Difference between revisions of "1984 USAMO Problems/Problem 1"
(Created page with '1984 USAMO Problem #1 Problem: 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>…') |
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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: | ||
| − | <math>ab = - 32</math> | + | *<math>ab = - 32</math> |
| − | <math>abcd = - 1984</math> | + | *<math>abcd = - 1984</math> |
| − | <math>cd = 62</math> | + | *<math>cd = 62</math> |
| − | <math>a + b + c + d = 18</math>* | + | *<math>a + b + c + d = 18</math>* |
| − | <math>ab + ac + ad + bc + bd + cd = k</math> | + | *<math>ab + ac + ad + bc + bd + cd = k</math> |
| − | <math>abc + abd + acd + bcd = - 200</math> | + | *<math>abc + abd + acd + bcd = - 200</math> |
Substituting given values obtains the following: | Substituting given values obtains the following: | ||
| − | <math>ac + ad + bc + bd = k - 62 + 32 = k - 30</math> | + | *<math>ac + ad + bc + bd = k - 62 + 32 = k - 30</math> |
| − | <math>(a + b)(c + d) = k - 30</math> | + | *<math>(a + b)(c + d) = k - 30</math> |
| − | <math>- 32c + 62b + 62a - 32d = - 200</math> | + | *<math>- 32c + 62b + 62a - 32d = - 200</math> |
| − | <math>31(a + b) - 16(c + d) = - 100</math> | + | *<math>31(a + b) - 16(c + d) = - 100</math> |
| − | + | ||
| + | Multiplying the * equation by 16 gives | ||
<math>16(a + b) + 16(c + d) = 288</math>. Adding it to the previous equation gives | <math>16(a + b) + 16(c + d) = 288</math>. Adding it to the previous equation gives | ||
| − | <math>47(a + b) = 188</math> | + | *<math>47(a + b) = 188</math> |
| − | <math>a + b = 4</math> | + | *<math>a + b = 4</math> |
| − | <math>c + d = 18 - 4 = 14</math> | + | *<math>c + d = 18 - 4 = 14</math> |
| − | <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> |
Revision as of 12:35, 11 August 2009
1984 USAMO Problem #1
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
