Let's solve each part of the problem step by step:

1. Calculate the tax (6% of $52):

Tax = $6\% \times 52 = \frac{6}{100} \times 52 = 3.12$

Tax = $3.12

2. Calculate the tip based on the pre-tax amount (25% of $52):

Tip = $25\% \times 52 = \frac{25}{100} \times 52 = 13.00$

Tip = $13.00

3. What is the total amount the five mathematicians will pay?

Total Amount = Pre-tax bill + Tax + Tip
Total Amount = $52 + 3.12 + 13.00 = 68.12$

Total Amount = $68.12

4. How much will each person pay if they divide the bill equally?

Amount per person = Total Amount ÷ 5
Amount per person = $68.12 ÷ 5 = 13.624$
Rounded to two decimal places: $13.62$

Each person pays $13.62

If you'd like a breakdown of any of these steps, let me know! Here are some related questions to deepen understanding:

1. How is a percentage calculated in general?
2. What would the total amount be if the tax rate were 10% instead of 6%?
3. If they left a 20% tip instead of 25%, how much would the tip be?
4. How would the amount per person change if there were 6 mathematicians instead of 5?
5. What happens if a service fee of 5% is added to the total amount?

Tip: Always calculate percentages on the original value unless instructed otherwise!