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4 expressions, 3 variables?

Find all positive real numbers \(a\), \(b\) and \(c\) that satisfy the conditions:

\[a+b+c=75 \\ abc=2013 \\ a\leq3 \\ c\geq61\]

Give proof. You can write all families of positive real solutions.

Note that \(a,b\) and \(c\) might not necessarily be integers.

Note by Sharky Kesa
10 months, 3 weeks ago

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Since \(a\le 3\), \(b+c\le 72\) Similarly \(a+b\ge 14\).From this we get, \(14-a\le b\le 72-c\) again since \(a\le 3\) and \(c \ge 61\) we get \(11 \le b\le 11\) implies that \(b=11\). Now remains \(a+c=64\) and \(ac=183\) which on solving gives \(a=3,c=61\) @Sharky Kesa Please see if I am geting wrong:) Siddharth Singh · 9 months, 1 week ago

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@Siddharth Singh Yup I have a pretty much same proof! Sualeh Asif · 9 months, 1 week ago

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a=3,b=11,c=61 Naitik Sanghavi · 10 months, 3 weeks ago

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@Naitik Sanghavi How do you know that is the only solution? How do you know there aren't any other real solutions? Where is your proof? You only wrote 1 integral solution. Sharky Kesa · 10 months, 3 weeks ago

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@Sharky Kesa @Sharky This can be proved to be the only real solution! Sualeh Asif · 9 months, 1 week ago

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