AMC 10 · 2014 · #24
Grade 9 algebracountingPick an answer.
Nobody can write out f₁₀₀ as a formula, so the plan splits the problem in two along its natural seam. Tool #9 (Solve an Easier Related Problem) replaces 'iterate a whole function 100 times' with 'iterate a single number 100 times under t↦ |t|-1', because after the first step the outcome depends only on the one number f₀(x). Tool #11 (Work Backwards) then runs that easier problem in reverse: instead of guessing what f₁₀₀ looks like from a picture of f₁,f₂,f₃, it asks which starting numbers are still sitting on 0 after exactly 100 steps, and answers that question with proof — the starting number must be an even whole number of size at most 100. Tool #7 (Identify Subproblems) then unfolds f₀ into three straight-line pieces by splitting the number line at x=-100 and x=100, so each allowed target value can be hit at most once per piece. Tool #2 (Make a Systematic List) finishes: list the allowed target values, count how many x each one has, and add. Counting the exact preimages beats extrapolating a pattern from a graph, because the whole difficulty of this problem lives in the two endpoints where two of the three pieces collide.
Unfold f₀ into three lines
The first function is three straight pieces.
Absolute value is just 'distance', so once you know which side of 100 and -100 you are on, every bar comes off and only straight lines are left.
6.NS.C.7Identify SubproblemsFollow one number through the rule
Following one number shows what survives.
The rule marches a number down toward 0 one unit per step, so only a whole number lands exactly on 0 instead of stepping past it into a permanent bounce.
The rule marches a number down toward zero one unit per step, so only the right parity lands exactly on zero.
▸ Why?
Each step changes the value by the same fixed amount, so the position after many steps is fully predictable.
▸ Why?
A number an odd distance from zero can only step over it, never onto it, so parity decides everything.
Turn f₁₀₀(x)=0 into a rule for f₀
The target becomes a short list of first values.
To be sitting on 0 after exactly 100 steps you must be close enough to reach it and have the right parity to be standing on it, not on -1.
9.F-IF.A.2Work BackwardsSolve f₀(x)=v on each piece
Each value is solved on each piece.
A horizontal line cuts the zigzag once per slanted piece, except at the very peak and the very valley where two pieces meet at the same point.
8.EE.C.7Identify SubproblemsCount the targets and add
Adding the counts gives 301, choice (C).
Almost every allowed height is hit three times; only the two extreme heights are hit twice, and that is the whole correction.
6.NS.C.6Make A Systematic ListThe rule 'take the size, then subtract 1' walks a number down to 0 one step at a time, so after 100 steps you are on 0 only if you started at an even whole number no bigger than 100 — then count the x that produce those starts: three each, except at the peak and the valley.
- Unfold f₀ into three lines
- Follow one number through the rule
- Turn f₁₀₀(x)=0 into a rule for f₀
- Solve f₀(x)=v on each piece
- Count the targets and add