TopsOJ Summer 2025 - Guts Round - Problem 25

Problem 25: Let N_{27} be the answer to problem 27, and n = N_{27}+7. How many positive integers b are there such that n < b < n^2, and when n^2 is written in base b as \overline{d_1d_2}, d_1 > d_2?

Problem 26: Let N_{25} be the answer to problem 25. How many pairs of integers (a,b) are there such that a \ge 1, 0 \le b < 10^{N_{25}}, and a \cdot 10^{N_{25}} + b = (a+b)^2?

Problem 27: Let N_{26} be the answer to problem 26. There are N_{26} mathematicians in a circle, numbered 1 to N_{26} in clockwise order. Each person can be admitting or non-admitting, and everyone starts off non-admitting. Turns move clockwise around the circle among currently non-admitting people, starting with person 1 and cycling back to the front while more than one person doesn't admit. On each person's turn, they must make somebody admit. However, each person also wants to make the highest numbered person admit. They will prioritize not admitting for as long as possible, and their secondary priority is to make the highest numbered person admit. Assuming optimal play from everyone, which person doesn't admit?

Find (N_{25}+N_{26}+N_{27})^2.