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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.

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Not Continued Fractions

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

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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
$$\begin{align}&(x-2013)(x-2012)-(x-2012)(x-2011)=0\\
\Rightarrow&(x^2-2013x-2012x+2012\times2013)-(x^2-2012x-2011x+2011\times2012)=0\\
\Rightarrow&x^2-2013x-2012x+2012\times2013-x^2+2012x+2011x-2011\times2012=0\\
\Rightarrow&-2x+2012\times2013-2012\times2011=0\\
\Rightarrow&-2x+2\times2012=0\\
\Rightarrow&2\times2012=2x\\
\Rightarrow&2012=x\end{align}$$
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