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DOTS Division
Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
Sixational
The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.
Age 14 to 16 Short
Challenge Level
Answer: $54$ kg
Body weighs $b$ kg, head $h$ kg, tail $9$ kg
$h=9+\frac13b$ and $b=h+9$, so $$\begin{align}b&=h+9\\
&=\left(9+\tfrac13b\right)+9\\
&=\tfrac13b+18\\
\Rightarrow\tfrac23b&=18\\
\Rightarrow\ \ b&=27\end{align}$$
So $h=9+\frac13\times27=18$ and the fish weighs $27+18+9=54$ kg
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.
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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.