NMTC doubt questions

I recently appeared nmtc final round junior and these are the questions in which i have doubt. Please upload the solution if you know the answer
Q.1 a,b,c are positive real numbers. Find the minimum value of $\frac{a+3c}{a+2b+c}$ + $\frac{4b}{a+b+2c}$ - $\frac{8c}{a+b+3c}$ I got the answer $\frac{2}{5}$ . Is it correct?

Q.2 show that for any natural number n, there is a positive integer all of whose digits are 5 or 0 and is divisible by n. I have totally no idea how this one is done , and i ended up just bluffing some answer in the end. thanks in advance for the solution

Easy Math Editor

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

Question 1: The minimum value is $12\sqrt2 - 17$ and it occurs when $(a,b,c) = \left( \dfrac32 - \sqrt2, \dfrac1{\sqrt2} - \dfrac12 , \dfrac1{\sqrt2} \right)$ .

Pi Han Goh - 4 years, 7 months ago

can you please post the detailed solution

Sayantan Dhar - 4 years, 7 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}`	$\boxed{123}$