Strange succession 
An ant starts from the origin of a cartesian plane and makes $n \ge 2$ steps of lengh $d_1, d_2, \cdots, d_n$.  Is there a condition (necessary and sufficient) on $d_i$'s for which the ant can come back to the origin after the $n$ steps? (The ant can move in any direction.)

I think the claim is $d_i \leq \sum\limits_{j ≠ i} d_j$ for all $i$, which of course is necessary, but I cannot prove it is sufficient.

Your condition is clearly necessary and there is no sufficient condition.  At each step, even if the distance is right you only reach the origin if the angle you choose is exactly right, representing a point out of $[0,2\pi)$.  As there are only a countable number of steps, there are only (at most) a countable number of chances to pick the right angle.  The measure of a countable set of points in a line segment is zero.
