Question

How many nonnegative integers can be written in the form

$\begin{aligned} &a_7 \cdot 3^7 + a_6 \cdot 3^6 + a_5 \cdot 3^5 \\ &\quad {}+ a_4 \cdot 3^4 + a_3 \cdot 3^3 + a_2 \cdot 3^2 \\ &\quad {}+ a_1 \cdot 3^1 + a_0 \cdot 3^0, \end{aligned}$

where $a_i \in \{-1, 0, 1\}$ for $0 \le i \le 7?$

$512$

$729$

$1094$

$3281$

$59{,}048$

Answer: D

Concepts:number base bijection symmetry

Difficulty rating: 1660

Solution:

Adding $1$ to every $a_i$ gives a bijection between these expressions and the base- $3$ numerals for $0$ through $3^8 - 1,$ so exactly $3^8 = 6561$ distinct integers occur. They are symmetric about $0$ (negating all $a_i$ negates the value), so besides $0$ itself, half are positive: $1 + \tfrac12(6561 - 1) = 3281$ nonnegative integers, namely $0$ through $3280.$

Thus, the correct answer is D.

2018 AMC 12A Problem 13

Answer: D

Difficulty rating: 1660

Solution:

Problem 13 in Other Years