  We formalize a combinatorial principle, called the 3XOR principle, due to
Feige, Kim and Ofek (2006), as a family of unsatisfiable propositional formulas
for which refutations of small size in any propositional proof system that
possesses the feasible interpolation property imply an efficient deterministic
refutation algorithm for random 3SAT with n variables and \Omega(n^{1.4})
clauses. Such small size refutations would improve the state-of-the-art (with
respect to the clause density) efficient refutation algorithm, which works only
for \Omega(n^{1.5}) many clauses (Feige and Ofek (2007)).
  We demonstrate polynomial-size refutations of the 3XOR principle in
resolution operating with disjunctions of quadratic equations with small
integer coefficients, denoted R(quad); this is a weak extension of cutting
planes with small coefficients. We show that R(quad) is weakly automatizable
iff R(lin) is weakly automatizable, where R(lin) is similar to R(quad) but with
linear instead of quadratic equations (introduced in Raz and Tzameret (2008)).
This reduces the problem of refuting random 3CNF with n variables and
\Omega(n^{1.4}) clauses to the interpolation problem of R(quad) and to the weak
automatizability of R(lin).
