Find all real solutions of the equation
$$(x^2-5x+5)^{(x^2-11x+30)} = 1$$

Well done to Mathilde from JMG le Clézio in Vanuatu, Nishad, Nayanika from the Tiffin Girls' School in the UK, Abhinav from St Olaves Grammar School in England and Santiago from Tecnologico de Monterrey High School in Mexico who found all six solutions. This is Mathilde's work (click to enlarge):

1. Find all the solutions to $(x^2 - 7x + 11)^{(x^2 - 13x + 42)} = 1$.
    How do these solutions compare to the first equation?  

Nishad, Abhinav and Santiago found all of the solutions. Here is Santiago's work (click to enlarge):

2. Can you find a Mega Quadratic Equation with solutions $3, 4, 5, 6, 7, 8$?
    How about $4, 5, 6, 7, 8, 9$?...

Nishad, Abhinav and Santiago found all of the solutions. Here is Abhinav's work (click to enlarge):

3. Can you explain why there are only $4$ solutions to $(x^2-5x+5)^{(x^2-4)}=1$?

Nishad, Abhinav and Santiago found all of the solutions. Here is Nishad's work:

The first question to ask is when is $f(x)^{g(x)} = 1$
Well we know that $a^0 = 1$ so $g(x)=0$ (*)
Next we know that $1^a = 1$ so $f(x)=0$ (**)
Now the final insight is $(-1)^{2a}=1$ (where $a$ is an integer) (***)

For the equation
$(x^2 -5x +5)^{(x+2)(x-2)}=1$

Using (*) we get the solutions $x=-2$ and $x=2$

Using (**) we get the solutions $x=4$ and $x=1$

Using (***) we get that $x=2$ but we have already used this solution and then we get $x=3$ but this then makes exponent $g(x)$ odd so this means that $x=3$ isn't a solution.

4. Can you explain why there are only $3$ solutions to $(x^2-6x+10)^{(x^2+x-2)}=1$?

Nishad, Abhinav and Santiago found all of the solutions. Here is Santiago's work (click to enlarge):

5. Can you find a Mega Quadratic equation with exactly $2$ solutions? $5$ solutions?

Nishad, Abhinav and Santiago found all of the solutions. Here is Abhinav's work:

With thanks to Don Steward, whose ideas formed the basis of this problem.