Melody

That is EXCELLENT Elijah !!

Not lucky, mostly consistent consciencious effort  

You do not need help with all of this.

Please discuss what you CAN do when you put your question up.

I await your response.

It is not a trick but pretty much any scientific calculator can do it.

You have a square.  All the sides are the same. the perimeter is 20 what is the side length?

Then what is the area of this square?

You have a another square.  All the sides are the same. the perimeter is 8 what is the side length?

Then what is the area of this square?

Here is an earier solution to this question by CPhill

You are very welcome.

It is not a proof though...

$k = x + y(1 - x) + z(1 - x)(1 - y)\\ k = x + y-xy + z(1-x-y+xy)\\ k = (x + y-xy) - z(x+y-xy)+z\\ k = (x + y-xy) (1- z)+z\\$

The maximum of x+y-xy is 1 when x=1 and y=1 

You can prove that by considering the case where  x=1-t   and y=1-m   where t and m are (0,1]

If z=0 then we have  1*1+0 = 1

So I get a max of 1 when  x=1,y=1 and z=0

There is also be a max of 1 when x=0, y=0 and z =1

I do not think that k can be bigger than 1

If the ratio of the perimeter of the squares is  5:2

Then .. What is the ratio of the sides of the squares.  Draw a pick and think about it.   I'll continue after you answer.

Hint:

5:2 is the same as   20:8    Do you see why?

20:8 is easier to work with.  

Let the squares have a perimeter of  20 and 8 respecively.  How long are the sides.?

You are welcome.