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2(abc+bcd+cda+dab)/(p^2+q^2+r^2+s^2) \in Z (HOMC 2010 Q6)

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July 25, 2019
number theoryquadraticsInteger

Problem Statement

Let a,ba,b be the roots of the equation x2−px+q=0x^2-px+q = 0 and let c,dc, d be the roots of the equation x2−rx+s=0x^2 - rx + s = 0, where p,q,r,sp, q, r,s are some positive real numbers. Suppose that M=2(abc+bcd+cda+dab)p2+q2+r2+s2M =\frac{2(abc + bcd + cda + dab)}{p^2 + q^2 + r^2 + s^2} is an integer. Determine a,b,c,da, b, c, d.
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