Difference between revisions of "1984 USAMO Problems/Problem 1"
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| − | + | ==Problem== | |
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In the polynomial <math>x^4 - 18x^3 + kx^2 + 200x - 1984 = 0</math>, the product of <math>2</math> of its roots is <math>- 32</math>. Find <math>k</math>. | In the polynomial <math>x^4 - 18x^3 + kx^2 + 200x - 1984 = 0</math>, the product of <math>2</math> of its roots is <math>- 32</math>. Find <math>k</math>. | ||
| − | Solution | + | ==Solution== |
Let the four roots be <math>a,b,c,d</math>. By Vieta's Formulas, we have the following: | Let the four roots be <math>a,b,c,d</math>. By Vieta's Formulas, we have the following: | ||
| Line 32: | Line 30: | ||
*<math>(a + b)(c + d) = 4(14) = 56 = k - 30</math> | *<math>(a + b)(c + d) = 4(14) = 56 = k - 30</math> | ||
*<math>k = \boxed{86}</math> | *<math>k = \boxed{86}</math> | ||
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| + | ==See Also== | ||
| + | |||
| + | {{USAMO box|year=1984|before=First<br>Problem|num-a=2}} | ||
| + | [[Category:Intermediate Algebra Problems]] | ||
Revision as of 19:55, 16 April 2010
Problem
In the polynomial 
, the product of 
of its roots is 
. Find 
.
Solution
Let the four roots be 
. By Vieta's Formulas, we have the following:
*
*
Substituting given values obtains the following:
Multiplying the * equation by 16 gives
. Adding it to the previous equation gives
See Also
| 1984 USAMO (Problems • Resources) | ||
| Preceded by First Problem |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
