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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.


For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

System Speak

Age 16 to 18
Challenge Level

How would you solve simultaneous equations if there were only two unknowns rather than five?
Can you find an expression for b in terms of a?
Can you find an expression for c in terms of b?
Can you combine these to give an expression for c in terms of a?