Difference between revisions of "1984 USAMO Problems/Problem 1"
(→Solution) |
(→Solution) |
||
| Line 18: | Line 18: | ||
Let <math>p=a+b</math> and <math>q=c+d</math>. Plugging our known values for <math>ab</math> and <math>cd</math> into the third Vieta equation, <math>-200 = abc+abd + acd + bcd = ab(c+d) + cd(a+b)</math>, we have <math>-200 = -32(c+d) + 62(a+b) = 62p-32q</math>. Moreover, the first Vieta equation, <math>a+b+c+d=18</math>, gives <math>p+q=18</math>. Thus we have two linear equations in <math>p</math> and <math>q</math>, which we solve to obtain <math>p=4</math> and <math>q=14</math>. | Let <math>p=a+b</math> and <math>q=c+d</math>. Plugging our known values for <math>ab</math> and <math>cd</math> into the third Vieta equation, <math>-200 = abc+abd + acd + bcd = ab(c+d) + cd(a+b)</math>, we have <math>-200 = -32(c+d) + 62(a+b) = 62p-32q</math>. Moreover, the first Vieta equation, <math>a+b+c+d=18</math>, gives <math>p+q=18</math>. Thus we have two linear equations in <math>p</math> and <math>q</math>, which we solve to obtain <math>p=4</math> and <math>q=14</math>. | ||
| − | Therefore, we have <math>(\underbrace{a+b}_4)(\underbrace{c+d}_{14}) = k-30, yielding k=4\cdot 14+30 = \boxed{86}</math>. | + | Therefore, we have <math>(\underbrace{a+b}_4)(\underbrace{c+d}_{14}) = k-30</math>, yielding <math>k=4\cdot 14+30 = \boxed{86}</math>. |
Solution #2 (sketch) | Solution #2 (sketch) | ||
Revision as of 21:28, 27 April 2014
Problem
In the polynomial 
, the product of 
of its roots is 
. Find 
.
Solution
Let the four roots be 
, 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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
