AMC 10 · 2023 · #16
Grade 11 algebraPick an answer.
The word "maximum" hands the finish to Tool #14 (Extreme Principle): once the height is written as a function of one quantity, the winner is whichever allowed value of that quantity pushes the function to its edge. Getting there starts with Tool #4 (Introduce a Variable) — an absolute value of a complex expression says nothing until z=x+yi turns it into ordinary real numbers. Tool #13 (Convert to Algebra) then squares the condition into a single polynomial equation in x and y, which looks hopeless at first sight. Tool #15 (Organize Information in More Ways) is what rescues it: the same block x²+x+1 hides inside both halves of that equation, and naming it turns a messy two-variable mess into a tidy quadratic in y². Tool #7 (Identify Subproblems) then splits the work in two that do not interfere — "for a fixed value of that block, how big can y² be" and "how small can the block itself be." Tool #3 (Eliminate Possibilities) closes the problem: √(m)/n could in principle be written many ways, and the coprime condition kills all but one pair.
Write it in parts
Write it in real and imaginary parts.
A complex number is just two real numbers wearing one name, so splitting z into x and y turns one hard condition into ordinary algebra.
11.N-CN.A.2Introduce A VariableTurn the size into an equation
The size becomes one equation.
Squaring the modulus trades one square root for one clean polynomial equation, and nothing is lost because both sides were already non-negative.
Squaring the size trades one square root for one clean polynomial equation, and nothing is lost.
▸ Why?
The size of a complex number is the hypotenuse over its two coordinates, so squaring it is a plain sum.
▸ Why?
Squaring both sides of a true equation between nonnegative numbers keeps it true and reversible.
Spot the repeated block
The same block appears twice.
When the same block shows up in every corner of an expression, name it — the two-variable problem quietly becomes a one-variable one.
9.A-SSE.A.2Organize Information In More WaysSolve for the squared height
A quadratic gives the squared height.
Freezing A turns the curve into a plain quadratic, and the discriminant collapsing to something linear is the sign this substitution was the right one.
9.A-REI.B.4Identify SubproblemsPush the block to its minimum
The height peaks when the block is smallest.
If a quantity only ever decreases as A grows, the maximum sits at the smallest A the problem allows — and completing the square names that smallest A instantly.
9.A-SSE.B.3Extreme PrincipleMatch the form and add
Adding the two numbers gives 21.
The coprime condition is not decoration — it is what makes the pair (m,n) unique, so the answer is not ambiguous.
11.N-RN.A.2Eliminate PossibilitiesGive z coordinates, square the size condition, and the same block x²+x+1 shows up everywhere — name it, and the curve turns into a quadratic in y² whose top sits exactly where that block is smallest.
- Write z in real and imaginary parts
- Turn the size condition into an equation
- Spot the hidden repeated block
- Solve for y squared in terms of A
- Push A down to its smallest value
- Match the form and read off m + n