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
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
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
Note that
is an integer from
through
........................ So, ......................... We construct the following table:
~BJHHar ~MRENTHUSIASM
Solution 3 (Answer Choices)
Suppose our polynomial is equal to
Then we are given that
Then the polynomials
,
also have
when
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 |
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