$\frac{1}{a(a - b)(a - c)} + \frac{1}{b(b - c)(b - a)} + \frac{1}{c(c - a)(c - b)} = \frac{1}{a(a - b)(a - c)} - \frac{1}{b(b - c)(a - b)} + \frac{1}{c(a - c)(b - c)} = \frac{bc(b - c) - ac(a - c) + ab(a - b)}{abc(a - b)(a - c)(b - c)} = \frac{b^2c - bc^2 - a^2c + ac^2 + a^2b - ab^2}{abc(a - b)(a - c)(b - c)} = \frac{b^2(c - a) - b(c^2 - a^2) + ac(c - a)}{abc(a - b)(a - c)(b - c)} = -\frac{b^2 - b(c + a) + ac}{abc(a - b)(b - c)} = -\frac{b^2 - bc - ab + ac}{abc(a - b)(b - c)} = -\frac{b(b - c) - a(b - c)}{abc(a - b)(b - c)} = -\frac{b - a}{abc(a - b)} = \frac{1}{abc}$

$\frac{x^2}{(x - y)(x - z)} + \frac{y^2}{(y - x)(y - z)} + \frac{z^2}{(z - x)(z - y)} = \frac{x^2}{(x - y)(x - z)} - \frac{y^2}{(x - y)(y - z)} + \frac{z^2}{(x - z)(y - z)} = \frac{x^2(y - z) - y^2(x - z) + z^2(x - y)}{(x - y)(x - z)(y - z)} = \frac{x^2y - x^2z - xy^2 + y^2z + xz^2 - yz^2}{(x - y)(x - z)(y - z)} = \frac{x^2(y - z) - x(y^2 - z^2) + yz(y - z)}{(x - y)(x - z)(y - z)} = \frac{(y - x)(x^2 - x(y + z) + yz)}{(x - y)(x - z)(y - z)} = \frac{x^2 - x(y + z) + yz}{(x - y)(x - z)} = \frac{x^2 - xy - xz + yz}{(x - y)(x - z)} = \frac{x(x - z) - y(x - z)}{(x - y)(x - z)} = \frac{(x - z)(x - y)}{(x - y)(x - z)} = \frac{x - y}{x - y} = 1$