# 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
2 years, 8 months 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:)

- 2 years, 6 months ago

Yup I have a pretty much same proof!

- 2 years, 6 months ago

a=3,b=11,c=61

- 2 years, 8 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.

- 2 years, 8 months ago

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

- 2 years, 6 months ago