Difference between revisions of "2005 AIME II Problems/Problem 13"
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== Problem == | == Problem == | ||
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Let <math> P(x) </math> be a polynomial with integer coefficients that satisfies <math> P(17)=10 </math> and <math> P(24)=17. </math> Given that <math> P(n)=n+3 </math> has two distinct integer solutions <math> n_1 </math> and <math> n_2, </math> find the product <math> n_1\cdot n_2. </math> | Let <math> P(x) </math> be a polynomial with integer coefficients that satisfies <math> P(17)=10 </math> and <math> P(24)=17. </math> Given that <math> P(n)=n+3 </math> has two distinct integer solutions <math> n_1 </math> and <math> n_2, </math> find the product <math> n_1\cdot n_2. </math> | ||
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== See also == | == See also == | ||
− | *[[2005 AIME II Problems/Problem 12| Previous problem]] | + | * [[2005 AIME II Problems/Problem 12| Previous problem]] |
− | *[[2005 AIME II Problems/Problem 14| Next problem]] | + | * [[2005 AIME II Problems/Problem 14| Next problem]] |
* [[2005 AIME II Problems]] | * [[2005 AIME II Problems]] | ||
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+ | [[Category:Intermediate Algebra Problems]] |
Revision as of 22:34, 7 September 2006
Problem
Let be a polynomial with integer coefficients that satisfies
and
Given that
has two distinct integer solutions
and
find the product
Solution
Define the polynomial . By the givens,
,
,
and
. Note that for any polynomial
with integer coefficients and any integers
we have
divides
. So
divides
, and so
must be one of the eight numbers
and so
must be one of the numbers
or
. Similarly,
must divide
, so
must be one of the eight numbers
or
. Thus,
must be either 19 or 22. Since
obeys the same conditions and
and
are different, one of them is 19 and the other is 22 and their product is
.