Any help now???

Let x = (x₁, x₂, ..., x_n) represent our solution, x_i is the velocity for i-th section (according to the hint, we will always ride with uniform velocity on one section).

It only makes sense to consider solutions with x_i > max(0, v_i). Also there's always an optimal solution where we use all our energy, because if we have some energy left, we can increase velocity somewhere and we'll arrive faster.

Let . We'll consider only solutions satisfying F(x) = 0 because we want to use all our energy. Let . We want to optimize g while respecting the constraint F(x) = 0.

Lagrange multipliers tell us that for optimal solution there exists some λ, such that , so for every i we have x_i²(x_i - v_i)k_i = C (where ).

For given C we can solve this equation using binsearch (left side is increasing if x_i > max(0, v_i). We can also observe that if we increase C, then the solution x(C) will also increase and so F(x(C)) will also increase. So we can binsearch for such C that F(x(C)) = 0 and this will give us optimal x.