AMC 10 · 2024 · #5
Grade 6 algebraPick an answer.
Tool #13 (Convert to Algebra) turns the verbal "flip plus to minus" rule into clean arithmetic: each flip of x subtracts 2x from the original total S = 2500, so the new sum is S - 2 · (sum of flipped terms) and we need this ＜ 0, i.e. the flipped terms must sum to more than 1250. Tool #16 (Change Focus) is the strategic insight that lets us minimize the count — instead of asking "how many flips?", ask "what's the maximum reduction per flip?" and you get the greedy choice 99, 97, 95, …. Tool #6 (Guess and Check) finishes by testing the two adjacent candidate values k = 14 and k = 15 in the closed-form sum.
Find the total
The first fifty odds add to 2500.
Pairing the first and last terms (1 + 99 = 100) and walking inward is the same equivalent-expressions trick Grade 6 introduces for arithmetic sums.
Pairing the first term with the last and walking inward makes every pair add to the same amount.
▸ Why?
In an evenly spaced list, moving inward raises one partner as much as it lowers the other.
▸ Why?
Consecutive whole numbers climb by the same fixed step, which is what makes the list evenly spaced.
What one flip costs
Each flip removes twice that number.
Letting F stand for "sum of flipped numbers" turns the verbal rule into a clean expression — the Grade 6 "letters stand for numbers" idea.
6.EE.A.2Convert To AlgebraSet the target inequality
The flipped numbers must exceed 1250.
Solving "strictly negative" for the flipped-sum gives the inequality F ＞ 1250 — a Grade 6 "write/solve inequality of the form x ＞ c" step.
6.EE.B.8Convert To AlgebraTake the biggest first
With k flips the best is k times a hundred minus k.
Picking the heaviest numbers first is the "do the most with the fewest" extreme-principle move; the arithmetic-series formula is a Grade 6 equivalent-expressions calculation.
6.EE.A.3Change Focus Count The ComplementCheck the boundary
Fourteen fails, 15 works.
Two two-digit multiplications bracket the threshold cleanly — Grade 5 multi-digit multiplication is enough.
5.NBT.B.5Guess And CheckThis AMC 12 problem only needs Grade 6 inequalities (turn "new sum negative" into F ＞ 1250) plus Grade 5 multi-digit multiplication — flip the biggest odd numbers first and you cross the line at flip 15!