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?

Coffee

To make 11 kilograms of this blend of coffee costs £15 per kilogram. The blend uses more Brazilian, Kenyan and Mocha coffee... How many kilograms of each type of coffee are used?

Thousands of X

Age 14 to 16 Short Challenge Level:

Answer: $2012$

By factorising

Notice that the two terms have a common factor of $x-2012$, so
\begin{align}&(x-2013)(x-2012)-(x-2012)(x-2011)=0\\ \Rightarrow&(x-2012)((x-2013)-(x-2011))=0\\ \Rightarrow&(x-2012)(x-2013-x+2011)=0\\ \Rightarrow&(x-2012)\times-2=0\\ \Rightarrow&x-2012=0\\ \Rightarrow&x=2012\end{align}

By expanding
Expanding both brackets and collecting like terms gives
\begin{align}&(x-2013)(x-2012)-(x-2012)(x-2011)=0\\ \Rightarrow&(x^2-2013x-2012x+2012\times2013)-(x^2-2012x-2011x+2011\times2012)=0\\ \Rightarrow&x^2-2013x-2012x+2012\times2013-x^2+2012x+2011x-2011\times2012=0\\ \Rightarrow&-2x+2012\times2013-2012\times2011=0\\ \Rightarrow&-2x+2\times2012=0\\ \Rightarrow&2\times2012=2x\\ \Rightarrow&2012=x\end{align}
You can find more short problems, arranged by curriculum topic, in our short problems collection.