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Baby Circle

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

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Absurdity Again

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

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Strange Rectangle 2

Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.

Ab Surd Ity

Stage: 5 Challenge Level: Challenge Level:1

For the first part try squaring the whole expression.

For the second part, you could write $a = {\root 3\of {2+\sqrt{5}}}$ and $b= {\root 3\of {2-\sqrt{5}}}$, and use the identity $$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + 3ab(a + b) + b^3.$$