## USACO 2024 US Open Contest, Platinum

## Problem 2. Splitting Haybales

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Farmer John wants to fairly split haybales between his two favorite cows Bessie and Elsie. He has $N$ ( $1\le N\le 2\cdot 10^5$) haybales sorted in non-increasing order, where the $i$-th haybale has $a_i$ units of hay ($2\cdot 10^5\ge a_1\ge a_2 \ge \dots \ge a_N \ge 1$).

Farmer John is considering splitting a contiguous range of haybales $a_l, \dots, a_r$ between Bessie and Elsie. He has decided to process the haybales in order from $l$ to $r$, and when processing the $i$-th haybale he will give it to the cow who currently has less hay (if it is a tie, he will give it to Bessie).

You are given $Q$ ($1\le Q\le 2\cdot 10^5$) queries, each with three integers $l,r,x$ ($1\le l\le r\le N$, $|x|\le 10^9$). For each query, output how many more units of hay Bessie will have than Elsie after processing haybales $l$ to $r$, if Bessie starts with $x$ more units than Elsie. Note that this value is negative if Elsie ends up with more haybales than Bessie.

#### INPUT FORMAT (input arrives from the terminal / stdin):

First line contains $N$.Second line contains $a_1\dots a_N$.

Third line contains $Q$.

Next $Q$ lines contain $l, r, x$.

#### OUTPUT FORMAT (print output to the terminal / stdout):

Output $Q$ lines, containing the answer for each query.#### SAMPLE INPUT:

2 3 1 15 1 1 -2 1 1 -1 1 1 0 1 1 1 1 1 2 1 2 -2 1 2 -1 1 2 0 1 2 1 1 2 2 2 2 -2 2 2 -1 2 2 0 2 2 1 2 2 2

#### SAMPLE OUTPUT:

1 2 3 -2 -1 0 1 2 -1 0 -1 0 1 0 1For the 1st query, Elsie starts with $2$ more hay than Bessie. Then, after processing haybale $1$, Bessie will receive $3$ hay, and Bessie will have $1$ more hay than Elsie.

For the 3rd query, Elsie and Bessie start with the same number of hay. After processing haybale $1$, Bessie will receive $3$ hay, and Bessie will have $3$ more hay than Elsie.

For the 9th query, Bessie starts with $1$ more hay than Elsie, then after processing the 1st haybale, has $2$ less hay than Elsie, and after processing the 2nd haybale, has $1$ less hay than Elsie.

#### SAMPLE INPUT:

5 4 4 3 1 1 7 1 1 20 1 2 20 1 5 20 1 1 0 1 5 0 1 4 0 3 5 2

#### SAMPLE OUTPUT:

16 12 7 4 1 2 1

In the 5th query, there are $5$ haybales to process. Bessie receives $4$ hay, then Elsie receives $4$ hay, then Bessie receives $3$ hay, then Elsie receives $1$ hay, then Elsie receives $1$ hay.

#### SCORING:

- Input 3: $Q\le 100$
- Inputs 4-6: At most $100$ distinct $a_i$
- Inputs 7-22: No additional constraints.

Problem credits: Benjamin Qi