Difference between revisions of "2018 AMC 10A Problems/Problem 18"
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This looks like balanced ternary, in which all the integers with absolute values less than <math>\frac{3^n}{2}</math> are represented in <math>n</math> digits. There are 8 digits. Plugging in 8 into the formula gives a maximum bound of <math>|x|=3280.5</math>, which means there are 3280 positive integers, 0, and 3280 negative integers. Since we want all nonnegative integers, there are <math>3280+1=\boxed{3281}</math> integers or <math>\boxed{D}</math>. | This looks like balanced ternary, in which all the integers with absolute values less than <math>\frac{3^n}{2}</math> are represented in <math>n</math> digits. There are 8 digits. Plugging in 8 into the formula gives a maximum bound of <math>|x|=3280.5</math>, which means there are 3280 positive integers, 0, and 3280 negative integers. Since we want all nonnegative integers, there are <math>3280+1=\boxed{3281}</math> integers or <math>\boxed{D}</math>. | ||
| + | |||
| + | <math>QED\blacksquare</math> | ||
Revision as of 17:09, 8 February 2018
How many nonnegative integers can be written in the form ![\[a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,\]](http://latex.artofproblemsolving.com/1/2/e/12e09c391eb94ff988e6dcfc490ddb6c72e79342.png)
where 
for 
?
Solution
This looks like balanced ternary, in which all the integers with absolute values less than 
are represented in 
digits. There are 8 digits. Plugging in 8 into the formula gives a maximum bound of 
, which means there are 3280 positive integers, 0, and 3280 negative integers. Since we want all nonnegative integers, there are 
integers or 
.
