AMC 10 · 2020 · #19
Grade 8 number-theoryPick an answer.
Tool #9 (Easier Problem): the exponents 289 and 17 are huge — replace them with the substitution x = 2¹⁷ to reveal a cleaner (x¹⁷+1)/(x+1), then drop further to the tiny prototype (x³+1)/(x+1) with x=2² to see the mechanism. Tool #5 (Pattern): the alternating polynomial quotient x¹⁶ - x¹⁵ + … + 1 is a clean pattern. Tool #7 (Subproblems): split each subtraction 2^A - 2^B into a string of 1-bits. Tool #2 (Systematic List): once we know each pair contributes 17 bits, list the pair-blocks and sum.
Substitute to simplify
Substituting gives a familiar polynomial division.
Naming x = 2¹⁷ turns a scary problem into the much friendlier (x¹⁷+1)/(x+1).
8.EE.A.1Solve An Easier Related ProblemWrite the quotient
The quotient alternates in sign.
The same alternating pattern that factors x³+1 factors x¹⁷+1 — just longer.
The same alternating pattern that factors a small sum of like powers factors a long one.
▸ Why?
A sum of like odd powers always carries the sum of the bases as a factor.
▸ Why?
Once it is a product, dividing by that factor leaves exactly the alternating quotient behind.
Return to powers of two
Undo the substitution.
Each polynomial term becomes one power of 2; signs alternate, ending with +1.
8.EE.A.1Solve An Easier Related ProblemPair the neighbours
Pair neighbours to kill the negatives.
Group +,- adjacent terms into 8 pairs; the rightmost +1 has no partner.
8.EE.A.1Identify SubproblemsExpand one pair
Each pair becomes a run of consecutive bits.
2¹⁷ - 1 in binary is seventeen 1s in a row — a tiny pattern reused 8 times.
8.EE.A.1Solve An Easier Related ProblemCount the blocks
Count the blocks and their width.
Each pair fills a clean 17-wide window of exponents — no overlaps.
8.EE.A.1Make A Systematic ListAdd it all up
Including the final one gives 137.
Eight 17-bit blocks plus one stand-alone bit = 137.
8.EE.A.1Make A Systematic ListThis AMC 12 problem only needs Grade 8 exponent rules you already know: rename x = 2¹⁷ to shrink the fraction to (x¹⁷+1)/(x+1), expand into 17 alternating powers of 2, pair them up so each pair becomes 17 consecutive 1-bits in binary, and add: 8 × 17 + 1 = 137.