AMC 10 · 2016 · #21
Grade 10 geometry-2dPick an answer.
An endless picture cannot be drawn all the way, so the picture has to become a formula. Tool #1 (Draw a Diagram) puts the square on coordinate axes with D at the origin, which turns "drop a perpendicular to CD" into "keep the x-coordinate, set y to 0" and turns diagonal BD into the line y = x. Tool #4 (Introduce a Variable) then handles all the rounds at once: call the current foot Q_i = (q, 0) and do the work one time, in terms of q. Tool #13 (Convert to Algebra) converts one full round into a single rule that sends q to the next value, and tool #5 (Look for a Pattern) reads a closed form out of that rule. Tool #7 (Identify Subproblems) splits the goal into two easy pieces, the area of one triangle and then the sum. Tool #15 (Organize Information in More Ways) rewrites each area as a difference of two fractions so the pieces cancel, and tool #9 (Solve an Easier Related Problem) finishes by summing the first N terms and watching what is left over shrink.
Put the square on coordinates
Coordinates make one round computable.
Choosing D as the origin makes both moving parts trivial: the diagonal becomes y=x and a perpendicular foot just erases the height.
10.G-GPE.B.4Draw A DiagramDo one round with a letter
One letter carries the whole round.
One letter stands in for every round at once, so the work is done once instead of infinitely many times.
9.A-CED.A.2Introduce A VariableCross the diagonal to find P_i
The crossing point comes out cleanly.
An intersection point is just the solution of the two line equations at once, and on y=x the point's height equals its distance from D along the base.
9.A-REI.C.6Convert To AlgebraRead the pattern of the feet
The feet follow a simple pattern.
The messy-looking rule q ↦ q/(q+1) is really just "add 1 to the denominator" once the fraction is written as 1/something.
9.F-IF.A.3Look For A PatternArea of one triangle
Each area is a product of neighbours.
With the base flat on the axis, the height is simply how high P_i is, and on the diagonal y=x that height is the next foot's distance from D.
10.G-GPE.B.7Identify SubproblemsRewrite each area as a difference
It rewrites as a difference.
A product of two consecutive denominators is exactly what you get when you subtract their reciprocals, so the term can be split for free.
Each area can be written as a difference of two reciprocals, so the sum collapses as it is added up.
▸ Why?
A number divided by itself undoes the multiplication, so each matched pair disappears.
▸ Why?
Adding a quantity and then subtracting it changes nothing, so only the two unmatched ends survive.
Add up and watch it collapse
The sum collapses to 1/4, choice (B).
Summing a finite chunk first turns an infinite question into a formula whose leftover piece visibly fades to nothing.
9.F-IF.B.4Solve An Easier Related ProblemWhen each piece of an endless sum can be written as one fraction minus the next, almost everything cancels and only the very first fraction is left.
- Put the square on coordinates
- Do one round with a letter
- Cross the diagonal to find P_i
- Read the pattern of the feet
- Area of one triangle
- Rewrite each area as a difference
- Add up and watch it collapse