satisfying:
![\[
a_7 \cdot 3^7 + a_6 \cdot 3^6 + a_5 \cdot 3^5 + a_4 \cdot 3^4 + a_3 \cdot 3^3 + a_2 \cdot 3^2 + a_1 \cdot 3^1 + a_0 \cdot 3^0
\]](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 
be a polynomial with integer coefficients such that 
and 
for some integer 
.
such that 
.
and 
Then 
in terms of 
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 
.
,
,
and 
are positive integers whose greatest common divisor is 
,
Y by
Is it just me or are the 2023 national sprint #21 and 2025 state target #4 strangely similar?
2023 Natioinal Sprint #21 wrote:
A right triangle with integer side lengths has perimeter 
feet and area ft^2. What is the arithmetic mean of all possible values of ?
feet and area ft^2. What is the arithmetic mean of all possible values of ?2025 State Target #4 wrote:
Suppose a right triangle has an area of 20 cm^2 and a perimeter of 40 cm. What is
the length of the hypotenuse, in centimeters?
the length of the hypotenuse, in centimeters?
This post has been edited 2 times. Last edited by mithu542, Apr 18, 2025, 6:04 PM
