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.

#Algebra #Sharky

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

a=3,b=11,c=61

naitik sanghavi - 5 years, 7 months ago

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 - 5 years, 7 months ago

@Sharky This can be proved to be the only real solution!

Sualeh Asif - 5 years, 5 months ago

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 - 5 years, 5 months ago

Yup I have a pretty much same proof!

Sualeh Asif - 5 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$