Symmetry, Substitutions and Surprises

In this feature you will be challenged to exploit the symmetries of a range of equations to derive elegant and efficient solutions to problems which, at first sight, might seem very difficult.
Solve the equations to identify the clue numbers in this Sudoku problem.
Five equations and five unknowns. Is there an easy way to find the unknown values?
Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.
What do you get when you raise a quadratic to the power of a quadratic?
Observe symmetries and engage the power of substitution to solve complicated equations.
Can you solve the system of equations to find the values of x, y and z?