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min and max of z if x+y+z=7 and xy+yz+zx=11

Source: 2012 Argentina OMA Finals L3 p1

January 15, 2023
algebrasystem of equationsinequalitiesSystem

Problem Statement

Determine if there are triplets (x,y,z)x,y,z) of real numbers such that {x+y+z=7xy+yz+zx=11\begin{cases} x+y+z=7 \\ xy+yz+zx=11\end{cases} If the answer is affirmative, find the minimum and maximum values ​​of zz in such a triplet.
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