= Solution
For a sequence $J$ in $\mathbb F_p^2$, write $(r\mid J)$ for the number of $r$-term subsequences whose sum is zero. We use the following consequence of the <Chevalley-Warning theorem>: if $|J|=3p-1$, then
$$
\boxed{1-(p\mid J)+(2p\mid J)\equiv0\pmod p.}
$$
Indeed, with one variable $x_i$ for each term $(a_i,b_i)$, apply Chevalley--Warning to
$$
\sum_i x_i^{p-1},
\qquad
\sum_i a_ix_i^{p-1},
\qquad
\sum_i b_ix_i^{p-1}.
$$
Their degree sum is $3(p-1)<3p-1$. A common zero has support of size $0,p$, or $2p$ modulo $p$, and each fixed support contributes $(p-1)^{|S|}$ assignments. Reducing modulo $p$ gives the displayed congruence.
Now let the given $3p$ terms have total sum zero. If no $p$ of them summed to zero, delete any one term and apply the congruence to the remaining $3p-1$ terms. It gives
$$
(2p\mid J)\equiv-1\pmod p,
$$
so those remaining terms contain a zero-sum $2p$-subsequence. Its complement in the original $3p$ terms has size $p$ and sum zero, contradicting the assumption. Therefore the required $p$ terms exist.
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