Another Congruence Proof I've been asked to attempt a proof of the following congruence.  It is found in a section of my textbook with Wilson's theorem and Fermat's Little theorem.  I've pondered the problem for a while and nothing interesting has occurred to me.
$1^23^2\cdot\cdot\cdot(p-4)^2(p-2)^2\equiv (-1)^{(p+1)/2} (\text{mod} p)$

HINT: For $p=11$:
$$\begin{align*}
10!&=(1\cdot10)(2\cdot9)(3\cdot8)(4\cdot7)(5\cdot6)\\
&=\Big(1\cdot(-1)\Big)\Big(-9\cdot9\Big)\Big(3\cdot(-3)\Big)\Big(-7\cdot7\Big)\Big(5\cdot(-5)\Big)\\
&=(-1^2)(-9^2)(-3^2)(-7^2)(-5^2)\\
&=(-1)^5\cdot1^2\cdot3^2\cdot5^2\cdot7^2\cdot9^2
\end{align*}$$
