aime26-02

Proposed solution

I need to find the region $\mathcal{T}$ where a sphere of radius 42 can be placed on top of the disk at point P and lie completely inside the hemisphere.

Let me set up coordinates with the center of the disk and hemisphere at the origin, with the disk in the xy-plane and the hemisphere above it.

The hemisphere has radius 200, so its equation is $x^2 + y^2 + z^2 = 200^2$ with $z \geq 0$.

Let's say we place a sphere of radius 42 at point P = (a,b,0) on the disk. For this sphere to lie completely inside the hemisphere, I need to ensure that the highest point of the sphere is below or at the surface of the hemisphere.

The center of the placed sphere will be at (a,b,42), and its highest point will be at (a,b,42+42) = (a,b,84).

For this point to be inside the hemisphere:
$a^2 + b^2 + 84^2 \leq 200^2$

This gives us:
$a^2 + b^2 \leq 200^2 - 84^2$
$a^2 + b^2 \leq 40000 - 7056$
$a^2 + b^2 \leq 32944$

So the region $\mathcal{T}$ is the set of points (a,b) such that $a^2 + b^2 \leq 32944$.

The area of $\mathcal{T}$ is:
$\pi \cdot 32944 = \pi \cdot 32944$

The area of the disk is:
$\pi \cdot 200^2 = \pi \cdot 40000$

Therefore, the ratio is:
$\frac{\text{Area of } \mathcal{T}}{\text{Area of disk}} = \frac{32944\pi}{40000\pi} = \frac{32944}{40000}$

Now I need to simplify this fraction by finding the GCD of 32944 and 40000.

Let me find $\gcd(32944, 40000)$.

First, let me factor these numbers:
$32944 = 16 \cdot 2059 = 16 \cdot 7 \cdot 293$
$40000 = 16 \cdot 2500 = 16 \cdot 4 \cdot 625 = 16 \cdot 4 \cdot 5^4 = 64 \cdot 5^4$

Wait, let me be more systematic:
$32944 = 16 \cdot 2059$
$40000 = 16 \cdot 2500$

So $\gcd(32944, 40000) = 16 \cdot \gcd(2059, 2500)$

Let me check if 2059 and 2500 share any factors:
$2059 = 7 \cdot 293$
$2500 = 4 \cdot 625 = 4 \cdot 5^4$

Since 2059 is divisible by 7 and 2500 is not, $\gcd(2059, 2500) = 1$.

Therefore: $\gcd(32944, 40000) = 16$

So: $\frac{32944}{40000} = \frac{2059}{2500}$

Let me verify that 2059 and 2500 are relatively prime:
- $2059 = 7 \cdot 293$
- $2500 = 2^2 \cdot 5^4$

Since they share no common factors, $\gcd(2059, 2500) = 1$.

Therefore, $p = 2059$ and $q = 2500$, so $p + q = 2059 + 2500 = 4559$.

Wait, let me double-check my arithmetic:
$200^2 - 84^2 = 40000 - 7056 = 32944$

This looks correct.

$\boxed{4559}$