2018 AMC 12B Problems/Problem 22
Contents
Problem
Consider polynomials 
of degree at most 
, each of whose coefficients is an element of 
. How many such polynomials satisfy 
?
Solution 1 (Stars and Bars)
Suppose that 
This problem is equivalent to counting the ordered quadruples 
where all of 
and 
are integers from 
through 
such that ![\[P(-1)=-a+b-c+d=-9.\]](http://latex.artofproblemsolving.com/c/7/f/c7f49b5a6d16402c6df0bb8f3d4ef443afb6e6a7.png)
Let 
and 
Note that both of 
and 
are integers from through 
Moreover, the ordered quadruples 
and the ordered quadruples 
have one-to-one correspondence.
We rewrite the given equation as 
or ![\[a'+b+c'+d=9.\]](http://latex.artofproblemsolving.com/e/8/0/e80a8be115b429c588370434b0a5c09bb6d6194c.png)
By the stars and bars argument, there are 
ordered quadruples 
~pieater314159 ~MRENTHUSIASM
Solution 2 (Casework)
Suppose that This problem is equivalent to counting the ordered quadruples where all of and are integers from through such that 
which rearranges to ![\[b+d+9=a+c.\]](http://latex.artofproblemsolving.com/f/c/5/fc5bc84eeac495c6d334862b85e8814da91fe252.png)
Note that 
is an integer from through 
........................ So, ......................... We construct the following table:
![\[\begin{array}{c||c|c|c|c|c||c} & & & & & & \\ [-2.5ex] \textbf{Row} & \boldsymbol{x_2} & \boldsymbol{x_3} & \boldsymbol{x_4} & \boldsymbol{x_5} & \boldsymbol{x_2x_3x_4 + x_3x_4x_5} & \textbf{Valid?} \\ [0.5ex] \hline & & & & & & \\ [-2ex] 1 & 1 & 1 & 2 & 2 & 0 & \checkmark \\ 2 & 1 & 2 & 1 & 2 & 0 & \checkmark \\ 3 & 1 & 2 & 2 & 1 & 2 & \\ 4 & 2 & 1 & 1 & 2 & 1 & \\ 5 & 2 & 1 & 2 & 1 & 0 & \checkmark \\ 6 & 2 & 2 & 1 & 1 & 0 & \checkmark \end{array}\]](http://latex.artofproblemsolving.com/f/a/a/faa97a9f53c43d17636b4b0e9c38e3258c5504fa.png)
~BJHHar ~MRENTHUSIASM
Solution 3 (Answer Choices)
Suppose our polynomial is equal to
![\[ax^3+bx^2+cx+d\]](http://latex.artofproblemsolving.com/1/6/1/1613f6d170713d792e670327c87ba120cd21399d.png)
Then we are given that
![\[9=b+d-a-c.\]](http://latex.artofproblemsolving.com/9/8/2/982e26dd99d04efb96badfa08966308ca5adeab6.png)
Then the polynomials ![\[cx^3+bx^2+ax+d\]](http://latex.artofproblemsolving.com/4/4/a/44ab370dd28eead33b3e5257d99f796aa784daf2.png)
, ![\[ax^3+dx^2+cx+b,\]](http://latex.artofproblemsolving.com/0/7/1/07158838b892172691eb2723619ae53e4e3e95cc.png)
![\[cx^3+dx^2+ax+b\]](http://latex.artofproblemsolving.com/9/b/0/9b09d9c448ed8b9867651cb72be6714d1005cb4f.png)
also have ![\[b+d-a-c=-9\]](http://latex.artofproblemsolving.com/1/1/3/113740f26f961875d421c788fd5300f72b9ec54e.png)
when ![\[x=-1.\]](http://latex.artofproblemsolving.com/c/8/2/c828c25d74f90767015791806ed1e1b5dd8f230f.png)
So the number of solutions must be divisible by 4. So the answer must be 
See Also
| 2018 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 21 |
Followed by Problem 23 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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