Problem 1 of the Irish Mathematical Olympiad 2007

Let r, s and t be the roots of the cubic polynomial p(x) = x^{3} - 2007x + 2002.

Determine the value of \displaystyle \frac{r 1}{r + 1} \cdot \frac{s 1}{s + 1} \cdot \frac{t 1}{t + 1}.

Solution by Steve Dinh, a.k.a. Vo Duc Dien

Expand \displaystyle \frac{r 1}{r + 1} \cdot \frac{s 1}{s + 1} \cdot \frac{t 1}{t + 1}= \frac{3rst 3 + rt + st + rs s r t}{rst + rt + st + rs + s + r + t + 1}. (*)

Since r, s and t are the roots, we can write p(x) as

(x r)(x s)(x t) = x^{3} -(s + r + t)x^{2} +(rt + st + rs)x - rst = x^{3} - 2007x + 2002.

or s + r + t = 0, rt + st + rs = -2007 and rst = -2002.

Substituting these into (*), we have

	\displaystyle \frac{r  1}{r + 1} \cdot \frac{s  1}{s + 1} \cdot \frac{t  1}{t + 1} = \frac{-3 \cdot 2002 - 3 - 2007}{- 2002 - 2007 + 1}  =  \frac{-8016}{-4008}  = 2.