AoPS Academy
![\[
a_7 \cdot 3^7 + a_6 \cdot 3^6 + a_5 \cdot 3^5 + a_4 \cdot 3^4 + a_3 \cdot 3^3 + a_2 \cdot 3^2 + a_1 \cdot 3^1 + a_0 \cdot 3^0
\]](http://latex.artofproblemsolving.com/0/c/e/0ce2c259010b678238f11d35fc1328ea76f945f6.png)
where 
for 
?
sides and 
sides, respectively. The total number of sides is 33, and the total number of diagonals is 243. What are the values of and ?
be a polynomial with integer coefficients such that 
and 
for some integer 
. Prove that there exists no integer 
such that 
.
for some real number 
and ![\[ a_n=na_{n-1}+(n-1)(n!(n-1)!-1) \]](http://latex.artofproblemsolving.com/4/1/f/41f0cc7b041a8bba056882b5c794503195c374fd.png)
for integers 
. Given that 
, and given that 
, where 
and 
are positive integers whose greatest common divisor is 
, compute 
and 
are in the coordinate plane such that 
. Let 
denote the locus of all points 
in the coordinate plane satisfying 
, and let 
be the midpoint of 
. Points 
and 
are on such that 
and 
. The value of 
can be expressed in the form 
, where 
and 
are relatively prime positive integers. Find 
.
, since we can extend our given equation to 
. Next, note that the range of 
is ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
, so we want to be sure our value of 
gives a value for 
in the range . The only working value is 
, giving 
. We now require 
, and we can solve for to get 
.
and 
Then 
in terms of 
Prove that
Let 
Prove that
satisfying:
has the properties that 
for all positive real values of 
for 
.
.
"expanded" by 
every time 
is from the closest power of 
)
.
and 
the function reaches a maximum of 
is 
and 
we will have another 
.
.
