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Double Time

Crack this code which depends on taking pairs of letters and using two simultaneous relations and modulus arithmetic to encode the message.

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.

Real(ly) Numbers

Age 16 to 18 Challenge Level:

If $x$, $y$ and $z$ are real numbers such that:


and $xy+yz+zx=3$,

what is the largest value that any one of these numbers can have?