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Cyclic sum of fractions with denominators (a-b)(a-c): evaluate

This type of question, asked 4 times: CDS 1 2021, CDS 2 2021, CDS 2 2022, CDS 2 2023. All 4 questions below, oldest first.

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  1. Question 1

    CDS 1 2021

    20
    What is 1bc(a−b)(a−c)+1ca(b−c)(b−a)+1ab(c−a)(c−b)\frac{1}{bc(a - b)(a - c)} + \frac{1}{ca(b - c)(b - a)} + \frac{1}{ab(c - a)(c - b)} equal to?

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  2. Question 2

    CDS 2 2021

    23
    What is x4(x2−y2)(x2−z2)+y4(y2−x2)(y2−z2)+z4(z2−x2)(z2−y2)\dfrac{x^4}{(x^2-y^2)(x^2-z^2)}+\dfrac{y^4}{(y^2-x^2)(y^2-z^2)}+\dfrac{z^4}{(z^2-x^2)(z^2-y^2)} equal to ?

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  3. Question 3

    CDS 2 2022

    10
    What is 1x(x−y)(x−z)+1y(y−z)(y−x)+1z(z−x)(z−y)\frac{1}{x(x - y)(x - z)} + \frac{1}{y(y - z)(y - x)} + \frac{1}{z(z - x)(z - y)} equal to ?

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  4. Question 4

    CDS 2 2023

    74
    If p=a2(b−a)(c−a)p = \frac{a^2}{(b - a)(c - a)}, q=b2(c−b)(a−b)q = \frac{b^2}{(c - b)(a - b)}, r=c2(a−c)(b−c)r = \frac{c^2}{(a - c)(b - c)}, then what is (p+q+r)2(p + q + r)^2 equal to ?

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