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Real(ly) Numbers
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 of the numbers can have?
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:
Take a look at the system of equations below:
$ab = 1$
$bc = 2$
$cd = 3$
$de = 4$
$ea = 6$
Can you find a set of values {a, b, c, d, e} that satisfy the system?Can you find more than one?
Working systematically. Mathematical reasoning & proof. Generalising. Simultaneous equations. Inequalities. Creating and manipulating expressions and formulae. Polynomial functions and their roots. Circle properties and circle theorems. 2D shapes and their properties. Expanding and factorising quadratics.
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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.