Olympiad problems

2018 India PRMO, 4

The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation.

2022 Vietnam TST, 5

Tags: combinatorics , base system

A fractional number $x$ is called [i][b]pretty[/b][/i] if it has finite expression in base$-b$ numeral system, $b$ is a positive integer in $[2;2022]$. Prove that there exists finite positive integers $n\geq 4$ that with every $m$ in $(\frac{2n}{3}; n)$ then there is at least one pretty number between $\frac{m}{n-m}$ and $\frac{n-m}{m}$

2025 India National Olympiad, P6

Tags: quadratic residue , base system , construction

Let $b \geqslant 2$ be a positive integer. Anu has an infinite collection of notes with exactly $b-1$ copies of a note worth $b^k-1$ rupees, for every integer $k\geqslant 1$. A positive integer $n$ is called payable if Anu can pay exactly $n^2+1$ rupees by using some collection of her notes. Prove that if there is a payable number, there are infinitely many payable numbers. [i]Proposed by Shantanu Nene[/i]