Prove that
and 
Prove that
be positive integers. Maximize 
if 
.
,
,
.
where 
and 
are the roots of the cubic equation 
satisfies ![$$\log_{\sqrt[3]3}(\log_3 x)+\log_3(\log_{27}x)+\log_{27}(\log_{\sqrt[3]3}x)=1.$$](http://latex.artofproblemsolving.com/4/f/9/4f9bef4f56017af4af3893afdfc10be5d986a462.png)
Compute 
.
should be equal to 
Prove that
Where 
Prove that
Let 
Prove that
Y by
If the average of the set 
is 
what is the value of 
?
Compute the two-digit base
number 
such that 
William is ordering bottles. There are eight colors of bottles: White, Red, Blue, Green, Orange, Purple, Yellow, and Charteruse. What is the probability he puts the red bottle first and the white bottle last?
A paper towel roll is a cylinder with another cylinder in the middle cut out. Trying to save money, a CEO of a paper towel company makes the inside radius increase by
He is then sued, and forced to lower the price to match the original ratio of paper towel to cost. By what percentage does he lower the cost?
Eleven elves are making christmas presents. Each makes the same number of presents, and the sum of the digits of the total number of presents is
Also, after two elves steal all of the presents they made, the remaining number of presents ends in 
How many presents did the two steal?
is 
what is the value of 
?Compute the two-digit base
number 
such that 
William is ordering bottles. There are eight colors of bottles: White, Red, Blue, Green, Orange, Purple, Yellow, and Charteruse. What is the probability he puts the red bottle first and the white bottle last?
A paper towel roll is a cylinder with another cylinder in the middle cut out. Trying to save money, a CEO of a paper towel company makes the inside radius increase by
He is then sued, and forced to lower the price to match the original ratio of paper towel to cost. By what percentage does he lower the cost?Eleven elves are making christmas presents. Each makes the same number of presents, and the sum of the digits of the total number of presents is
Also, after two elves steal all of the presents they made, the remaining number of presents ends in 
How many presents did the two steal?
so 
is not the only multiple of 
with a digit sum of 
so 
Then we have 
This indicates that 
so 
are both possible solutions. However, 
yields 
and 
is not a digit in base 
So our only solution is 
isn't possible, as 
