Olympiad problems

1968 Bulgaria National Olympiad, Problem 3

Prove that a binomial coefficient $\binom nk$ is odd if and only if all digits $1$ of $k$, when $k$ is written in binary, are on the same positions when $n$ is written in binary. [i]I. Dimovski[/i]

2009 VJIMC, Problem 1

Tags: bases , number theory

A positive integer $m$ is called self-descriptive in base $b$, where $b\ge2$ is an integer, if i) The representation of $m$ in base $b$ is of the form $(a_0a_1\ldots a_{b-1})_b$ (that is $m=a_0b^{b-1}+a_1b^{b-2}+\ldots+a_{b-2}b+a_{b-1}$, where $0\le a_i\le b-1$ are integers). ii) $a_i$ is equal to the number of occurences of the number $i$ in the sequence $(a_0a_1\ldots a_{b-1})$. For example, $(1210)_4$ is self-descriptive in base $4$, because it has four digits and contains one $0$, two $1$s, one $2$ and no $3$s.

1979 Bundeswettbewerb Mathematik, 3

Tags: other bases , consecutive , bases , number theory

In base $10$ there exist two-digit natural numbers that can be factorized into two natural factors such that the two digits and the two factors form a sequence of four consecutive integers (for example, $12 = 3 \cdot 4$). Determine all such numbers in all bases.

2000 Slovenia National Olympiad, Problem 1

Tags: number theory , bases

Find all prime numbers whose base $b$ representations (for some $b$) contain each of the digits $0,1,\ldots,b-1$ exactly once. (Digit $0$ may appear as the first digit.)

2015 AoPS Mathematical Olympiad, 6

Tags: number theory , bases

Find the five-hundredth-smallest positive integer that can be written using only the digits $1$, $3,$ and $5$ in base $7$? [i]Proposed by CaptainFlint[/i]

2024 AMC 12/AHSME, 11

Tags: bases

There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base-$b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$? $\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }21$

2024 AMC 10, 18

Tags: bases

There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base-$b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$? $\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }21$

1998 Slovenia National Olympiad, Problem 1

Tags: bases , number theory

Let $n$ be a positive integer. If the number $1998$ is written in base $n$, a three-digit number with the sum of digits equal to $24$ is obtained. Find all possible values of $n$.

2025 AIME, 1

Tags: bases

Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$