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Symmetric Inequality Problem

I am struggling with understanding https://nrich.maths.org/251.

To restate the problem,

If \(x\), \(y\) and \(z\) are real numbers such that: \(\begin{cases}x+y+z=5 \\ xy+yz+zx=3\end{cases}\) , What is the largest value that any one of these numbers can have?

In particular, I do not understand the first solution given, and while the second I am getting a grip with (creates a quadratic uses the discriminant inequality since \(x\), \(y\) and \(z\) are real numbers), would like to ask whether any classical inequalities can be used here, as I would be personally more satisfied with this.

The problem I had with applying inequalities I knew was that \(x\), \(y\) and \(z\) could be any real numbers, not just positive.

Any help/discussion would be much appreciated!

Note by Arthur Conmy
1 month ago

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i m getting \(\frac{13}{3} \) as maximum value and -1 as minimum value Vilakshan Gupta · 1 month ago

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@Vilakshan Gupta Yep same here! What was your method? Sathvik Acharya · 1 month ago

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@Sathvik Acharya but u said u are getting \(\frac{13}{2} \) . I used the same method which i referred to in the link i gave. Vilakshan Gupta · 1 month ago

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@Vilakshan Gupta Oops! Sorry I am getting \(\frac{13}{3}\). I actually did not check the link out. Let me check it now Sathvik Acharya · 1 month ago

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@Sathvik Acharya Ok. Tell me about your method please Vilakshan Gupta · 1 month ago

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@Vilakshan Gupta Cauchy Schwarz inequality Sathvik Acharya · 1 month ago

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@Sathvik Acharya same method as mine then? or in a different way? Vilakshan Gupta · 1 month ago

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Hey buddy is the answer \(\frac{13}{2}\)? Sathvik Acharya · 1 month ago

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okay ! great Vilakshan Gupta · 1 month ago

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actually u have put a dot by mistake in front of 251 Vilakshan Gupta · 1 month ago

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The link u have referred is not opening, it says the page not found.However, u can see my solution to this problem : https://brilliant.org/problems/almost-vietas/#!/solution-comments/171217/ Vilakshan Gupta · 1 month ago

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@Vilakshan Gupta Thanks Vilakshan. I leant from both your and Sharky's answers. Arthur Conmy · 1 month ago

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@Vilakshan Gupta Fixed btw Arthur Conmy · 1 month ago

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@Arthur Conmy yup Vilakshan Gupta · 1 month ago

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