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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?

Overturning Fracsum

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

Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

Leonardo's Problem

Age 14 to 18 Challenge Level:

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

All the coins 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 coins that he had taken.
Ben returned a third of the coins that he had taken.
Chris returned one sixth of the coins that 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 coins that really belonged to them.

What is the smallest number of coins that this strange transaction will work for?

How much did each person grab from the pile?