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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?

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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.

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Three Ways

Stage: 5 Challenge Level: Challenge Level:1

Given that $x + y = -1$ find the largest value of $xy$

  1. by coordinate geometry.
  2. by calculus.
  3. by algebra.