AMC 10 · 2019 · #4
Grade 6 number-theoryPick an answer.
Tool #9 (Easier Problem): try a smaller target sum first, like 'consecutive integers summing to 5'. The trick of including negatives so most terms cancel pops out: -4, -3, …, 4, 5 sums to 5 and has 10 terms. Tool #1 (Diagram): a number line shows the symmetric block from -k to +k summing to 0, with a tail {k+1, …, 45} paying for the 45. Tool #5 (Pattern): once we see the smaller case, generalize — use -44 to 45 for sum 45, giving 90 terms. Tool #3 (Eliminate): the choices climb 9, 25, 45, 90, 120. Anything ≤ 45 ignores negatives. 120 is too long because the symmetric trick caps at 90.
Get a feel with a small case
A short example shows the cancelling.
Negatives cancel positives in pairs — the only 'survivor' is the top number.
6.NS.C.6Solve An Easier Related ProblemWhat a symmetric block does
A symmetric block sums to zero.
Pattern from the warm-up — sum survives, length doubles.
In a block running symmetrically about zero, negatives cancel positives in pairs and only the top survives.
▸ Why?
Each number and its opposite add to nothing, so every such pair drops out of the total.
▸ Why?
Pairing from the two ends inward matches every term with its opposite, leaving exactly one unmatched.
Build the longest run
Set the two ends so the leftover is the target.
Same pattern as the warm-up, just bigger: from -(S-1) to S gives 2S terms.
6.NS.C.6Look For A PatternBound it with the formula
The term count must divide the doubled target.
Length must divide 90 — and 90 itself works.
6.EE.B.7Solve An Easier Related ProblemMatch the choices
The largest that passes is 90.
Among the choices, only divisors of 90 are achievable; biggest one is 90.
4.OA.B.4Eliminate PossibilitiesThis AMC 12 problem only needs Grade 6 "negatives on the number line" you already know — start at -44 and run up to 45 so every pair (-k, k) cancels and only 45 is left, giving 90 terms.