A lattice point is a point whose coordinates are both integers. How many lattice points are on the boundary or inside the region bounded by $y=|x|$ and $y=-x^2+6$?
I thought there were two points, but that isn't right. What am I missing?
2026-03-28 17:41:13.1774719673
How many lattice points are on the boundary or inside the region bounded by $y=|x|$ and $y=-x^2+6$?
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Remark(I): The two equations intersects at $(\pm 2, 2)$.
Remark(II): The points $(\pm 2, 2); \ (\pm 1, 5); \ (0, 6)$ lies on the boundary of the parabola $y=-x^2+6$.
Remark(III): The points $(\pm 2, 2); \ (\pm 1, 1); \ (0, 0)$ lies on the boundary of $y= | x |$.
Remark(IV): Consider the line $y=1$; we know that the point $(1,5)$ lies on the boundary of the parabola $y=-x^2+6$; also we know that the point $(1,1)$ lies on the boundary of $y= | x |$.
So all the other lattice points , between these two points; which lies on the line $y=1$; are in the interior, i.e. all the points $(1, 2); \ (1, 3); \ (1, 4)$ are in the interior.
Remark(V): Consider the line $y=-1$; we know that the point $(-1,5)$ lies on the boundary of the parabola $y=-x^2+6$; also we know that the point $(-1,1)$ lies on the boundary of $y= | x |$.
So all the other lattice points , between these two points; which lies on the line $y=-1$; are in the interior, i.e. all the points $(-1, 2); \ (-1, 3); \ (-1, 4)$ are in the interior.
Remark(VI): Consider the line $y=0$; we know that the point $(0,6)$ lies on the boundary of the parabola $y=-x^2+6$; also we know that the point $(0,0)$ lies on the boundary of $y= | x |$.
So all the other lattice points , between these two points; which lies on the line $y=0$; are in the interior, i.e. all the points $(0, 1); \ (0, 2); \ (0, 3) \ (0, 4); \ (0, 5); $ are in the interior.
The points on the boundary of are as follows:
$$(\pm 2, 2); \ (\pm 1, 5); \ (0, 6); \\ (\pm 1, 1); \ (0, 0). $$
The interior points are as follows:
$$(\pm 1, 2); \ (\pm 1, 3); \ (\pm 1, 4); \\ (0, 1); \ (0, 2); \ (0, 3); \ (0, 4); \ (0,5). $$
So there are $19$ such points.