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We take three pieces of random lengths from the interval $[1,n]$, and then guarantee that they can form a triangle (ie that the triangle inequality is satisfied). That is to say we say that the sum of the smaller two lengths are greater than the sum of the longest length.

What is the expected area of the triangle which we can guarantee forming, in terms of $n$?

Note: the three lengths do not sum of $n$ as in other variations of the broken stick problem.

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    $\begingroup$ Do you really expect a nice closed form answer? The triple integral is easy to set up (Heron's formula+limits mixing the length restriction with the triangle inequality) and the numeric integration for fixed $n$ is also not a big deal but good luck with finding an analytic expression even when $n\in[1,2]$, in which case the triangle inequality is never a constraint. $\endgroup$
    – fedja
    Commented Jul 22, 2017 at 21:33
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    $\begingroup$ You might have better luck with the expected value of the square of the area. The portion with all sides over $n/2$ might dominate. $\endgroup$
    – Aaron Meyerowitz
    Commented Jul 23, 2017 at 5:24

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