AMC 10 · 2019 · #25
Grade 10 geometry-2dPick an answer.
Tracking actual points for 15 rounds is hopeless, but the question only asks about angles. Tool #4 (Introduce a Variable): name the angles x_n, y_n, z_n so each round becomes a number update instead of a construction. Tool #1 (Draw a Diagram): the two right angles at an altitude foot put four points on one circle, which is what converts the picture into an angle equation. Tool #5 (Look for a Pattern): that equation, x_n = 180° - 2x_n-1, is the same every round. Tool #9 (Easier Related Problem): instead of the angle itself, follow only its gap from 60° — the gap obeys the much simpler rule "multiply by -2". Tool #14 (Extreme Principle): the triangle turns obtuse the first moment the largest gap crosses 30°, so the whole problem collapses to one inequality about powers of 2.
See what one round builds
Look at the new triangle one round builds.
The three altitude feet are the only new points, so the whole process is one repeated construction.
10.G-CO.A.1Draw A DiagramName the angles, not the points
Name the angles, not the points.
The question asks only about shape, and three angles pin down a triangle's shape completely.
10.G-CO.C.10Introduce A VariableTwo right angles make a circle
Two right angles put four points on one circle.
Any two points that view a segment at 90° sit on the circle with that segment as diameter.
Any two points that view a segment at a right angle sit on the circle with that segment as diameter.
▸ Why?
The midpoint of that segment is equally far from both viewing points, so all of them share one radius.
▸ Why?
An angle at the circle measures the far arc, and a right angle is exactly the half-circle arc.
One rule, repeated every round
The same rule repeats every round.
Every round runs the identical construction, so the identical angle equation must come out.
10.G-CO.C.9Look For A PatternFollow the gap from 60 degrees
The gap from sixty doubles each round.
Measuring from the point that never moves turns a messy rule into plain doubling.
9.F-BF.A.1Solve An Easier Related ProblemPlug in the three starting gaps
Plug in the three starting gaps.
A tiny 0.001 head start doubles every round, and doubling eats any head start fast.
8.EE.A.1Look For A PatternConfirm the rule stayed legal
Confirm the rule stayed legal.
A rule proved for acute triangles is only trustworthy if every triangle before the answer is still acute.
10.G-CO.C.10Extreme PrincipleFind the first round past 90
The first round past ninety degrees is 15.
The gap crosses 30° the moment a power of 2 clears 30000.
9.A-CED.A.1Extreme PrincipleEach round replaces every angle x by 180° - 2x, which leaves 60° alone and doubles the distance from 60° while flipping its sign — so a gap of 0.001° needs 0.001 × 2ⁿ ＞ 30, and the first time that happens is n = 15.
- See what one round builds
- Name the angles, not the points
- Two right angles make a circle
- One rule, repeated every round
- Follow the gap from 60 degrees
- Plug in the three starting gaps
- Confirm the rule stayed legal
- Find the first round past 90