Problem 1640

1. Let $P(x) = (x - a)(x - b)(x - c)(x - d)(x - e)$. This is a polynomial of degree 5.
2. The given equation can be rewritten as:
$$(x - a)(x - b)(x - c)(x - d) + (x - a)(x - b)(x - c)(x - e) + (x - a)(x - b)(x - d)(x - e) + (x - a)(x - c)(x - d)(x - e) + (x - b)(x - c)(x - d)(x - e) = 0$$
3. Notice that each term in the equation is a product of four linear factors, which can be seen as the derivative of $P(x)$. Specifically, the polynomial in question is $P'(x)$, the derivative of $P(x)$.
4. Since $a, b, c, d, e$ are distinct real numbers, $P(x)$ has five distinct real roots. By Rolle's Theorem, between every pair of consecutive roots of $P(x)$, there must be at least one root of $P'(x)$.
5. Since there are five distinct roots of $P(x)$, there are four intervals between these roots. By Rolle's Theorem, there must be at least one root of $P'(x)$ in each of these four intervals.
6. Therefore, $P'(x)$ has at least four distinct real roots.
7. Since $P'(x)$ is a polynomial of degree 4, it can have at most four roots. Thus, $P'(x)$ has exactly four distinct real roots.

$\blacksquare$