AMC 10 · 2005 · #20
Grade 9 algebraPick an answer.
The two group sums are not independent — they always add up to the total of the whole set. So we name one of them s (Tool #4), which turns a two-variable minimisation into a one-variable quadratic we can complete the square on. The whole problem then collapses to a boundary question (Tool #14): how close to the balance point can a group sum get? Answering it honestly requires proving that perfect balance is impossible, and for that a systematic list of the possible pair sums (Tool #2), organised by parity, eliminates every case (Tool #3). Finding one split that scores well is not a proof of minimality on its own; the impossibility half is what makes the answer certain.
The letters use every number
The two totals always add to a fixed number.
Nothing is left out and nothing is repeated, so the two sums are locked to a fixed total.
7.NS.A.1Introduce A VariableReduce to one variable
So the expression depends on a single unknown.
Choosing the first group decides the second, so one number carries all the information.
7.EE.B.4Introduce A VariableComplete the square
Completing the square shows only the distance from the balance point matters.
Splitting a fixed total as evenly as possible is what makes a sum of squares small, and each step away from even costs a fixed jump.
Splitting a fixed total as evenly as possible is what makes the sum of squares smallest, and each step away costs more.
▸ Why?
For a fixed total, the parts sit best when they are equal, and pulling them apart always makes the combination worse.
▸ Why?
Whatever one group gains the other loses, so the total never moves and only the split does.
Ask whether a group can sum to 4
A parity count narrows the shapes a balanced group could take.
Odd numbers only combine in pairs to make an even total, which cuts the search down to three shapes.
2.OA.C.3Eliminate PossibilitiesRule out all-odd and all-even
The all-odd and all-even shapes both fail.
With only four odds and four evens available, these two extreme shapes each describe just one group.
7.NS.A.1Make A Systematic ListRule out two odd plus two even
The mixed shape fails too, so the balance point is unreachable.
The odd pairs land either far below or well above the target, and the even pairs are never large or negative enough to close the gap.
7.NS.A.1Make A Systematic ListBound it, then hit the bound
A split attains the next bound, so the minimum is 34, choice (C).
A minimum needs two things: a reason nothing can go lower, and one example that gets there.
9.A-SSE.A.2Extreme PrincipleWhen two sums must add to a fixed total, keeping them as equal as possible makes the sum of their squares smallest — and if perfect balance is impossible, the next-closest split wins.
- The letters use every number
- Reduce to one variable
- Complete the square
- Ask whether a group can sum to 4
- Rule out all-odd and all-even
- Rule out two odd plus two even
- Bound it, then hit the bound