P6588 "JROI-1" Vectors

Foreword: Although SCR has already been merged into JROI, as the person in charge of JROI, I still want to thank the SCR problem team for their **selfless** dedication, out of respect for the setters. We will not make major changes to the background story of the problem; we will only add a **small amount of connecting** text. -------------- The beginner hotpot is boiling, so of course it is time to play a game. Little L is a middle school student who likes playing Identity V. On this day, he had just self-studied the basic operations of vectors. While playing, he looked at the mechanism walls he drew with different lengths and directions (he was playing Mad Eyes), and came up with a wonderful idea.

Little L has $n$ vectors $\overrightarrow{a_1},\overrightarrow{a_2}\ldots\overrightarrow{a_n}$, and he wants you to help him answer the following two questions. + For a given $l,r$, compute $$\sum\limits_{i=l}^{r-1}\sum\limits_{j=i+1}^{r}\overrightarrow{a_i}\cdot\overrightarrow{a_j}$$ + For a given $l,r$, compute $$\sum\limits_{i=l}^{r-1}\sum\limits_{j=i+1}^{r}\overrightarrow{a_i}\oplus\overrightarrow{a_j}$$ As time goes by, these vectors will also keep changing. Little L hopes that after changes happen, you can still give the answers.

The first line contains two integers $n,m$, representing the number of vectors and the number of operations. The next $n$ lines each contain two integers $x,y$. The $i$-th line represents the vector $\overrightarrow{a_i}$. The next $m$ lines each start with an integer $opt$ indicating the operation type, followed by several integers describing the operation. There are five types of operations as follows. 1. Input three integers $i,x,y(1\leq i\leq n)$, add $(x,y)$ to $\overrightarrow{a_i}$. 1. Input three integers $i,x,y(1\leq i\leq n)$, subtract $(x,y)$ from $\overrightarrow{a_i}$. 1. Input two integers $i,t(1\leq i\leq n)$, modify $\overrightarrow{a_i}$ to $t\overrightarrow{a_i}$. 1. Input two integers $l,r(1\leq l

For every operation of type 4 and type 5, output one line containing one integer, the answer to this operation.

#### Explanation for Sample 1 After the first two operations, the three vectors are $(4,7),(4,5),(-2,4)$. Then the query result is $4\times(-2)+5\times4=12$. After the next operation, the three vectors are $(4,7),(12,15),(-2,4)$. The query result is $(4\times15-7\times12)+[4\times4-7\times(-2)]+[12\times4-15\times(-2)]=-24+30+78=84$. ----------- #### Constraints **This problem uses bundled testdata.** + Subtask 1 ( $20\%$ ): $n,m\leq 100$. + Subtask 2 ( $30\%$ ): there is no operation type 5. + Subtask 3 ( $50\%$ ): no special requirements. For $100\%$ of the data, $2\leq n\leq 10^5$, $1\leq m\leq 10^5$, and **it is guaranteed that for the vector $\overrightarrow{a_i}$ at any time, $-1000\leq x_i,y_i\leq 1000$**. ----------- #### About Vector Operations For vectors $\overrightarrow{a},\overrightarrow{b}$ and a constant $\lambda$, suppose the coordinate representations of $\overrightarrow{a},\overrightarrow{b}$ are $(x_a,y_a),(x_b,y_b)$: + $\overrightarrow{a}+\overrightarrow{b}=(x_a+x_b,y_a+y_b)$ + $\overrightarrow{a}-\overrightarrow{b}=(x_a-x_b,y_a-y_b)$ + $\lambda\overrightarrow{a}=(\lambda x_a,\lambda y_a)$ + $\overrightarrow{a}\cdot\overrightarrow{b}=x_ax_b+y_ay_b$ + $\overrightarrow{a}\oplus\overrightarrow{b}=x_ay_b-x_by_a$ Translated by ChatGPT 5