This is an olympiad question?

1[hide]$ 1 =\left(\frac{2}{2}\right)^{2}$[/hide] 2[hide]$ 2 = 2+2-2$[/hide] 3[hide]$ 3 =\frac{2}{2}+2$[/hide] 4[hide]$ 4 = (2+2)\cdot (!2)$[/hide] 5[hide]$ 5 = !2+2+2$[/hide] 6[hide]$ 6 = 2\cdot 2+2$[/hide] 7[hide]$ 7 = !(2+2)-2$[/hide] 8[hide]$ 8 = 2\cdot 2\cdot 2$[/hide] 9[hide]$ 9 = !( (2+2)\cdot (!2) )$[/hide] 10[hide]$ 10 = (2+2)!!+2$[/hide] 11[hide]$ 11 = !(2+2)-2$[/hide] 12

Note that I have used the multifactorial and the subfactorial. If these are not allowed (the problem does not specify), then it would be nice to have a list of what operations were allowed.

Yes, it really is And your answer is correct, I think(not checked carefully).

t0rajir0u wrote: This is an olympiad question? Note that I have used the multifactorial and the subfactorial. If these are not allowed (the problem does not specify), then it would be nice to have a list of what operations were allowed.

Or also : 11

And, just for fun : $ \forall n > 0$ : $ n =\frac{\ln(\frac{\ln(2)}{\ln(\sqrt{\sqrt{\sqrt{\cdots\sqrt{2}}}})})}{\ln{2}}$ with exactly $ n$ $ \sqrt{.}$

I just realized I was neglecting some other common symbols.

Example

I'm pretty sure the sub- and multi-factorial answers can be rewritten in this form as well. I'll do it later.

Official Solution: Some possible answers are shown in the following: \[ \begin{array}{ccc} 1=2^{2-2}&2=2+2-2&3=2+\frac 22\\ 4=\sqrt{2\times2}^2&5=2\times2+\lfloor\sqrt{2}\rfloor&6=2+2+2\\ 7=\lceil\sqrt{(2+2)!}\rceil+2&8=2\times2\times2&9=(2+\lfloor\sqrt{2}\rfloor)^2\\ 10=2\times\lceil\sqrt{(2+2)!}\rceil&11=\frac{22}{2}&12=\frac{(2+2)!}2\\ \end{array}\]

t0rajir0u: For 11, you wrote $ !(2 + 2) - 2$, when it's actually $ !(2 + 2)\ \textcolor{red}{ + }\ 2$.

What is ! sign in front of a number?? behind the number is factorial, but i havent seen in front of a number...

The subfactorial: $ !n = n! \sum^{n}_{k=0} \frac{(-1)^{k}}{k!}$

ahh thanks