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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?

Always Two

Age 14 to 18 Challenge Level:

Find all the triples of numbers $a$, $b$, $c$
such that each one of them plus the product
of the other two is always $2$.

Quadratic equations. Surds. Expanding and factorising quadratics. Creating and manipulating expressions and formulae. Simultaneous equations. Mathematical reasoning & proof. Graphs. Polynomial functions and their roots. Golden ratio. Inequalities.