Question 3
Let $p\ge17$ be a prime. Prove that $t=3$ is the largest positive integer with the following property: for arbitrary integers $a,b,c,d$ such that $p\nmid abc$ and $p\mid a+b+c$, there exist integers $x,y,z\in\{0,1,\ldots,\lfloor p/t\rfloor-1\}$ for which $p\mid ax+by+cz$.
Editorial notice: The printed statement omits the term $+d$. As written it is false, since $x=y=z=0$ also works for $t=4$. The intended corrected requirement is $p\mid ax+by+cz+d$; the model solution explicitly proves that corrected version.
Editorial notice: The printed statement omits the term $+d$. As written it is false, since $x=y=z=0$ also works for $t=4$. The intended corrected requirement is $p\mid ax+by+cz+d$; the model solution explicitly proves that corrected version.