2018 AMC 10A Problems/Problem 18
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 
.
