CEMC Cayley, 2014

Solution 1

We start from the OUTPUT and work back to the INPUT.

Since the OUTPUT 32 is obtained from adding 16 to the previous number, the previous number is $32-16=16$.

Since 16 is obtained by dividing the previous number by 2, then the previous number is $2 \times 16$ or $32$.

Since 32 is obtained by subtracting 8 from the INPUT, then the INPUT must have been $32+8=40$.

Solution 2

Suppose that the INPUT is $x$.

Subtracting 8 gives $x-8$.

Dividing this result by 2 gives $\frac{1}{2}(x-8)$ or $\frac{1}{2}x-4$.

Adding 16 to this result gives $(\frac{1}{2}x-4)+16=\frac{1}{2}x+12$, which is the OUTPUT.

If the OUTPUT is 32, then $\frac{1}{2}x+12=32$ or $\frac{1}{2}x=20$ and so $x=40$.

Therefore, the INPUT must have been 40.