2005 AMC 12A Problems/Problem 24
Problem
Let . For how many polynomials
does there exist a polynomial
of degree 3 such that
?
Solution
We can write the problem as
.
Since and
,
. Thus,
, so
.
Hence, we conclude ,
, and
must each be
,
, or
. Since a quadratic is uniquely determined by three points, there can be
different quadratics
after each of the values of
,
, and
are chosen.
However, we have included which are not quadratics: lines. Namely,
Clearly, we could not have included any other constant functions. For any linear function, we have because
is y-value of the midpoint of
and
. So we have not included any other linear functions. Therefore, the desired answer is
.
See also
2005 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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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