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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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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

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Bang's Theorem

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

Leonardo's Problem

Stage: 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?