Question 23
Fix positive integers $m$ and $n$. Suppose that $a_{1}, a_{2}, \ldots, a_{m}$ are reals, and that pairwise distinct vectors $v_{1}, \ldots, v_{m} \in \mathbb{R}^{n}$ satisfy
$$
\sum_{j \neq i} a_{j} \frac{v_{j}-v_{i}}{\left\|v_{j}-v_{i}\right\|^{3}}=0
$$
for $i=1,2, \ldots, m$.
Prove that
$$
\sum_{1 \leq i\lt j \leq m} \frac{a_{i} a_{j}}{\left\|v_{j}-v_{i}\right\|}=0
$$
$$
\sum_{j \neq i} a_{j} \frac{v_{j}-v_{i}}{\left\|v_{j}-v_{i}\right\|^{3}}=0
$$
for $i=1,2, \ldots, m$.
Prove that
$$
\sum_{1 \leq i\lt j \leq m} \frac{a_{i} a_{j}}{\left\|v_{j}-v_{i}\right\|}=0
$$