Difference between revisions of "2003 AIME II Problems/Problem 15"
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== Problem == | == Problem == | ||
| + | Let | ||
| + | <center><math>P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).</math></center> | ||
| + | Let <math>z_{1},z_{2},\ldots,z_{r}</math> be the distinct zeros of <math>P(x),</math> and let <math>z_{K}^{2} = a_{k} + b_{k}i</math> for <math>k = 1,2,\ldots,r,</math> where <math>i = \sqrt { - 1},</math> and <math>a_{k}</math> and <math>b_{k}</math> are real numbers. Let | ||
| + | <center><math>\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},</math></center> | ||
| + | where <math>m,</math> <math>n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math> | ||
== Solution == | == Solution == | ||
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== See also == | == See also == | ||
| − | + | {{AIME box|year=2003|n=II|num-b=14|after=Last Question}} | |
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Revision as of 13:42, 21 November 2007
Problem
Let
Let 
be the distinct zeros of 
and let 
for 
where 
and 
and 
are real numbers. Let
where 
and 
are integers and is not divisible by the square of any prime. Find 
Solution
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See also
| 2003 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Last Question | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
