F. Symmetric Projections

Codeforces

IDCF886F

Time2000ms

Memory256MB

Difficulty—

geometry

English · Original

Chinese · Translation

You are given a set of _n_ points on the plane. A line containing the origin is called good, if projection of the given set to this line forms a symmetric multiset of points. Find the total number of good lines. Multiset is a set where equal elements are allowed. Multiset is called symmetric, if there is a point _P_ on the plane such that the multiset is [centrally symmetric](https://en.wikipedia.org/wiki/Point_reflection) in respect of point _P_. ## Input The first line contains a single integer _n_ (1 ≤ _n_ ≤ 2000) — the number of points in the set. Each of the next _n_ lines contains two integers _x__i_ and _y__i_ ( - 106 ≤ _x__i_, _y__i_ ≤ 106) — the coordinates of the points. It is guaranteed that no two points coincide. ## Output If there are infinitely many good lines, print _\-1_. Otherwise, print single integer — the number of good lines. [samples] ## Note Picture to the first sample test: ![image](https://espresso.codeforces.com/eedc60313be8684bd6169b8b23f0f0afd92479a8.png) In the second sample, any line containing the origin is good.

给定平面上的一组 #cf_span[n] 个点。一条经过原点的直线被称为“好线”，如果将给定点集投影到这条直线上后，形成的点的多重集是对称的。求好线的总数。多重集是指允许重复元素的集合。一个多重集被称为对称的，如果存在平面上的一个点 #cf_span[P]，使得该多重集关于点 #cf_span[P] 中心对称。第一行包含一个整数 #cf_span[n] (#cf_span[1 ≤ n ≤ 2000]) —— 点集中的点数。接下来的 #cf_span[n] 行，每行包含两个整数 #cf_span[xi] 和 #cf_span[yi] (#cf_span[ - 106 ≤ xi, yi ≤ 106]) —— 点的坐标。保证没有两个点重合。如果存在无穷多条好线，请输出 _-1_。否则，输出一个整数 —— 好线的数量。第一个样例的图示：在第二个样例中，任意一条经过原点的直线都是好线。 ## Input 第一行包含一个整数 #cf_span[n] (#cf_span[1 ≤ n ≤ 2000]) —— 点集中的点数。接下来的 #cf_span[n] 行，每行包含两个整数 #cf_span[xi] 和 #cf_span[yi] (#cf_span[ - 106 ≤ xi, yi ≤ 106]) —— 点的坐标。保证没有两个点重合。 ## Output 如果存在无穷多条好线，请输出 _-1_。否则，输出一个整数 —— 好线的数量。 [samples] ## Note 第一个样例的图示：在第二个样例中，任意一条经过原点的直线都是好线。

Samples

Input #1

Output #1

Input #2

2
4 3
1 2

Output #2

\-1

API Response (JSON)

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  "problem": {
    "name": "F. Symmetric Projections",
    "description": {
      "content": "You are given a set of _n_ points on the plane. A line containing the origin is called good, if projection of the given set to this line forms a symmetric multiset of points. Find the total number of ",
      "description_type": "Markdown"
    },
    "platform": "Codeforces",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "CF886F"
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  "statements": [
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      "statement_type": "Markdown",
      "content": "You are given a set of _n_ points on the plane. A line containing the origin is called good, if projection of the given set to this line forms a symmetric multiset of points. Find the total number of ...",
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      "statement_type": "Markdown",
      "content": "给定平面上的一组 #cf_span[n] 个点。一条经过原点的直线被称为“好线”，如果将给定点集投影到这条直线上后，形成的点的多重集是对称的。求好线的总数。\n\n多重集是指允许重复元素的集合。\n\n一个多重集被称为对称的，如果存在平面上的一个点 #cf_span[P]，使得该多重集关于点 #cf_span[P] 中心对称。\n\n第一行包含一个整数 #cf_span[n] (#cf_span[1 ≤ n ...",
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