Understand the problem

Find the 3-digit number whose ratio with the sum of its digits is minimal.

Albania TST 2013

Topic
Number Theory, Inequalities.
Easy
Suggested Book
Problem Solving Strategies by Arthur Engel

Do you really need a hint? Try it first!

Suppose that the number is $(abc)_{10}$. Then we would like to minimise $\frac{100a+10b+c}{a+b+c}$. Try to minimise for one variable at a time.
Note that, $\frac{100a+10b+c}{a+b+c}=1+\frac{9(11a+b)}{a+b+c}$. In this expression, it is possible to minimise for $c$ independent of $a,b$.
From the previous hint, the expression is minimised for $c=9$. Show that the ratio can now be rewritten as $1+\frac{10a-9}{a+b+9}$. Minimise for $b$.
In the previous hint we see that the minimising value of $b$ is also 9. Finally, the ratio may be written as $10-\frac{189}{a+18}$. This is minimised for $a=1$. Hence the answer is 199.

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