AMC 10 · 2025 · #14
Grade 6 algebraPick an answer.
The question asks for the greatest n, so the answer sits at a boundary: the last spot where a term is still positive. That is the Extreme Principle. To find that boundary I first turn each average into a sum, spot the pattern that every term drops by 2, then push n until a term would go negative.
Turn averages into sums
The first three sum to 6075.
An average is just the total shared out equally, so multiplying the average by the count gives the total back.
An average is just the total shared out equally, so multiplying it by the count gives the total back.
▸ Why?
An average is a total divided by how many there are, so the two are two readings of one thing.
▸ Why?
That total is the listed numbers added together, so a new term simply adds one more piece.
Write the sum of k terms
The first k sum to k times 2028 minus k.
Each step lowers the running average by exactly 1, so the average is a simple straight-line count downward from 2025.
6.SP.A.3Look For A PatternRecover a single term
Differencing gives the term 2029 minus two k.
The gap between two running totals is exactly the newest number added, so subtracting the totals uncovers that term.
6.EE.A.3Convert To AlgebraPush to the boundary
Positivity forces n at most 1014.
The list can only grow as long as the shrinking terms stay above zero; the last positive term marks the hard stop.
6.EE.B.5Extreme PrincipleConfirm the list works
Such a list really exists, so 1014.
A boundary answer only counts if you can actually build one example that reaches it.
6.SP.B.5Guess And CheckTurn each average into a total, watch the terms fall by 2 each step, and stop counting the moment a term would drop below 1.
- Turn averages into sums
- Write the sum of k terms
- Recover a single term
- Push to the boundary
- Confirm the list works