Difference between revisions of "2018 AMC 10A Problems/Problem 18"

Revision as of 02:42, 25 January 2019

Problem

How many nonnegative integers can be written in the form $a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$ where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$ ?

$\textbf{(A) } 512 \qquad \textbf{(B) } 729 \qquad \textbf{(C) } 1094 \qquad \textbf{(D) } 3281 \qquad \textbf{(E) } 59,048$

Solution 1

This looks like balanced ternary, in which all the integers with absolute values less than $\frac{3^n}{2}$ are represented in $n$ digits. There are 8 digits. Plugging in 8 into the formula for the balanced ternary gives a maximum bound of $|x|=3280.5$ , which means there are 3280 positive integers, 0, and 3280 negative integers. Since we want all nonnegative integers, there are $3280+1=\boxed{3281}$ integers or $\boxed{D}$ .

Solution 2

Note that all numbers formed from this sum are either positive, negative or zero. The number of positive numbers formed by this sum is equal to the number of negative numbers formed by this sum, because of symmetry. There is only one way to achieve a sum of zero, if all $a_i=0$ . The total number of ways to pick $a_i$ from $i=0, 1, 2, 3, ... 7$ is $3^8=6561$ . $\frac{6561-1}{2}=3280$ gives the number of possible negative integers. The question asks for the number of nonnegative integers, so subtracting from the total gives $6561-3280=\boxed{3281}$ . (RegularHexagon)

Solution 3

Note that the number of total possibilities (ignoring the conditions set by the problem) is $3^8=6561$ . So, E is clearly unrealistic.

Note that if $a_7$ is 1, then it's impossible for $a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$ to be negative. Therefore, if $a_7$ is 1, there are $3^7=2187$ possibilities. (We also must convince ourselves that these $2187$ different sets of coefficients must necessarily yield $2187$ different integer results.)

As A, B, and C are all less than 2187, the answer must be $\boxed{(D) 3281}$

Solution 4

Note that we can do some simple casework: If $a_7=1$ , then we can choose anything for the other 7 variables, so this give us $3^7$ . If $a_7=0$ and $a_6=1$ , then we can choose anything for the other 6 variables, giving us $3^6$ . If $a_7=0$ , $a_6=0$ , and $a_5=1$ , then we have $3^5$ . Continuing in this vein, we have $3^7+3^6+\cdots+3^1+3^0$ ways to choose the variables' values, except we have to add 1 because we haven't counted the case where all variables are 0. So our total sum is $\boxed{(D) 3281}$ . Note that we have counted all possibilities, because the largest positive positive power of 3 must be greater than or equal to the largest negative positive power of 3, for the number to be nonnegative.

Solution 5

The key is to realize that this question is basically taking place in $a\in\{0,1,2\}$ if each value of $a$ was increased by $1$ , essentially making it into base $3$ . Then the range would be from $0\cdot3^7+$ $0\cdot3^6+$ $0\cdot3^5+$ $0\cdot3^4+$ $0\cdot3^3+$ $0\cdot3^2+$ $0\cdot3^1+$ $0\cdot3^0=$ $0$ to $2\cdot3^7+$ $2\cdot3^6+$ $2\cdot3^5+$ $2\cdot3^4+$ $2\cdot3^3+$ $2\cdot3^2+$ $2\cdot3^1+$ $2\cdot3^0=$ $3^8-1=$ $6561-1=$ $6560$ , yielding $6561$ different values. Since the distribution for all $a_i\in \{-1,0,1\}$ the question originally gave is symmetrical, we retain the $3280$ positive integers and one $0$ but discard the $3280$ negative integers. Thus, we are left with the answer, $\boxed{\textbf{(D)} 3281}\qquad$ . ∎ --anna0kear

Solution 6

First, set $a_i=0$ for all $i\geq1$ . The range would be the integers for which $[-1,1]$ . If $a_i=0$ for all $i\geq2$ , our set expands to include all integers such that $-4\leq\mathbb{Z}\leq4$ . Similarly, when $i\geq3$ we get $-13\leq\mathbb{Z}\leq13$ , and when $i\geq4$ the range is $-40\leq\mathbb{Z}\leq40$ . The pattern continues until we reach $i=7$ , where $-3280\leq\mathbb{Z}\leq3280$ . Because we are only looking for positive integers, we filter out all $\mathbb{Z}<0$ , leaving us with all integers between $0\leq\mathbb{Z}\leq3280$ , inclusive. The answer becomes $\boxed{(D)3281}$ . ∎ --anna0kear

Solution 7

To get the number of integers, we can get the highest positive integer that can be represented using $a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$ where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$ .

Note that the least nonnegative integer that can be represented is $0$ , when all $a_i=0$ . The highest number will be the number when all $a_i=1$ . That will be $3^7+3^6+3^5+3^4+3^3+3^2+3^1+3^0=\frac{3^8-1}{3-1}$ $=3280$

Therefore, there are $(3280+1)$ nonnegative integers that can be represented. Our answer is $\boxed{\textbf{(D) } 3281}$

@@ Line 45: / Line 45: @@
 Note that the least nonnegative integer that can be represented is <math>0</math>, when all <math>a_i=0</math>. The highest number will be the number when all <math>a_i=1</math>. That will be <cmath>3^7+3^6+3^5+3^4+3^3+3^2+3^1+3^0=\frac{3^8-1}{3-1}</cmath> <cmath>=3280</cmath>
-Therefore, there are <math>(3280+1)</math> nonnegative integers that can be represented. Our answer is $\boxed{}
+Therefore, there are <math>(3280+1)</math> nonnegative integers that can be represented. Our answer is <math>\boxed{\textbf{(D) } 3281}</math>
 ==See Also==

2018 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

2018 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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