Even if 3 of them will be correct he passes to onsite. I think FHC organizators should really think about fairness such structure of competition, because everybody have different kind of CPU and RAM and if competitor have access to powerful machine he would be able to solve problems with less effective algorithms and logic. I believe that "fair competition" implies equal condition for all competitors.

Could you please provide some details on the line sweeping part?

A doubles FFT is fine, if you multiply a vector of distances between adjacent points by a reversed self.

EDIT: entire solution (without FFT and other helper functions) below

            long[] seg = new long[n];
            for (int i = 0; i + 1 < n; ++i) {
                seg[i] = pos[i + 1] - pos[i];
            }
            seg[n - 1] = pos[0] + l - pos[n - 1];
            long[] rseg = new long[n];
            for (int i = 0; i < n; ++i) {
                rseg[i] = seg[n - 1 - i];
            }
            long[] prod = FastFourierTransform.mul(seg, rseg);
            long[] total = new long[n];
            for (int i = 0; i < prod.length; ++i) {
                total[(i + 1) % n] = (total[(i + 1) % n] + prod[i]) % MODULO;
            }

            res = 0;
            for (int i = 0; i <= k; ++i) {
                long pr = comb(n - i, k - i) * invcomb(n, k) % MODULO;
                if (i != 0) {
                    pr = pr * 2 % MODULO;
                }
                res = (res + total[i] * pr) % MODULO;
            }
            res = res * invs[4] % MODULO;
            res = res * invs[l] % MODULO;

I know a similar idea: using a doubles FFT with an integer FFT — doubles FFT may get precision error, and you can use the mod result of integer FFT to correct it.

My solution with fft without any modules(aka 1 module) got OK.

UPD: Just noticed, Petr described same solution

Yeah, it's certainly something for next year though. I was surprised by the amount of people that wanted to view the rounds on mobile because

1. You presumably aren't going to submit any solutions on mobile
2. I don't think I heard of anybody wanting to use mobile last year

But in retrospect, the use cases are pretty obvious. In untimed rounds you might want to read the problems on the bus or something before you're at a computer and ready to code. In any round you might want to use a mobile device as a separate screen to show the problems while you're coding on a main screen. Etc.

So certainly we'll work on this for next year.