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The number of ways is 520.

Here is the answer: The largest possible value of c is 100.

The distance between the circumcenters of Triangle ABC and Triangle ADC is 6*sqrt(3).

To find the length of the tangent from the origin to the circle, we first need to determine the equation of the circle. We can do this by using the three given points to form a system of equations.

Let the equation of the circle be:

(x-a)^2 + (y-b)^2 = r^2

Substituting the coordinates of the three points into this equation, we get a system of three equations:

(3-a)^2 + (4-b)^2 = r^2

(6-a)^2 + (8-b)^2 = r^2

(5-a)^2 + (13-b)^2 = r^2

Solving this system of equations, we find that the center of the circle is at (-2, -4/3) and the radius is 13√(28/9).

Now, to find the length of the tangent from the origin to the circle, we can use the Pythagorean theorem. Let's denote the length of the tangent as t, the distance from the origin to the center of the circle as d, and the radius of the circle as r. We have:

d^2 + t^2 = r^2

We can calculate d using the distance formula:

d = sqrt((-2)^2 + (-4/3)^2) = sqrt(68/9)

Substituting the values of d and r into the Pythagorean theorem equation, we get:

(68/9) + t^2 = (13√(28/9))^2

Solving for t, we find that the length of the tangent is equal to 10.

Therefore, the length of the segment tangent from the origin to the circle is 10.

Let's break down the problem:

Total Possible Outcomes:

We have numbers from 500 to 999, inclusive.

That's a total of 999 - 500 + 1 = 500 numbers.

Favorable Outcomes (Even Numbers with Distinct Digits):

Even Numbers: The last digit must be 0, 2, 4, 6, or 8. That's 5 choices.

Distinct Digits:

For the first digit (hundreds place), we have 5 choices (5, 6, 7, 8, or 9, as we can't use 0).

For the second digit (tens place), we have 8 choices left (since we've used one digit).

So, for each of the 5 choices for the last digit, we have 5 * 8 = 40 choices for the first two digits.

Therefore, the total number of favorable outcomes is 5 (last digit choices) * 40 (first two digit choices) = 200.

Probability:

Probability = Favorable Outcomes / Total Outcomes

Probability = 200 / 500

Probability = 2/5

So, the probability of choosing an even integer between 500 and 999 with distinct digits is 2/5.

The answer is x = 50, y = -47.