I think this has something to do with the theory of martingales. The famous monkey abracadabra problem is similar to this. mathproblems123.wordpress.com/2010/09/16/…

Very beautiful proof.

I tried to approach the symmedian problem, and I was wondering what other approaches are there for problems like this, because the using the approach used for bisectors I couldn't get very far, because of the relatively complicated expression of the symmedian length. I just want to know if there are other possible approaches. I tried searching the Internet, but I didn't find anything related.

$\lambda$ is the Lebesgue measure on the real line?

Thank you. I will check the details later, but I suppose you are correct. :)

It is not a homework problem. I had the problem discussed with some of my friends during a math contest. We weren't able to prove or disprove it. This was two years ago... I haven't posted any details, because I didn't manage to get very far with it. It is obvious that $a_n \to 0$. Then if the sequence $(\varepsilon_1+..+\varepsilon_n)$ is bounded, the result is obvious. That's what I could get so far. I remember proving that if the sequence $(\varepsilon_1+..+\varepsilon_n)$ is bounded from below or above, the result also follows, but I don't remember exactly how I did that.

This is clearly true for absolute convergent series.