1984 USAMO Problems/Problem 1
Problem
In the polynomial 
, the product of 
of its roots is 
. Find 
.
Solution
Let the four roots be 
and 
, so that 
. From here we show two methods; the second is more slick, but harder to see.
Solution #1
Using Vieta's formulas, we have:
$\begin{align*}a+b+c+d &= 18,\\ ab+ac+ad+bc+bd+cd &= k,\\ abc+abd+acd+bcd &=-200,\\ abcd &=-1984.\\ \end{align*}$ (Error compiling LaTeX. Unknown error_msg)
From the last of these equations, we see that 
. Thus, the second equation becomes 
, and so 
. The key insight is now to factor the left-hand side as a product of two binomials: 
, so that we now only need to determine 
and 
rather than all four of 
.
Let 
and 
. Plugging our known values for 
and 
into the third Vieta equation, 
, we have 
. Moreover, the first Vieta equation, 
, gives 
. Thus we have two linear equations in 
and 
, which we solve to obtain 
and 
.
Therefore, we have 
, yielding 
.
Solution #2 (sketch)
We start as before: and 
. We now observe that a and b must be the roots of a quadratic, 
, where r is a constant (secretly, r is just -(a+b)=-p from Solution #1). Similarly, c and d must be the roots of a quadratic 
.
Now
$<cmath> \begin{align*}x^4-18x^3+kx^2+200x-1984 &= (x^2+rx-32)(x^2+sx+62)\\ & = x^4+(r+s)x^3+(62-32+rs)x^2+(62s-32r)x-1984.\end{align*} </cmath>$ (Error compiling LaTeX. Unknown error_msg)
Equating the coefficients of 
and 
with their known values, we are left with essentially the same linear equations as in Solution #1, which we solve in the same way. Then we compute the coefficient of 
and get 
See Also
| 1984 USAMO (Problems • Resources) | ||
| Preceded by First Problem |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
