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Roots and Coefficients
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?
Pair Squares
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.
Escriptions
For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.
Age 16 to 18 Challenge Level:
If $x$, $y$ and $z$ are real numbers such that:
$x+y+z=5$
and $xy+yz+zx=3$,
what is the largest value that any one of these numbers can have?
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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.