Score : $600$ points

问题描述

我们有一序列 $k$ 个数：$d_0,d_1,...,d_{k - 1}$。

按顺序处理以下 $q$ 个查询：

• 第 $i$ 个查询包含三个整数 $n_i$, $x_i$ 和 $m_i$。令由 $n_i$ 个数构成的序列 $a_0,a_1,...,a_{n_i - 1}$ 如下所示：[ \begin{aligned} a_j = \begin{cases} x_i & ( j = 0 ) \ a_{j - 1} + d_{(j - 1)~\text{mod}~k} & ( 0 < j \leq n_i - 1 ) \end{cases}\end{aligned} ] 输出满足条件 $(a_j~\text{mod}~m_i) < (a_{j + 1}~\text{mod}~m_i)$ 的所有 $j$ 的数量，其中 $0 \leq j < n_i - 1$。

这里 $(y~\text{mod}~z)$ 表示两个整数 $y$ 和 $z~(z > 0)$ 中 $y$ 除以 $z$ 后的余数。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

We have a sequence of $k$ numbers: $d_0,d_1,...,d_{k - 1}$.

Process the following $q$ queries in order:

• The $i$-th query contains three integers $n_i$, $x_i$, and $m_i$. Let $a_0,a_1,...,a_{n_i - 1}$ be the following sequence of $n_i$ numbers: [ \begin{aligned} a_j = \begin{cases} x_i & ( j = 0 ) \ a_{j - 1} + d_{(j - 1)~\textrm{mod}~k} & ( 0 < j \leq n_i - 1 ) \end{cases}\end{aligned} ] Print the number of $j~(0 \leq j < n_i - 1)$ such that $(a_j~\textrm{mod}~m_i) < (a_{j + 1}~\textrm{mod}~m_i)$.

Here $(y~\textrm{mod}~z)$ denotes the remainder of $y$ divided by $z$, for two integers $y$ and $z~(z > 0)$.

Constraints

• All values in input are integers.
• $1 \leq k, q \leq 5000$
• $0 \leq d_i \leq 10^9$
• $2 \leq n_i \leq 10^9$
• $0 \leq x_i \leq 10^9$
• $2 \leq m_i \leq 10^9$

Input

Input is given from Standard Input in the following format:

$k$ $q$

$d_0$ $d_1$ $...$ $d_{k - 1}$

$n_1$ $x_1$ $m_1$

$n_2$ $x_2$ $m_2$

$:$

$n_q$ $x_q$ $m_q$

Output

Print $q$ lines.

The $i$-th line should contain the response to the $i$-th query.

Sample Input 1

3 1
3 1 4
5 3 2

Sample Output 1

1

For the first query, the sequence {$a_j$} will be $3,6,7,11,14$.

• $(a_0~\textrm{mod}~2) > (a_1~\textrm{mod}~2)$
• $(a_1~\textrm{mod}~2) < (a_2~\textrm{mod}~2)$
• $(a_2~\textrm{mod}~2) = (a_3~\textrm{mod}~2)$
• $(a_3~\textrm{mod}~2) > (a_4~\textrm{mod}~2)$

Thus, the response to this query should be $1$.

Sample Input 2

7 3
27 18 28 18 28 46 1000000000
1000000000 1 7
1000000000 2 10
1000000000 3 12

Sample Output 2

224489796
214285714
559523809

update @ 2024/3/10 17:16:32