| # 1540_C1. Converging Array (Easy Version) |
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| ## Problem Description |
| This is the easy version of the problem. The only difference is that in this version q = 1. You can make hacks only if both versions of the problem are solved. |
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| There is a process that takes place on arrays a and b of length n and length n-1 respectively. |
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| The process is an infinite sequence of operations. Each operation is as follows: |
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| * First, choose a random integer i (1 ≤ i ≤ n-1). |
| * Then, simultaneously set a_i = min\left(a_i, \frac{a_i+a_{i+1}-b_i}{2}\right) and a_{i+1} = max\left(a_{i+1}, \frac{a_i+a_{i+1}+b_i}{2}\right) without any rounding (so values may become non-integer). |
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| See notes for an example of an operation. |
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| It can be proven that array a converges, i. e. for each i there exists a limit a_i converges to. Let function F(a, b) return the value a_1 converges to after a process on a and b. |
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| You are given array b, but not array a. However, you are given a third array c. Array a is good if it contains only integers and satisfies 0 ≤ a_i ≤ c_i for 1 ≤ i ≤ n. |
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| Your task is to count the number of good arrays a where F(a, b) ≥ x for q values of x. Since the number of arrays can be very large, print it modulo 10^9+7. |
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| Input |
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| The first line contains a single integer n (2 ≤ n ≤ 100). |
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| The second line contains n integers c_1, c_2 …, c_n (0 ≤ c_i ≤ 100). |
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| The third line contains n-1 integers b_1, b_2, …, b_{n-1} (0 ≤ b_i ≤ 100). |
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| The fourth line contains a single integer q (q=1). |
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| The fifth line contains q space separated integers x_1, x_2, …, x_q (-10^5 ≤ x_i ≤ 10^5). |
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| Output |
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| Output q integers, where the i-th integer is the answer to the i-th query, i. e. the number of good arrays a where F(a, b) ≥ x_i modulo 10^9+7. |
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| Example |
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| Input |
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| 3 |
| 2 3 4 |
| 2 1 |
| 1 |
| -1 |
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| Output |
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| 56 |
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| Note |
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| The following explanation assumes b = [2, 1] and c=[2, 3, 4] (as in the sample). |
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| Examples of arrays a that are not good: |
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| * a = [3, 2, 3] is not good because a_1 > c_1; |
| * a = [0, -1, 3] is not good because a_2 < 0. |
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| One possible good array a is [0, 2, 4]. We can show that no operation has any effect on this array, so F(a, b) = a_1 = 0. |
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| Another possible good array a is [0, 1, 4]. In a single operation with i = 1, we set a_1 = min((0+1-2)/(2), 0) and a_2 = max((0+1+2)/(2), 1). So, after a single operation with i = 1, a becomes equal to [-1/2, 3/2, 4]. We can show that no operation has any effect on this array, so F(a, b) = -1/2. |
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| ## Contest Information |
| - **Contest ID**: 1540 |
| - **Problem Index**: C1 |
| - **Points**: 1500.0 |
| - **Rating**: 2700 |
| - **Tags**: dp, math |
| - **Time Limit**: {'seconds': 5, 'nanos': 0} seconds |
| - **Memory Limit**: 256000000 bytes |
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| ## Task |
| Solve this competitive programming problem. Provide a complete solution that handles all the given constraints and edge cases. |
