gnistory

1. Understand the Reflection

General Form: A reflection in the complex plane can be represented by the function: h(z) = α * conj(z) + β where: * z is the complex number being reflected

* conj(z) is the complex conjugate of z * α is a complex number that determines the scaling and rotation * β is a complex number that determines the translation

Key Insight: The line of reflection is perpendicular to the vector represented by α.

2. Determine the Vector α

α = 5/13 + 12/13 * i

3. Find the Line of Reflection

Perpendicular Vector: The vector perpendicular to α is found by rotating α by 90 degrees counterclockwise.

We can achieve this by multiplying α by i: i * α = i * (5/13 + 12/13 * i) = -12/13 + 5/13 * i

Line Equation: The line of reflection passes through the origin and is parallel to the vector -12/13 + 5/13 * i.

Equation: y = (5/12) * x or 5x - 12y = 0

Therefore, the line of reflection for the function h(z) = α*conj(z) + β is 5x - 12y = 0.

Note: The β term (2 - 3i) in the function h(z) translates the reflection, but it does not change the line of reflection itself. The line of reflection remains the same regardless of the translation.

This is a fun thought experiment! Here's a breakdown of the calculations involved:

1. Number of Possible Arrangements

The number of ways to arrange 26 letters is given by the factorial of 26 (written as 26!).

26! is a huge number: approximately 4.03 x 10^26

2. Probability of Success

Assuming each arrangement is equally likely, the probability of getting A-Z in a single throw is 1 / 26!

3. Expected Number of Throws

The expected number of throws to achieve a specific outcome in a series of independent trials is 1 / probability of success.

Therefore, the expected number of throws to get A-Z is 26!

4. Time Calculation

Time per throw: 1.5 seconds

Total time: Expected number of throws * Time per throw

Total time in years: Total time in seconds / (60 seconds/minute * 60 minutes/hour * 24 hours/day * 365 days/year)

5. Result

The expected time to get A-Z is incredibly long.

It would take on the order of 10^26 years.

There are 4 ordered pairs (a,b) that work.

The roots of x^2 + bx + c are 17 and -25.

The value of n is 68.

The solution is p = -7, q = 13, r = 10.

The possible values of a are -15, -10, 10, and 15.

You can use modular arithmetic.

Considering the number 100, the tens digit of 7^1993 is 4.

By the Law of Cosines, a_1 = 6 - 2*sqrt(2) and a_2 = 6 + 2*sqrt(2).

Solving the system, we get r = -8 and s = 9.

The answer is -41.

There are 38 primes from 200 to 400.

The length of AB is 10*sqrt(3).