Difference between revisions of "2023 AIME I Problems/Problem 9"
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Find the number of cubic polynomials <math>p(x) = x^3 + ax^2 + bx + c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> | Find the number of cubic polynomials <math>p(x) = x^3 + ax^2 + bx + c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> | ||
− | are integers in <math> | + | are integers in <math>\{ -20, -19, -18, \dots , 18, 19, 20 \}</math>, such that there is a unique integer |
− | + | <math>m \neq 2</math> with <math>p(m) = p(2).</math> | |
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− | |||
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− | <math>m \neq 2</math> with <math>p(m) = p(2) | ||
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==Solution== | ==Solution== |
Revision as of 15:57, 8 February 2023
Find the number of cubic polynomials , where
,
, and
are integers in
, such that there is a unique integer
with
Solution
It can be easily noticed that is independent of the condition
, and can thus safely take all
possible values between
and
.
There are two possible ways for to be the only integer satisfying
:
has a double root at
or a double root at
.
Case 1: has a double root at
:
In this case, , or
. Thus
ranges from
to
. One of these values,
corresponds to a triple root at
, which means
. Thus there are
possible values of
. (It can be verified that
is an integer).
Case 2: has a double root at
:
See the above solution. This yields possible combinations of
and
.
Thus, in total we have combinations of
.
-Alex_Z