Polygon and Points

AtCoder

IDabc296_g

Time3000ms

Memory256MB

Difficulty—

There is a convex $N$\-gon $S$ in the two-dimensional coordinate plane where the positive $x$\-axis points to the right and the positive $y$\-axis points upward. The vertices of $S$ have the coordinates $(X_1,Y_1),\ldots,(X_N,Y_N)$ in counter-clockwise order. For each of $Q$ points $(A_i,B_i)$, answer the following question: is that point inside or outside or on the boundary of $S$? ## Constraints * $3 \leq N \leq 2\times 10^5$ * $1 \leq Q \leq 2\times 10^5$ * $-10^9 \leq X_i,Y_i,A_i,B_i \leq 10^9$ * $S$ is a strictly convex $N$\-gon. That is, its interior angles are all less than $180$ degrees. * $(X_1,Y_1),\ldots,(X_N,Y_N)$ are the vertices of $S$ in counter-clockwise order. * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $X_1$ $Y_1$ $\vdots$ $X_N$ $Y_N$ $Q$ $A_1$ $B_1$ $\vdots$ $A_Q$ $B_Q$ [samples]

Samples

Input #1

Output #1

ON
IN
OUT

The figure below shows $S$ and the given three points. The first point is on the boundary of $S$, the second is inside $S$, and the third is outside $S$.
![image](https://img.atcoder.jp/abc296/828da6ca52e6b48a908ad06fa59eb9cb.png)

Input #2

Output #2

ON
ON
ON

API Response (JSON)

{
  "problem": {
    "name": "Polygon and Points",
    "description": {
      "content": "There is a convex $N$\\-gon $S$ in the two-dimensional coordinate plane where the positive $x$\\-axis points to the right and the positive $y$\\-axis points upward. The vertices of $S$ have the coordinat",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc296_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a convex $N$\\-gon $S$ in the two-dimensional coordinate plane where the positive $x$\\-axis points to the right and the positive $y$\\-axis points upward. The vertices of $S$ have the coordinat...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments