A11297. Turtle vs. Rabbit Race: Optimal Trainings

Isaac begins his training. There are $n$ running tracks available, and the $i$ -th track ( $1 \le i \le n$ ) consists of $a_i$ equal-length sections.

Given an integer $u$ ( $1 \le u \le 10^9$ ), finishing each section can increase Isaac's ability by a certain value, described as follows:

- Finishing the $1$ -st section increases Isaac's performance by $u$ .
- Finishing the $2$ -nd section increases Isaac's performance by $u-1$ .
- Finishing the $3$ -rd section increases Isaac's performance by $u-2$ .
- $\ldots$
- Finishing the $k$ -th section ( $k \ge 1$ ) increases Isaac's performance by $u+1-k$ . (The value $u+1-k$ can be negative, which means finishing an extra section decreases Isaac's performance.)

You are also given an integer $l$ . You must choose an integer $r$ such that $l \le r \le n$ and Isaac will finish each section of each track $l, l + 1, \dots, r$ (that is, a total of $\sum_{i=l}^r a_i = a_l + a_{l+1} + \ldots + a_r$ sections).

Answer the following question: what is the optimal $r$ you can choose that the increase in Isaac's performance is maximum possible?

If there are multiple $r$ that maximize the increase in Isaac's performance, output the smallest $r$ .

To increase the difficulty, you need to answer the question for $q$ different values of $l$ and $u$ .

The first line of input contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases.

The descriptions of the test cases follow.

The first line contains a single integer $n$ ( $1 \le n \le 10^5$ ).

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \le a_i \le 10^4$ ).

The third line contains a single integer $q$ ( $1 \le q \le 10^5$ ).

The next $q$ lines each contain two integers $l$ and $u$ ( $1 \le l \le n, 1 \le u \le 10^9$ ) — the descriptions to each query.

The sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ . The sum of $q$ over all test cases does not exceed $2 \cdot 10^5$ .

For each test case, output $q$ integers: the $i$ -th integer contains the optimal $r$ for the $i$ -th query. If there are multiple solutions, output the smallest one.

For the $1$ -st query in the first test case:

- By choosing $r = 3$ , Isaac finishes $a_1 + a_2 + a_3 = 3 + 1 + 4 = 8$ sections in total, hence his increase in performance is $u+(u-1)+\ldots+(u-7)=8+7+6+5+4+3+2+1 = 36$ .
- By choosing $r = 4$ , Isaac finishes $a_1 + a_2 + a_3 + a_4 = 3 + 1 + 4 + 1 = 9$ sections in total, hence his increase in performance is $u+(u-1)+\ldots+(u-8)=8+7+6+5+4+3+2+1+0 = 36$ .

Both choices yield the optimal increase in performance, however we want to choose the smallest $r$ . So we choose $r = 3$ .

For the $2$ -nd query in the first test case, by choosing $r = 4$ , Isaac finishes $a_2 + a_3 + a_4 = 1 + 4 + 1 = 6$ sections in total, hence his increase in performance is $u+(u-1)+\ldots+(u-5)=7+6+5+4+3+2 = 27$ . This is the optimal increase in performance.

For the $3$ -rd query in the first test case:

- By choosing $r = 5$ , Isaac finishes $a_5 = 5$ sections in total, hence his increase in performance is $u+(u-1)+\ldots+(u-4)=9+8+7+6+5 = 35$ .
- By choosing $r = 6$ , Isaac finishes $a_5 + a_6 = 5 + 9 = 14$ sections in total, hence his increase in performance is $u+(u-1)+\ldots+(u-13)=9+8+7+6+5+4+3+2+1+0+(-1)+(-2)+(-3)+(-4) = 35$ .

Both choices yield the optimal increase in performance, however we want to choose the smallest $r$ . So we choose $r = 5$ .

Hence the output for the first test case is $[3, 4, 5]$ .