AMC 10 · 2003 · #24
Grade 9 algebraPick an answer.
The sum of three absolute values is a chain of segments whose slopes climb -3,-1,1,3. The line has slope -2, which sits strictly between the first two. That single fact is the whole problem: tool #16 (Change Focus) says to stop comparing two graphs and instead watch the gap between them, g(x)=|x-a|+|x-b|+|x-c|-(2003-2x). Adding 2x pushes every slope up by 2, so g has slopes -1,1,3,5: it falls, then rises, with its only turn at x=a. Tool #14 (Extreme Principle) then finishes it — a function that falls and then rises hits zero exactly once only when its lowest value is zero, which turns "exactly one solution" into a plain equation. Tool #1 (Draw a Diagram) keeps the four-piece shape in view, and tool #4 (Introduce a Variable) names that gap function so the slopes can be tracked.
Collapse the system to one equation
Substituting the line leaves one equation in x, so meetings are its roots.
Two graphs meet where their y-values agree, so the pair of equations becomes one equation in x.
8.EE.C.8Introduce A VariableRead the four slopes of the V-chain
The bent chain has slopes -3, -1, +1, +3 across its four stretches.
Each absolute value switches from falling to rising at its own corner, so the total slope steps up by 2 at a, at b, and at c.
9.F-IF.B.4Draw A DiagramWatch the gap instead of the two graphs
Adding the line lifts every slope by two, so the gap turns upward at the first corner.
Tilting the picture until the line becomes flat turns "where do they cross?" into "where is this one function zero?".
9.F-IF.B.4Change Focus Count The ComplementOne root only when the bottom touches zero
Such a shape meets zero exactly once only when its minimum is zero.
A V-shaped graph meets a horizontal level twice, never, or exactly once — and "once" happens only at its lowest point.
A V-shaped graph meets a horizontal level exactly once only at its very lowest point.
▸ Why?
The graph falls to a bottom and then rises, so any level above the bottom is reached twice, once on each side.
▸ Why?
That bottom value is the separate distances added together, so it can be computed without drawing anything.
Compute the bottom value
Evaluating there gives the clean condition b + c = 2003, with a dropping out.
The two distances from a out to b and c are exactly the height the line has to make up at x=a.
9.A-REI.B.3Introduce A VariablePush c as low as the condition allows
With b smaller than c, that forces c = 1002 at least, and it is attained, choice (C).
Two different positive integers adding to 2003 are closest to even when they are 1001 and 1002, so the larger one cannot dip below 1002.
9.A-CED.A.3Extreme PrincipleWhen a V-shaped graph meets a line, subtract the line first: the pair crosses exactly once only when the bottom of the V lands right on it.
- Collapse the system to one equation
- Read the four slopes of the V-chain
- Watch the gap instead of the two graphs
- One root only when the bottom touches zero
- Compute the bottom value
- Push c as low as the condition allows