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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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Solve the system of equations to find the values of x, y and z: xy/(x+y)=1/2, yz/(y+z)=1/3, zx/(z+x)=1/7

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Leonardo's Problem

Age 14 to 18 Challenge Level:

Three people (Alan, Ben and Chris), collectively own a certain number of gold sovereigns. Respectively they own a half, one third and one sixth of the total number.

All the sovereigns were piled on a table and each of them grabbed a part of the pile so that none were left.

After a short time:
Alan returned half of the sovereigns that he had taken.
Ben returned a third of what he had taken.
Chris returned one sixth of what she had taken.

Finally each of the three got an equal share of the amount that had been returned to the table.

Surprisingly, each person had exactly the number of sovereigns that really belonged to them.

What is the smallest number of sovereigns that this strange transaction will work for? How much did each person grab from the pile?