Half-SAT intractability proof
The clue that you got is a bit strange, since true clauses are not enough. Here is one possible solution. Given a CNF formula with $m$ clauses, let $a$ be a new variable, add the clause $a \lor \lnot a$, and add $m+1$ copies of the clause $a$. I claim that if this formula is half-satisfiable then the original formula is satisfiable.
An assignment in which $a$ is true satisfies at least $m+2$ out of the $2m+2$ clauses, which is more than half the clauses. Therefore any assignment half-satisfying the new formula must assign false to $a$, leading to the conclusion that the $m$ original clauses must all be satisfied.