AMC 10 · 2018 · #13

Grade 7 number-theory

Archetype Overlap / Complement Counting └ Sub-archetype Complement by Subtracting Forbidden Arrangements

base-conversionexponents symmetry-argumentcomplementary-counting ↑ Prerequisites: base-conversion

📏 Medium solution 💡 2 insights

Problem

Each of eight coefficients is chosen from minus one, zero, and one, and they build a number by multiplying the powers of three from the seventh down to the zeroth. Count how many different nonnegative integers can come out.

Pick an answer.

(A)

512

(B)

729

(C)

1094

(D)

3281

(E)

59,048

How to solve

Strategy Change Focus / Count the Complement

Tool #2 (Systematic List) counts the raw number of coefficient tuples: 3 choices per digit, 8 digits, so 3⁸ tuples. Tool #14 (Extreme Principle) proves each tuple lands on a different integer by bounding the lower digits against the leading one. Tool #16 (Change Focus) is the key move: instead of counting nonnegative values head-on, use the sign-flip symmetry — positives and negatives come in equal numbers, with exactly one zero, so the nonnegative count is just (total + 1) / 2. Tool #3 (Eliminate) confirms the matching choice and rules out the decoys.

1STEP 1

Count all coefficient tuples

There are three to the eighth tuples.

3⁸ = 6561 coefficient tuples

2STEP 2

Each tuple gives a distinct integer

Each tuple gives a distinct value.

3⁰ + 3¹ + … + 3^k-1 = (3^k - 1)/2 ＜ 3^k

3STEP 3

Use sign-flip symmetry

Flipping signs pairs positives with negatives.

a_i → -a_i sends v → -v, so #positive = #negative

4STEP 4

Solve for the nonnegative count

The lone zero in the middle must be added back.

2P + 1 = 6561 → P = 3280, P + 1 = 3281

5STEP 5

Match the answer choice

The result is 3281.

3281 → (D)

Answer

3281

The range argument and the symmetry agree. The largest value is all coefficients +1: 3⁷ + … + 3⁰ = (3⁸ - 1)/2 = 3280, and the smallest is -3280. Since every tuple is distinct, the expression hits every integer from -3280 to 3280 — that is 6561 integers, matching 3⁸. The nonnegative ones run 0, 1, 2, …, 3280, which is 3281 numbers, confirming (D).

💡Key takeaway

There are 3⁸ = 6561 ways to pick the digits, each gives a different number from -3280 to 3280, and since positives and negatives match up evenly with one zero in the middle, the nonnegative ones number (6561 + 1)/2 = 3281.

Count all coefficient tuples
Each tuple gives a distinct integer
Use sign-flip symmetry
Solve for the nonnegative count
Match the answer choice

Count all coefficient tuples

Each tuple gives a distinct integer

Use sign-flip symmetry

Solve for the nonnegative count

Match the answer choice

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