AMC 10 · 2010 · #5
Grade 7 algebraPick an answer.
The word doing all the work is 'guaranteed'. It quantifies over every remaining play of the tournament at once, which looks like infinitely many cases, so Tool #14 (Extreme Principle) is the primary move: replace all of them by the single least favourable one. That replacement is legitimate here, and it is worth saying why instead of assuming it — each archer's final total is a sum of independent shot values, so Chelsea's total is smallest exactly when every free shot of hers sits at its floor, and the opponent's is largest exactly when every shot of his sits at its ceiling. Tool #4 (Introduce a Variable) then names Chelsea's unknown current score so that the comparison can be written down, and it immediately shows that name cancelling. Tool #13 (Convert to Algebra) turns the guarantee into one inequality in n. The step that the algebra alone does not supply is tightness: an inequality that fails on paper only rules out a value of n if the failing scenario is one the rules permit, so Tool #14 returns to build that scenario explicitly with legal scores. Finally Tool #3 (Eliminate Possibilities) uses the fact that extra bullseyes never hurt to show the working values of n run upward without gaps, which is what makes 'the minimum' meaningful. Tool #16 (Change Focus / Count the Complement) supplies the independent cross-check in the review by counting the shots that are not bullseyes.
Turn 'guaranteed' into one worst case
Guaranteed becomes one specific worst case.
A promise is only as strong as the single worst day the rules still allow.
6.EE.B.5Extreme PrincipleName the totals; the current score cancels
Naming the totals makes the current score cancel.
Both totals are built on the same unknown starting score, so subtracting them throws it away and leaves only the gap.
Both totals are built on the same unknown starting score, so subtracting them throws it away and leaves the gap.
▸ Why?
Shifting two quantities by the same amount leaves their difference exactly as it was.
▸ Why?
Winning is a comparison of the two totals, and a comparison only ever cares about that difference.
Solve the inequality and keep it strict
The winning condition is a strict inequality.
Swapping one 4 for a bullseye buys exactly 6 points of margin, and the hole is deep enough to need more than 41 of those swaps.
7.EE.B.4Convert To AlgebraShow the losing case is really allowed
The losing case really is allowed, so smaller counts fail.
One legal scenario is enough to break a guarantee, so the scenario has to be built, not just imagined.
7.EE.B.3Extreme PrincipleConfirm 42 works and nothing smaller can
Checking the neighbours gives 42, choice (C).
An extra bullseye never hurts, so the working values run upward without gaps and the first success is the answer.
6.NS.C.7Eliminate Possibilities'Guaranteed' means you have to survive the worst case, so score your opponent at his best and yourself at your floor — here every bullseye you swap in for a 4 buys 6 points of cushion, and digging out of 250 takes 42 of them.
- Turn 'guaranteed' into one worst case
- Name the totals; the current score cancels
- Solve the inequality and keep it strict
- Show the losing case is really allowed
- Confirm 42 works and nothing smaller can