On congruences involving products of variables from short intervals

We prove several results which imply the following consequences. For any ε > 0 and any sufficiently large prime p, if $\mathcal{I}_{1}, …, \mathcal{I}_{13}$ are intervals of cardinalities $\mid\mathcal{I_{j}}\mid > p^{1/4+\varepsilon}$ and $abc \not\equiv 0(mod p)$, then the congruence $$ax_1 … x_6 + bx_7 … x_{13} \equiv c (mod \textit{p})$$ has a solution with $x_j \in \mathcal{I_j}$. There exists an absolute constant $n_0 \in \mathbb{N}$ such that for any 0 < ε < 1 and any sufficiently large prime p, any quadratic residue λ modulo p can be represented in the form $x_1 … x_{n0} \equiv\lambda$ (mod p), $x_i \in \mathbb{N}$, $x_i \leq p^{1/(4e^{2/3})+\varepsilon}$. For any $\varepsilon > 0$, there exists $n = n(\varepsilon) \in \mathbb{N}$ such that for any sufficiently large $m \in \mathbb{N}$ the congruence $x_1 … x_{n} \equiv 1$ (mod m), $x_{i} \in \mathbb{N}$, $x_i \leq m^{\varepsilon}$ has a solution with $x_1 \neq 1$.