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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 P(x) = x^3+3x^2+4x-11 = (x-a)(x-b)(x-c) = 0, we have , , and . Then

t = -(a+b)(b+c)(c+a) = -(s-a)(s-b)(s-c) = -(-3-a)(-3-b)(-3-c)

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 &= 11 + 3(4) + 9(-3) + 27 = 23\end{align*}

See also

1996 AIME (ProblemsResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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