2023 WSMO Speed Round Problems/Problem 6

Problem

Let $ABC$ be an equilateral triangle of side length $6.$ Points $A_1,A_2,A_3,B_1,B_2,B_3,C_1,C_2,C_3$ are chosen such that $A_1,A_2,A_3$ divide $BC$ into four equal segments, $B_1,B_2,B_3$ divide $AC$ into four equal segments, and $C_1,C_2,C_3$ divide $AB$ into four equal segments. If $i,j,k$ are chosen from the set ${1,2,3}$ independently and randomly, the expected area of $A_iB_jC_k$ is $\frac{a\sqrt{b}}{c},$ for squarefree $b$ and relatively prime positive integers $a$ and $c.$ Find $a+b+c.$

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2023 WSMO Speed Round Problems/Problem 6

Problem

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