Maximize the determinant Over the class $S$ of symmetric $n$ by $n$ matrices such that the diagonal entries are +1 and off diagonals are between $-1$ and $+1$ (inclusive/exclusive), is 
$$\max_{A \in S} \det A = \det(I_n)$$
?  

For the three by three case I have the following. Let $A$ be generally defined as
$$A= \pmatrix{
 1& x& y\\
 x& 1&z\\
y&z& 1\\
}.
$$
Then the determinant is 
$$|A| = 
\matrix{
1& z \\
z& 1 \\} - x\matrix{
x& z \\
y& 1 \\} + y\matrix{
x& 1 \\
y& z \\}
 = 1-z^2 - x(x-zy) + y(xz-y) =$$
$$= 1-x^2-y^2-z^2 +xyz-xyz= 1-(x^2+y^2+z^2)  \le 1 = \det(I_n)$$
