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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?

Pair Squares

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.


For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

Overturning Fracsum

Age 16 to 18
Challenge Level

Solve the following system of equations to find the values of $x$, $y$ and $z$. $${\frac {xy}{x+y}}=\frac 1 2$$ $${\frac {yz} {y+z}} =\frac 1 3$$ $${\frac{xz}{x+z}} = \frac 1 7$$

Can you show how the first equation can be arranged to get $\dfrac 1 x + \dfrac 1 y =2$?