amparvardi wrote: $100$-gon made of $100$ sticks. Could it happen that it is not possible to construct a polygon from any lesser number of these sticks? Are they all the same size?

Certainly not, they can be different (and actually, they should be different).

If one stick is longer than all the others combined then its impossible.

I think this is possible if we take one largest stick measuring $2^{100}$ units and 99 others measuring $2^{2}, 2^{2}, 2^{3}, 2^{4}, ..., 2^{97}, 2^{98}, 2^{99} + 1.$ In this case, the sum of the 99 smaller ones would be $ 2^{100} + 1$, so a 100-gon would be constructible. If we remove any of the smaller ones, the largest would exceed the sum of the 98 smaller ones, so the largest would have to be removed. The "new largest" measures now $2^{99} + 1$. But the sum of the 98 smaller ones would be exactly $ 2^{99}$ and would not exceed the largest; thus concluding the proof that it is not possible to construct a 99-gon. If we again remove the largest, having then 98 sticks, the sum of the smaller ones would equal the largest ($2^{98}$), and no 98-gon would be constructible. Analogously, no other n-gon would be constructible for n < 100.