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

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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?

Little and Large

Age 16 to 18 Challenge Level:

point X, within a rectangle.

The point $X$ moves around inside a rectangle of dimension $p$ units by $q$ units. The distances of $X$ from the vertices of the rectangle are $a$, $b$, $c$ and $d$ units. What are the least and the greatest values of

$a^2 + b^2 + c^2 + d^2$

and where is the point $X$ when these values occur?