ISL 1999 C7

Let $p >3$ be a prime number. For each nonempty subset $T$ of $\{0,1,2,3, \ldots , p-1\}$ , let $E(T)$ be the set of all $(p-1)$ -tuples $(x_1, \ldots ,x_{p-1} )$ , where each $x_i \in T$ and $x_1+2x_2+ \ldots + (p-1)x_{p-1}$ is divisible by $p$ and let $|E(T)|$ denote the number of elements in $E(T)$ . Prove that

$$|E(\{0,1,3\})| \geq |E(\{0,1,2\})|$$

with equality if and only if $p = 5$ .

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