Back From Summer '19 P4: Wesley And Cake

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Points: 25 (partial)
Time limit: 1.0s
Memory limit: 64M

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Problem type
A cake fit for a king ~wleung_bvg.

Everybody knows that cake comes in two shapes, circular or rectangular.
Everybody except for Wesley.

In his defence:

You wouldn't buy a rectangular shaped pizza would you?

Thus Wesley has only ever cut his cake in one way.

  • Imagine a cartesian grid over the center of the cake labelled at (0,0).
  • Make M cuts over the line formed by y=m_i x within the cake's boundaries.

With Wesley's birthday coming up, his friends have decided to play a little prank on him. They have purchased a square shaped cake with side length 2N and would like to know the side length of the largest axis-aligned square obtainable that does not intersect with any cut. A square intersects a cut if there is a non-zero area of the square on both sides of the cut. This means that touching a cut does not count as an intersection.

Slope will be given as a_i, b_i where m_i = \large \frac{a_i}{b_i}

Input Specifications

The first line will contain two integers, N and M (1 \le N \le 10^3, 1 \le M \le 10^5), half the length of the cake, and the number of cuts Wesley makes.

The next M lines will each contain two integers, a_i, b_i\ (1 \le |a_i|, b_i \le 10^3), the numerator and denominator of the slope of the line used to determine the cut. a_i and b_i are guaranteed to be coprime, and the pair (a_i, b_i) is guaranteed to be unique.

Output Specifications

Output two positive integers, n and d space-separated on one line. This means the side length of the largest obtainable axis-aligned square is \large \frac{n}{d}. Note that n and d must be coprime.

Subtasks

Subtask 1 [30%]

M \le 2

Subtask 2 [70%]

No further constraints.

Sample Input 1

1 2
-1 1
1 1

Sample Output 1

2 3

Explanation For Sample 1

The following image shows the cake:

The red square is the cake, and the blue and green lines shows the two cuts. The black line shows one of the many different largest squares obtainable.

Sample Input 2

2 2
-1 2
1 1

Sample Output 2

3 2

Sample Input 3

1000 2
-1000 1
1000 1

Sample Output 3

2000000 2001

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