Olympiad selection for grade 10th part III

Sorry to be discouraging, but I fail to see how this constitutes a proper proof (or nice solution) at all.

if x=1.5

y=1

z=0.5

then x/z +z/y +3y = 6.5

there are many cases for which the results will be more than 5.

This is logically unsound. Just because you fail to find cases of which the expression gives a minimum of $<5$ doesn't mean that you can claim that minimum occurs when $x=y=z$ .

You are essentially arguing using the notion of "proof by exhaustion". This method works when there are little cases to inspect. However, given that there are infinitely many positive real solutions to the equation $x+y+z=3$ , you can't possibly inspect all of them! (unless you have a supercomputer, which you don't while solving problems in an Olympiad!)

To illustrate my point, suppose that you want to prove the fallacious statement that all rabbits are white. Just because you find that all rabbits are white in your neighbourhood, that doesn't mean that all rabbits in this universe are white. (You can't possibly inspect all of them!)

However, if you are able to publish a paper/ discover a result about the genetic composition of the rabbit species which necessitates all of them to be white, there is no need to go around looking at rabbits to make sure that your claim is right.

The publish a paper/ discover a result analogy above is similar to the notion of most Olympiad inequalities proofs. A solid proof needs to convince the reader that case-checking is unnecessary, especially when the range is so large that it becomes impractical, eg. $x,y,z \in \mathbb{R^{+}}$

Nice solution!