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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?


Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

Without Calculus

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

Biggest Bendy

Age 16 to 18 Challenge Level:

You may wish to try the problem Bendy Quad first.


Four rods are hinged at their ends to form a quadrilateral with fixed side lengths.
Show that the quadrilateral has a maximum area when it is cyclic.