Difference between revisions of "2004 AMC 12B Problems/Problem 23"
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The [[polynomial]] <math>x^3 - 2004 x^2 + mx + n</math> 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 <math>n</math> are possible? | The [[polynomial]] <math>x^3 - 2004 x^2 + mx + n</math> 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 <math>n</math> are possible? | ||
− | <math>\mathrm{(A)}\ 250,000 | + | <math>\mathrm{(A)}\ 250,\!000 |
− | \qquad\mathrm{(B)}\ 250,250 | + | \qquad\mathrm{(B)}\ 250,\!250 |
− | \qquad\mathrm{(C)}\ 250,500 | + | \qquad\mathrm{(C)}\ 250,\!500 |
− | \qquad\mathrm{(D)}\ 250,750 | + | \qquad\mathrm{(D)}\ 250,\!750 |
− | \qquad\mathrm{(E)}\ 251,000</math> | + | \qquad\mathrm{(E)}\ 251,\!000</math> |
== Solution == | == Solution == | ||
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and we need the number of possible products <math>t = rs = r(1002 - r)</math>. | and we need the number of possible products <math>t = rs = r(1002 - r)</math>. | ||
− | Since <math>r > 0</math> and <math>t > 0</math>, it follows that <math>0 < t = r(1002-r) < 501^2 = 251001</math>, with the endpoints not achievable because the roots must be distinct. Because <math>r</math> cannot be an integer, there are <math>251000 - 500 = 250,500\ \mathrm{(C)}</math> possible values of <math>n = -1002t</math>. | + | Since <math>r > 0</math> and <math>t > 0</math>, it follows that <math>0 < t = r(1002-r) < 501^2 = 251001</math>, with the endpoints not achievable because the roots must be distinct. Because <math>r</math> cannot be an integer, there are <math>251000 - 500 = 250,\!500\ \mathrm{(C)}</math> possible values of <math>n = -1002t</math>. |
== See also == | == See also == |
Revision as of 22:05, 6 January 2009
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
Let the roots be , and let
. Then
![$(x - r)(x - s)(x - (r + s))$](http://latex.artofproblemsolving.com/9/a/a/9aa6b4356e1b571b8c6b0273d4a5ee73ddd93031.png)
![$= x^3 - (r + s + r + s) x^2 + (rs + r(r + s) + s(r + s))x - rs(r + s) = 0$](http://latex.artofproblemsolving.com/1/8/f/18f8cb41503afe71a27d8834fd83f6ee33165a95.png)
and by matching coefficients, . Then our polynomial looks like
and we need the number of possible products
.
Since and
, it follows that
, with the endpoints not achievable because the roots must be distinct. Because
cannot be an integer, there are
possible values of
.
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 |