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Absurdity Again

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

Just Touching

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

Pair Squares

Age 16 to 18
Challenge Level

You can choose any square numbers to start with but you'll find that only certain combinations of odd and even numbers will work. You'll need to solve three simultaneous linear equations in three unknowns to find formulae for the three integers in terms of the three square numbers that you started with.

For the second part you'll need a number which can be written as the sum of two squares in at least three different ways. You could use this example:

$1^2 + 132^2 = 28^2 + 129^2 = 36^2 + 127^2 = 55^2 + 120^2 = 63^2 + 116^2 = 80^2 + 105^2 = 17425$