### 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?

# Bina-ring

##### Stage: 5 Challenge Level:

Thank you Andrei from Tudor Vianu National College, Bucharest, Romania for this solution.

First I looked at the values of $A$ and $B$ for the first values of $n$ in the expansion of $(1 + \sqrt 2)^n = A + B\sqrt 2$. The results I obtained are written in the following table:
 $n$ $A$ $B$ 1 1 1 2 3 2 3 7 5 4 17 12 5 41 29

In the cases studied, I observe that $A$ is odd for all $n$, and that $B$ is odd for $n$ odd and even for $n$ even. Now, I shall generalise this result by induction for any $n$:

First I assume that $A(n)$ and $B(n)$ are odd and I prove that $A(n+1)$ is odd and $B(n+1)$ is even. This corresponds to $n$ odd. I shall consider $A(n) = 2p+1$ and $B(n) = 2q+1$.

So, $(1 + \sqrt 2)^n = (2p+1) + (2q+1)\sqrt 2$. $$(1 + \sqrt 2)^{n+1} = (1 + \sqrt 2)^n (1+\sqrt 2) = ((2p+1) + (2q+1)\sqrt 2)(1+ \sqrt 2) = (2p+1 + 2(2q+1)) + \sqrt 2 (2q+1+2p+1).$$ So, $A(n+1) = 2(p+2q+1) + 1$ and $B(n+1) = 2(p+q+1).$ These values that I have obtained for A(n+1) and B(n+1) confirm my prediction.

In a similar manner I assume $A(n)$ odd and $B(n)$ even:
$A(n) = 2p+1$, $B(n) = 2q$. I obtain:
$A(n+1) = 2(p+2q) + 1$ and $B(n+1) = 2(p+q) + 1$.
Now, my proof by induction is complete.
The results are summarized in the following table:
 $A$ even $A$ odd $B$ even - $n$ even $B$ odd - $n$ odd

Working in a similar manner for $(a + \sqrt p)^n$, I summarize my results in the following table:
 $n$ $A$ $B$ $a$ odd, $p$ odd all $n$ even exception $n=1, A=1$ even exception $n=1, B=1$ $a$ even, $p$ even all $n$ even even exception $n=1, B=1$ $a$ odd, $p$ even odd even odd odd odd even $a$ even, $p$ odd odd even even odd odd even