Given prime $p\ge 11$, $S$ is a subset of $\mathbb{Z}_p\times\mathbb{Z}_p$ with $3p-3$ elements. Prove: $S$ has a subset $T$ with $p-1$ elements, such that$\sum_{x\in T}x\equiv (0,0)\pmod{p}$.

I've considered proving every element in $\mathbb{Z}_p\times\mathbb{Z}_p$ can be such a sum, which definitely holds. Then we can find a way to let the sum go through the complete residue system, which seems difficult. Also, it shows that if we change $p-1$ into $2p-2$, it still holds. But those work slightly, since I'm not clear about the use of $p$ is a prime.

Given prime $p\ge 11$, $S$ is a subset of $\mathbb{Z}_p\times\mathbb{Z}_p$ with $3p-3$ elements. Prove: $S$ has a subset $T$ with $p-1$ elements, such that$\sum_{x\in T}x\equiv (0,0)\pmod{p}$.

Given prime $p\ge 11$, $S$ is a subset of $\mathbb{Z}_p\times\mathbb{Z}_p$ with $3p-3$ elements. Prove: $S$ has a subset $T$ with $p-1$ elements, such that$\sum_{x\in T}x\equiv (0,0)(\mod p)$.

I've considered proving every element in $\mathbb{Z}_p\times\mathbb{Z}_p$ can be such a sum, which definitely holds. Then we can find a way to let the sum go through the complete residue system, which seems difficult. Also, it shows that if we change $p-1$ into $2p-2$, it still holds. But those work slightly, since I'm not clear about the use of $p$ is a prime.

Given prime $p\ge 11$, $S$ is a subset of $\mathbb{Z}_p\times\mathbb{Z}_p$ with $3p-3$ elements. Prove: $S$ has a subset $T$ with $p-1$ elements, such that$\sum_{x\in T}x\equiv (0,0)\pmod{p}$.

I've considered proving every element in $\mathbb{Z}_p\times\mathbb{Z}_p$ can be such a sum, which definitely holds. Then we can find a way to let the sum go through the complete residue system, which seems difficult. Also, it shows that if we change $p-1$ into $2p-2$, it still holds. But those work slightly, since I'm not clear about the use of $p$ is a prime.