1984 USAMO Problems/Problem 1

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1984 USAMO Problem #1

Problem:

In the polynomial $x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$, the product of $2$ of its roots is $- 32$. Find $k$.

Solution:

Let the four roots be $a,b,c,d$. By Vieta's Formulas, we have the following:

  • $ab = - 32$
  • $abcd = - 1984$
  • $cd = 62$
  • $a + b + c + d = 18$*
  • $ab + ac + ad + bc + bd + cd = k$
  • $abc + abd + acd + bcd = - 200$

Substituting given values obtains the following:

  • $ac + ad + bc + bd = k - 62 + 32 = k - 30$
  • $(a + b)(c + d) = k - 30$
  • $- 32c + 62b + 62a - 32d = - 200$
  • $31(a + b) - 16(c + d) = - 100$

Multiplying the * equation by 16 gives

$16(a + b) + 16(c + d) = 288$. Adding it to the previous equation gives

  • $47(a + b) = 188$
  • $a + b = 4$
  • $c + d = 18 - 4 = 14$
  • $(a + b)(c + d) = 4(14) = 56 = k - 30$
  • $k = \boxed{86}$
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