AoPS Academy
is centered at the apex of a right square pyramid with base length 
and height 
The volume of the space where the sphere overlaps with the pyramid can be expressed in the form 
Find 
has how many digits?
has how many digits?
has how many digits?
is called 
-scared if all of the factors of are not factors of . Find the number of 
-scared numbers less than 
. Note that in the context of this problem, does NOT count as a factor.
is called -scared if all of the factors of are not factors of . Find the number of -scared numbers less than .
is called -scared if all of the factors of are not factors of . Find the number of -scared numbers less than .
:
Volume of a right square pyramid with base length 
and height 
has 
digits, and 
has 
,
.
digits (excluding the 
has 
digits, you can figure it out from there. So there are 
digits
.
(5 digits) because I just have that memorized. 
for a grand total of 
digits.
when we put this to the 16th power we get 2^32*5^16 we can make 16 0's because we have until 5^16 so we just have to add the 2^16 at the end and when we do this..... we get 
then we already know that it has to be 5 digits so we get 
be the number of candies the twins get combined, let 
and 
be the number of candies the other people get. 
,
,
(13 ways)
(11 ways)
(9 ways)
(7 ways)
,
(5 ways)
,
(3 ways)
,
(1 way)
digits
Taking 
on the inequality,
Thus, 
satisfies 
and 
.
.
.
,
,
,
gives us 
implying 
.
and plugging in 
gives us 
shows that 
as the function. Then, 
so the equation of the line must be 
-
.
and 
which gives the ordered pair 
,
has 
Then, 
Since this is a linear function and every time we add 
value, we add 
.
