Always Two

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

Surds

Find the exact values of x, y and a satisfying the following system of equations: 1/(a+1) = a - 1 x + y = 2a x = ay

Black and White Socks

Age 14 to 16 Short Challenge Level:

There are $20+n$ socks in the drawer altogether, so $\dfrac{n}{20+n}=\dfrac{1}{n}$.
$$\begin{split}&\frac{n}{20+n}=\frac{1}{n}\\ &\Rightarrow n=\frac{1(20+n)}{n}\\ &\Rightarrow n^2=20+n\\ &\Rightarrow n^2-n-20=0\\ &\Rightarrow (n+4)(n-5)=0\\ &\Rightarrow n+4=0\hspace{3mm}\text{or}\hspace{3mm} n-5=0\\&\Rightarrow n=-4\hspace{7mm}\text{or}\hspace{3mm}n=5\end{split}$$ $n$ must be positive as there cannot be a negative number of white socks, therefore there are $5$ white socks in the drawer.