Imagine that we have successfully processed first i - 1 bowls, i.e. we know height of the bottom y_j for every bowl j (1 ≤ j < i). Now we are going to place i-th bowl. For each j-th already placed bowl, we will calculate the relative height of the bottom of i-th bowl above the bottom of j-th bowl, assuming that there are no other bowls. Lets denote this value by Δ_i, j. It is obvious that height of the new bowl is equal to the maximal of the following values: y_i = max(y_j + Δ_i, j).

Now I will describe how to calculate Δ_i, j. Firstly, consider two trivial cases:

I. r_i ≥ R_j: bottom of i-th bowl rests on the top of j-th. Then Δ_i, j = h_j.
II. R_i ≤ r_j: bottom of i-th bowl reaches the bottom of j-th. Then Δ_i, j = 0.

Then there are three slightly more complicated cases.

1. r_i > r_j: bottom of i-th bowl gets stuck somewhere between the top and the bottom of j-th,
touching it's sides. From the similarity of some triangles we get that .

2. R_i ≤ R_j: top of i-th bowl gets stuck somewhere between the top and the bottom of j-th,
touching it's sides. From the similarity of some triangles we get that .

3. R_i > R_j: sides of i-th bowl touch the top of j-th in it's upper points. Then .

Note that, for example, cases 1 and 2 do not exclude each other, so the final value of Δ_i, j is equal to the maximum of the values, computed in all three cases.

Note that if the calculated value of Δ_i, j is negative, the result should be 0. Thanks to adamax for pointing it.