Question 67
Let $p\ge17$ be a prime. Prove that $t=3$ is the greatest positive integer with the following property: for all integers $a,b,c,d$ such that $p\nmid abc$ and $p\mid a+b+c$, there exist
$$x,y,z\in\{0,1,\ldots,\lfloor p/t\rfloor-1\}$$
such that $p\mid ax+by+cz+d$. Here $\lfloor p/t\rfloor$ denotes the greatest integer not exceeding $p/t$.
$$x,y,z\in\{0,1,\ldots,\lfloor p/t\rfloor-1\}$$
such that $p\mid ax+by+cz+d$. Here $\lfloor p/t\rfloor$ denotes the greatest integer not exceeding $p/t$.