AMC 10 · 2021 · #22
Grade 6 arithmeticPick an answer.
AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #7 (Subproblems) — split the count and the sum: removed sheets contribute removed-pages count = 2c and removed-page sum = a known arithmetic-series formula. Tool #13 (Algebra) — set up one equation in a (starting sheet) and c (count) from "mean of remaining = 19", and use 325 = small factorization to enumerate cases. Tool #6 (Guess & Check) — once the equation reduces to c(linear in a, c) = 325 = 5² · 13, test each divisor of 325 as c. Tool #3 (Eliminate) — answer choices 10, 13, 15, 17, 20 are themselves a tight set; only the divisors of 325 among them survive.
Let c sheets from sheet a be removed; the 2c removed pages are consecutive, summing to c(4a + 2c - 3).
Sum of 2c consecutive integers starting at 2a-1 — arithmetic-series formula gives c · (first+last).
6.EE.A.2Identify SubproblemsSetting the remaining mean to 19 over the total sum 1275 and clearing the fraction collapses to c(4a + 2c - 41) = 325.
Translate "mean = 19" into a single equation, then collect the c on one side as a factor.
6.EE.B.7Convert To AlgebraSince 325 = 5² · 13, its divisors that stay under 25 sheets leave only c ∈ {1, 5, 13} to test.
325 is small enough to factor by hand — list its divisors and keep the ones ≤ 24.
6.NS.B.4Eliminate PossibilitiesTesting c = 1 gives 4a = 364, so a = 91 — impossible with only 25 sheets, so drop it.
Plug c=1 into the linear equation and check whether a falls in [2, 24].
6.EE.B.7Guess And CheckTesting c = 5 gives a = 24, but then the last removed sheet is 28 ＞ 25 — out of range, so drop it.
c=5 forces the block to spill past sheet 25 — out of range.
6.EE.B.7Guess And CheckTesting c = 13 gives a = 10, removing sheets 10 to 22 — both ends interior, matching "from the middle", so it works.
c = 13 gives integer a = 10 inside the valid range — the unique surviving case.
6.EE.B.7Guess And CheckRemoved pages 19 to 44 sum to 819, so remaining 456 over 24 pages gives mean = 19, confirming the count.
Always plug the chosen c back into the original mean condition before declaring victory.
6.SP.A.3Eliminate PossibilitiesThis AMC 10 problem only needs Grade 6 mean and the equation tools you already know — write the total page sum (1275) and the removed-sum formula (c(4a + 2c - 3)), set the remaining mean to 19 to get c(4a + 2c - 41) = 325, then factor 325 = 5² · 13 and test c ∈ {1, 5, 13} — only c = 13 gives a valid block (sheets 10 to 22, pages 19 to 44).