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

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~OlutosinNGA
 
~OlutosinNGA
  
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Notice that there are <math>3^8</math> options for <math>a_7, a_6, \cdots a_0</math> since each <math>a_i</math> can take the value of <math>-1</math> or <math>0</math> or <math>1</math>. Now we want to find how many of them are positive and then we can add one in the end to account for <math>0</math>(they are asking for non-negative).  
 
Notice that there are <math>3^8</math> options for <math>a_7, a_6, \cdots a_0</math> since each <math>a_i</math> can take the value of <math>-1</math> or <math>0</math> or <math>1</math>. Now we want to find how many of them are positive and then we can add one in the end to account for <math>0</math>(they are asking for non-negative).  
  

Revision as of 10:37, 1 January 2021

The following problem is from both the 2018 AMC 12A #13 and 2018 AMC 10A #18, so both problems redirect to this page.

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

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 $\boxed{\textbf{D}}$.

Solution

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 non-negative integers, so subtracting from the total gives $6561-3280=\boxed{\textbf{(D) } 3281}$. (RegularHexagon, KLBBC minor changes)

Solution

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 $\boxed{\textbf{(D) } 3281}$

Solution

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 $\boxed{\textbf{(D) } 3281}$. 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

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

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 $\boxed{\textbf{(D) } 3281}$. --anna0kear

Solution

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 $3280$ positive integers and $(3280+1)$ nonnegative integers (while including $0$) that can be represented. Our answer is $\boxed{\textbf{(D) } 3281}$

~OlutosinNGA

Solution

Notice that there are $3^8$ options for $a_7, a_6, \cdots a_0$ since each $a_i$ can take the value of $-1$ or $0$ or $1$. Now we want to find how many of them are positive and then we can add one in the end to account for $0$(they are asking for non-negative).

By symmetry(look out for these on the contest), we see that exactly half of them are positive. So $\lfloor{\tfrac{3^8}{2}}\rfloor = 3280.$ So now we will add $1$ because of the $0$ to account for the non-negative solutions.

So our final answer is $3280 + 1 = 3281$ which is $\boxed{\textbf{(D) } 3281}$.

Video Solution

https://youtu.be/0xmbHDcUI2w

~IceMatrix

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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