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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.

Escriptions

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

Quadratic Harmony

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

Real(ly) Numbers

Age 16 to 18
Challenge Level

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 one of these numbers can have?


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