Score : $500$ points

问题陈述

在平面的第一象限中绘制了 $N$ 个数字7。

第 $i$ 个7是一个图形，由连接点 $(x_i-1,y_i)$ 和 $(x_i,y_i)$ 的线段以及连接点 $(x_i,y_i-1)$ 和 $(x_i,y_i)$ 的线段组成。

你可以选择从这 $N$ 个7中删除零个或多个。

在最优删除后，从原点完全可见的7的最大数量是多少？

这里，第 $i$ 个7从原点完全可见当且仅当：

• 以原点、点 $(x_i-1,y_i)$、点 $(x_i,y_i)$ 和点 $(x_i,y_i-1)$ 为顶点构成的四边形（不包括边界）不与其他7相交。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ 7's drawn in the first quadrant of a plane.

The $i$-th 7 is a figure composed of a segment connecting $(x_i-1,y_i)$ and $(x_i,y_i)$, and a segment connecting $(x_i,y_i-1)$ and $(x_i,y_i)$.

You can choose zero or more from the $N$ 7's and delete them.

What is the maximum possible number of 7's that are wholly visible from the origin after the optimal deletion?

Here, the $i$-th 7 is wholly visible from the origin if and only if:

• the interior (excluding borders) of the quadrilateral whose vertices are the origin, $(x_i-1,y_i)$, $(x_i,y_i)$, $(x_i,y_i-1)$ does not intersect with the other 7's.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq x_i,y_i \leq 10^9$
• $(x_i,y_i) \neq (x_j,y_j)\ (i \neq j)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$x_1$ $y_1$

$x_2$ $y_2$

$\hspace{0.45cm}\vdots$

$x_N$ $y_N$

Output

Print the maximum possible number of 7's that are wholly visible from the origin.

Sample Input 1

3
1 1
2 1
1 2

Sample Output 1

2

If the first 7 is deleted, the other two 7's ― the second and third ones ― will be wholly visible from the origin, which is optimal.

If no 7's are deleted, only the first 7 is wholly visible from the origin.

Sample Input 2

10
414598724 87552841
252911401 309688555
623249116 421714323
605059493 227199170
410455266 373748111
861647548 916369023
527772558 682124751
356101507 249887028
292258775 110762985
850583108 796044319

Sample Output 2

10

It is best to keep all 7's.

update @ 2024/3/10 09:53:39