1979 USAMO Problems/Problem 3

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Problem

$a_1, a_2, \ldots, a_n$ 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 $a \plus{ } b \plus{ } c$ (Error compiling LaTeX. Unknown error_msg) is divisible by $3$ is at least $\frac14$.