither piece can contribute anything at all.
Read off the requested value
The value is negative two.
The 6 was planted to match |z|², which turns the fraction into the conjugate and the whole expression into a quantity already known.
11.N-CN.A.3Organize Information In More WaysReplace each |w|² by ww, name the two real quantities z + z and zz, complete the square until a sum of squares equals zero — then 6/z is simply z.
- Trade moduli for conjugate products
- Name the two real quantities
- Expand each term in s and p
- Assemble one real equation
- Complete the square twice
- A zero sum forces both squares
- Read off the requested value