2018 AMC 12B Problems/Problem 22
Problem
Consider polynomials of degree at most
, each of whose coefficients is an element of
. How many such polynomials satisfy
?
Solution 1
Suppose that This problem is equivalent to counting the ordered quadruples
where all of
and
are integers from
through
We are given 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
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
Solution 3
As before, . This is
. Rephrased, how many two sums of integers from 0-9 have a difference of 9. Make a chart of pairs between these two sets:
Observe how there is one way to sum 2 numbers to 0 and two ways to 1, 3 ways to 2, and so on. At 9, there are 10 possible ways. Recall that only integers between 0-9 are valid. Now observe how there is 1 way to to sum to 18 in this fashion (9+9), 2 ways to sum to 17, and so forth again (to optionally prove that this pattern holds, apply stars and bars up to 9 and notice the symmetry).
The answer then is the number of ways to write each component of each pair. This is or, since it's symmetrical between sum of 4 and 5,
. Use summation rules to finally get
.
~BJHHar
See Also
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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