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M = ab + cd when a^2+b^2=4, c^2+d^2=4, ac+bd = 2 (HOMC 2009 J Q6)

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July 25, 2019
algebrasystem of equations

Problem Statement

Suppose that 44 real numbers a,b,c,da, b,c,d satisfy the conditions {a2+b2=4c2+d2=4ac+bd=2\begin{cases} a^2 + b^2 = 4\\ c^2 + d^2 = 4 \\ ac + bd = 2 \end{cases} Find the set of all possible values the number M=ab+cdM = ab + cd can take.
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