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# Minimizing 3 variable expression

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The positive real numbers a, b, c, and x, satisfy the equation "x=a+5b+15c=abc". Find the smallest possible value of x.

I don't want a full solution or an answer as I would wish to solve the problem myself. Can I please have a hint?

Jan 9, 2022
edited by Eesoog  Jan 9, 2022

#2
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I would give you a hint if I had produced anything that made sense. Jan 9, 2022
#3
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If abc represents the number 100a + 10b +c, then a, b and c must be integers.

Set a + 5b + 15c = 100a + 10b + c

Manipulate this to get c in terms of a and b, then see what the minimum values of a and b must be for c to be the smallest possible positive integer.

Jan 9, 2022
#4
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I don't think the question is referring abc as 100a + 10b + c since a, b, and c are positive real numbers, not integers from 0-9.

I think abc is a*b*c.

=^._.^=

catmg  Jan 9, 2022
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I think that perhaps using AM-GM gets the right answer.

(a + 5b + 15c)/3 >= cbrt(75*abc)

abc = a+5b+15c = x

x/3 >= cbrt(75*x)

And from there you can use algebra to simplify and solve. :))

=^._.^=

Jan 9, 2022