  In this paper we confirm three conjectures of Z.-W. Sun on determinants. We
first show that any odd integer $n>3$ divides the determinant
$$\left|(i^2+dj^2)\left(\frac{i^2+dj^2}n\right)\right|_{0\le i,j\le (n-1)/2},$$
where $d$ is any integer and $(\frac{\cdot}n)$ is the Jacobi symbol. Then we
prove some divisibility results concerning $|(i+dj)^n|_{0\le i,j\le n-1}$ and
$|(i^2+dj^2)^n|_{0\le i,j\le n-1}$, where $d\not=0$ and $n>2$ are integers.
Finally, for any odd prime $p$ and integers $c$ and $d$ with $p\nmid cd$, we
determine completely the Legendre symbol $(\frac{S_c(d,p)}p)$, where
$S_c(d,p):=|(\frac{i^2+dj^2+c}p)|_{1\le i,j\le(p-1)/2}$.
