1996 AIME Problems/Problem 5
From AoPSWiki
Problem
Suppose that the roots of
are
,
, and
, and that the roots of
are
,
, and
. Find
.
Solution
By Vieta's formulas on the polynomial
, we have
,
, and
. Then

This is just the definition for
.
Alternatively, we can expand the expression to get
![\begin{align*}t &= -(-3-a)(-3-b)(-3-c)\\ &= (a+3)(b+3)(c+3)\\ &= abc + 3[ab + bc + ca] + 9[a + b + c] + 27\\t &am...](http://alt2.artofproblemsolving.com/Forum/latexrender/pictures/f/c/a/fca4907e3052aae6f47fb7fe5425d14e93059ded.gif)
A third solution arises if it is seen that each term in the expansion of
has a total degree of 3. Another way to get terms with degree 3 is to multiply out
. Expanding both of these expressions and comparing them shows that:
See also
| 1996 AIME (Problems • Resources) | ||
| Preceded by Problem 4 | Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||







