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?

Ab Surd Ity

Find the value of sqrt(2+sqrt3)-sqrt(2-sqrt3)and then of cuberoot(2+sqrt5)+cuberoot(2-sqrt5).

Absurdity Again

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

Fibonacci Fashion

Age 16 to 18 Challenge Level:

Given $F_n={1\over\sqrt5}(\alpha^n-\beta^n)$ where $\alpha$ and $\beta$ are solutions of the quadratic equation $x^2-x-1=0$ and $\alpha > \beta$ show that

(1)$\alpha\beta =-1$, $\alpha + \beta = 1$

(2)${ 1\over \alpha}+{1\over \alpha^2} = {1\over \beta} + {1\over \beta^2}=1$

(3)$F_1=F_2=1$ and $F_n + F_{n+1} = F_{n+2}$ and hence $F_n$ is the $n$th Fibonacci number and

(4) the sum $1 + F_1 + F_2 + \ldots F_n$ gives another Fibonacci number.