AMC 10 · 2018 · #16
Grade 12 geometry-2dPick an answer.
The question asks for a minimum over a finite set of choices, which is exactly the Extreme Principle: find the shape that pushes the area as far down as it can go, then prove nothing beats it. Getting there needs three cheaper moves first. The +6 is decoration, so I substitute w = z+6 and solve the easier equation w⁸ = 81 centered at the origin (Solve an Easier Related Problem). Polar form with a named radius and angle turns that equation into eight evenly spaced points, and drawing them shows a regular octagon (Introduce a Variable, Draw a Diagram). Then, because the octagon's symmetry means only the gaps between chosen vertices matter, I can list every essentially different triangle — there are only five (Make a Systematic List) — compare them, and finally check the winning value against the answer choices (Eliminate Possibilities).
Slide the picture to the origin
Slide the picture to the origin.
Sliding a shape across the plane never changes its area, so solve the version that sits neatly around the origin.
11.N-CN.A.2Solve An Easier Related ProblemPin every root to one circle
All roots sit on one circle.
Taking a power multiplies the distance from the origin, so one size equation locks all eight roots onto the same circle.
Taking a power multiplies the distance from the origin, so one size equation locks every root onto one circle.
▸ Why?
A complex number is a point with a length and a direction, and multiplying multiplies the lengths.
▸ Why?
An exponent counts how many times a factor is used, so the length is raised to that same power.
Space the eight roots evenly
The eight roots are evenly spaced.
The eighth power multiplies the angle by eight, so the roots have to be spaced one eighth of a full turn apart.
12.N-CN.B.5Draw A DiagramOnly the gaps matter
Only the gaps matter.
The octagon's own symmetry means a triangle is fully described by how many steps separate its three corners.
10.G-CO.A.3Make A Systematic ListTurn each shape into a product of sines
Each shape becomes a product of sines.
Two sides and the angle between them decide the area, and on a circle all three are read straight off the arcs.
11.G-SRT.D.9Introduce A VariableCompare the five and take the minimum
Compare the five and take the smallest.
Pushing two of the three gaps down to the smallest size they can take makes the thinnest possible sliver.
11.F-TF.A.2Extreme PrincipleMeasure the three consecutive vertices
Measure the three consecutive vertices directly.
Picking the trio that straddles the imaginary axis makes the base flat, so the height is just one subtraction.
10.G-GPE.B.7Draw A DiagramSimplify and match a choice
Tidying gives three root two over two, minus three halves.
Both radical products collapse to whole numbers, leaving a plain difference.
11.N-RN.A.2Eliminate PossibilitiesSlide the picture so the roots ring the origin, see that they form a regular octagon, then grab three vertices sitting right next to each other — the thinnest sliver wins.
- Slide the picture to the origin
- Pin every root to one circle
- Space the eight roots evenly
- Only the gaps matter
- Turn each shape into a product of sines
- Compare the five and take the minimum
- Measure the three consecutive vertices
- Simplify and match a choice
More like this
Same archetype — closest grade level first.
