If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of
these numbers must be 1. Now for the complexity! When are the other
numbers real and when are they complex?
The sum of any two of the numbers 2, 34 and 47 is a perfect square.
Choose three square numbers and find sets of three integers with
this property. Generalise to four integers.
For any right-angled triangle find the radii of the three escribed
circles touching the sides of the triangle externally.
Alan of Madras College, St Andrew's and Tom of Bosworth College, Desford, Leicestershire, sent in really elegant solutions to show the maximum value taken by any one of these variables.
Firstly Alan's solution
Thus 13/3 is the maximum real value of one of the solutions of f ( a ) = 0 which corresponds exactly to the given equations, that is 13/3 is the maximum real value for any of the variables where all three satisfy the given equations.
The same method gives the minimum value . By lifting the graph the minimum possible value of x is made to occur when y = z = 3 which gives x = -1. So x = -1 is the minimum real value possible for any of the variables satisfying the given equations.
Tom uses a different method. He treats the two equations as simultaneous equations and one of the variables as if we know its value. Since the variables are interchangeable, this can be any of them. We can choose one of the variables to be bigger than the others and the limits we obtain on this variable can be applied to all of them.
So 13/3 is the maximum value any of the numbers can take and -1 is the minimum value any of the variables can take.