2004 AMC 12B Problems/Problem 23
Problem
The polynomial 
has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of 
are possible?
Solution 2
Letting the roots be 
, 
, and 
, where 
, we see that by Vieta's Formula's, 
, and so 
. Therefore, 
is a factor of 
. Letting 
gives that 
because 
. Letting 
and noting that 
for some 
, we see that is the sum of the roots of 
, and , and so 
. Now, we have that 
has roots and , and we wish to find the number of possible values of 
. By the quadratic formula, we see that ![\[\frac{1002 \pm \sqrt{1002^2 - 4a}}{2} = 501 \pm \sqrt{501^2 - a}\]](http://latex.artofproblemsolving.com/9/1/0/910c7c622afd81998d260e3e8d4fe2b7fa178550.png)
are the two values of noninteger positive real numbers and , neither of which is equal to 
. This information gives us that 
, and so since 
is evidently not a square, we have 
possible values of 
.
Solution 3 (cheese)
Observe that the answer clearly must have something to do with the number 
, and we see that 
is a multiple of , so there is a very high probability that it is the correct answer.
See also
| 2004 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 22 |
Followed by Problem 24 |
| 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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