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Algebra Difficulty 5.6 AIME, harder Prove it

Example 9. If a,b,ca, b, c are pairwise distinct rational numbers. Prove that 1(a−b)2+1(b−c)2+1(c−a)2\sqrt{\frac{1}{(a-b)^{2}}+\frac{1}{(b-c)^{2}}+\frac{1}{(c-a)^{2}}} is a rational number. (1991, Beijing Mathematical Competition)

Solution

To prove that w=(a−b)2(b−c)2+(b−c)2(c−a)2+(c−a)2(a−b)2w=(a-b)^{2}(b-c)^{2}+(b-c)^{2}(c-a)^{2}+(c-a)^{2}(a-b)^{2} is a perfect square, we need to show that:
w=[(a−b)(b−c)+(b−c)(c−a)+(c−a)(a−b)]2−2(a−b)(b−c)⋅(c−a)(a−b+b−c+c−a)=[(a−b)(b−c)+(b−c)(c−a)+(c−a)(a−b)]2. \begin{aligned} w & =[(a-b)(b-c)+(b-c)(c-a) \\ & +(c-a)(a-b)]^{2}-2(a-b)(b-c) \\ & \cdot(c-a)(a-b+b-c+c-a) \\ & =[(a-b)(b-c)+(b-c)(c-a) \\ & +(c-a)(a-b)]^{2} . \end{aligned}
Rational numbers.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.