1979 USAMO 第 3 题

$a_1, a_2, \ldots, a_n$ is an arbitrary sequence of positive integers. A member of the sequence is picked at random. Its value is $a$. Another member is picked at random, independently of the first. Its value is $b$. Then a third value, $c$. Show that the probability that $a + b +c$ is divisible by $3$ is at least $\frac14$.

$a_1, a_2, \ldots, a_n$ 是任意正整数序列。序列中的一个成员是随机挑选的。它的价值是$a$。另一位成员是随机挑选的，与第一个成员无关。它的价值是$b$。然后是第三个值，$c$。证明 $a + b +c$ 能被 $3$ 整除的概率至少为 $\frac14$。