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A sum of squares equals zero, so each square is zero

This type of question, asked 3 times: CDS 1 2020, CDS 1 2021, CDS 2 2023. All 3 questions below, oldest first.

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  1. Question 1

    CDS 1 2020

    17
    If x2+9y2=6xyx^{2} + 9y^{2} = 6xy, then what is y:xy : x equal to ?

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  2. Question 2

    CDS 1 2021

    3
    How many pairs of (x,y)(x, y) can be chosen from the set {2,3,6,8,9}\{2, 3, 6, 8, 9\} such that xy+yx=2\frac{x}{y} + \frac{y}{x} = 2, where x≠yx \ne y?

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  3. Question 3

    CDS 2 2023

    83
    If a,b,c,x,y,za, b, c, x, y, z are real numbers such that (a+b+c)2−3(ab+bc+ca)+3(x2+y2+z2)=0(a + b + c)^2 - 3(ab + bc + ca) + 3(x^2 + y^2 + z^2) = 0, then which one of the following is correct ?

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