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 $y=-x^2+6$?
The graph of the two equations is shown below:

[asy]
Label f;

f.p=fontsize(4);

xaxis(-3,3,Ticks(f, 2.0));

yaxis(-1,7,Ticks(f, 2.0));

real f(real x)

{

return abs(x);

}

draw(graph(f,-3,3), linewidth(1));
real g(real x)

{

return -x^2+6;

}

draw(graph(g,-2.5,2.5), linewidth(1));
[/asy]

We first find the $x$ values at which the two equations intersect. When $x\ge 0$, $y=|x|=x$. Plugging this into the second equation to eliminate $y$, we get $x=-x^2+6\Rightarrow x^2+x-6=0$. Factoring the left hand side gives $(x+3)(x-2)=0$, so $x=2$ (since we stated the $x$ was non-negative). By symmetry, the $x$ value of the left intersection is $x=-2$. So we just have to consider the integer $x$ values between these two bounds and find all integer $y$ values that make the point $(x,y)$ fall inside the region.

For $x=-2$, there is 1 point that works: $(-2,2)$. For $x=-1$, the value of $y=|x|$ is $y=1$ and the value of $y=-x^2+6$ is $y=5$, so all $y$ values between 1 and 5 inclusive work, for a total of 5 points. For $x=0$, the value of $y=|x|$ is $y=0$ and the value of $y=-x^2+6$ is $y=6$, so all $y$ values between 0 and 6 inclusive work, for a total of 7 points. By symmetry, when $x=1$, there are 5 points that work, and when $x=2$, there is 1 point that works.

In total, there are $1+5+7+5+1=\boxed{19}$ lattice points in the region or on the boundary.
19
Level 5
Algebra
Problem: 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 $y=-x^2+6$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the number of lattice points within the region bounded by the equations, we first need to understand the shapes represented by these equations. The equation $y=|x|$ describes a V-shaped graph that opens upwards, with its vertex at the origin (0,0). The lines intersect the y-axis at (0,0) and extend outwards, forming a boundary for the region.

The equation $y=-x^2+6$ describes a downward-opening parabola with its vertex at (0,6). This parabola intersects the y-axis at (0,6) and the x-axis at the points where $-x^2 + 6 = 0$, which simplifies to $x^2 = 6$. Thus, the x-intercepts are at $x = \sqrt{6}$ and $x = -\sqrt{6}$, approximately at $x \approx 2.45$ and $x \approx -2.45$.

Next, we determine the points of intersection between the two curves. Setting $|x| = -x^2 + 6$, we consider two cases: $y = x$ and $y = -x$. 

For $y = x$:
1. Set $x = -x^2 + 6$.
2. Rearranging gives $x^2 + x - 6 = 0$.
3. Factoring yields $(x - 2)(x + 3) = 0$, giving $x = 2$ and $x = -3$.

For $y = -x$:
1. Set $-x = -x^2 + 6$.
2. Rearranging gives $x^2 - x - 6 = 0$.
3. Factoring yields $(x - 3)(x + 2) = 0$, giving $x = 3$ and $x = -2$.

The intersection points are thus (2,2), (-3,3), (3,-3), and (-2,2). 

Next, we analyze the region bounded by these curves. The vertices of the region are at the intersection points, and we need to find all integer coordinate points (lattice points) that lie within or on the boundary of this region.

We check integer values of $x$ from -3 to 3:
- For $x = -3$: $y = 3$ (point is on the boundary).
- For $x = -2$: $y = 2$ (point is on the boundary).
- For $x = -1$: $y = 1$ (point is on the boundary).
- For $x = 0$: $y = 0$ (point is on the boundary).
- For $x = 1$: $y = 1$ (point is on the boundary).
- For $x = 2$: $y = 2$ (point is on the boundary).
- For $x = 3$: $y = -3$ (point is on the boundary).

Next, we check the integer values of $y$ for each integer $x$:
- For $x = -3$, $y$ can be 0, 1, 2, 3.
- For $x = -2$, $y$ can be 0, 1, 2.
- For $x = -1$, $y$ can be 0, 1.
- For $x = 0$, $y$ can be 0.
- For $x = 1$, $y$ can be 0, 1.
- For $x = 2$, $y$ can be 0, 1, 2.
- For $x = 3$, $y$ can be -3, -2, -1, 0.

Counting all the unique lattice points gives:
- From $x = -3$: 4 points.
- From $x = -2$: 3 points.
- From $x = -1$: 2 points.
- From $x = 0$: 1 point.
- From $x = 1$: 2 points.
- From $x = 2$: 3 points.
- From $x = 3$: 4 points.

Adding these counts results in a total of 19 unique lattice points within the region bounded by the two curves. Thus, the final answer is 19.<|END_OF_LATENT|> The final answer is $19$.