Problem 504

[rhirsh]

I'd like some clarification on the terminology "strictly contain" lattice points for Problem 504 (View Problem). Assuming the a=b=c=d=1, then would that quadrilateral "strictly contain" only 1 point (i.e. (0,0)), or would it have 5 (i.e. (1,0),(0,1),(-1,0),(0,-1),(0,0))? Basically I am not sure if points exactly "on the line" should be counted or not, and I cant find a good definition of what "strictly contain" means. Thanks!

[Marcus_Andrews]

Strict containment = lattice points on the lines of the quadrilateral are not counted.

[redmilk525]

I got correct answer when m=4 but got wrong when m=100.
Could somebody can help me verify that if ans=656 when m=10.
Thanks.

[nicolas.patrois]

Wrong.

[MYNick]

I get the right answer for m=4 too, but for m=100 not.
For m=10, is the answer 862?

[nicolas.patrois]

Yes.

[pj6444]

Hmmm, I'm getting the right value for m = 10 but not for m = 100.

[wozzo]

A(a, 0), B(0, b), C(−c, 0), D(0, −d), where 1 ≤ a, b, c, d ≤ m and a, b, c, d, m are integers.

Since the above line describes point C as being (-c, 0), where c is an integer, are we also to assume c is a positive integer? That is that point C can only be placed on the negative portion of the x axis? If so why not describe a,b,c,d, & m as natural numbers?

[hk]

Are there any non-positive numbers a,b,c,d for which 1 ≤ a, b, c, d ≤ m ?

[Ellenion]

I had the same question . Maybe the admins could edit the problem and define what "strictly contains" means on the problem?

[Animus]

A polygon strictly contains any point that is located within the polygon and not on its boundary.

So neither count the vertices of the rectangle nor the lattice points located on the sides of the rectangle.

[enigmaticcam]

Just to confirm, if a=b=c=d=2, then because there are four points that lie exactly on the diagonal edges of the quadrilateral, they would not be counted and so this would strictly contain only 5 lattice points?

[DJohn]

Yes. A point is not strictly inside if it's on an edge.

[Bezerkely]

Hmmm, I'm getting the right value for m = 10 but not for m = 100.

I was also getting the right value for m = 10 but not for m = 100. I switched my method of calculating the number of interior points from using floating point operations to integer operations only and got the right answer for m = 100.