Olympiad problems

2011 Kurschak Competition, 1

Let $a_1, a_2,...$ be an infinite sequence of positive integers such that for any $k,\ell\in \mathbb{Z_+}$, $a_{k+\ell}$ is divisible by $\gcd(a_k,a_\ell)$. Prove that for any integers $1\leqslant k\leqslant n$, $a_na_{n-1}\dots a_{n-k+1}$ is divisible by $a_ka_{k-1}\dots a_1$.

2024 Baltic Way, 16

Tags: perfect power , number theory , divisor

Determine all composite positive integers $n$ such that, for each positive divisor $d$ of $n$, there are integers $k\geq 0$ and $m\geq 2$ such that $d=k^m+1$.

2016 Japan MO Preliminary, 2

Tags: number theory

For $1\leq n\leq 2016$, how many integers $n$ satisfying the condition: the reminder divided by $20$ is smaller than the one divided by $16$.

2022 Iberoamerican, 2

Tags: number theory

Let $S=\{13, 133, \cdots\}$ be the set of the positive integers of the form $133 \cdots 3$. Consider a horizontal row of $2022$ cells. Ana and Borja play a game: they alternatively write a digit on the leftmost empty cell, starting with Ana. When the row is filled, the digits are read from left to right to obtain a $2022$-digit number $N$. Borja wins if $N$ is divisible by a number in $S$, otherwise Ana wins. Find which player has a winning strategy and describe it.

2020 APMO, 3

Tags: combinatorics , number theory

Determine all positive integers $k$ for which there exist a positive integer $m$ and a set $S$ of positive integers such that any integer $n > m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways.

1973 Bulgaria National Olympiad, Problem 3

Tags: number theory , php

Let $a_1,a_2,\ldots,a_n$ are different integer numbers in the range $[100,200]$ for which: $a_1+a_2+\ldots+a_n\ge11100$. Prove that it can be found at least number from the given in the representation of decimal system on which there are at least two equal digits. [i]L. Davidov[/i]

2025 Turkey Team Selection Test, 9

Tags: number theory , number theory with sequences , modular arithmetic , modular congruences

Let $n$ be a positive integer. For every positive integer $1 \leq k \leq n$ the sequence ${\displaystyle {\{ a_{i}+ki\}}_{i=1}^{n }}$ is defined, where $a_1,a_2, \dots ,a_n$ are integers. Among these $n$ sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by $n$?

1992 IMO Longlists, 63

Tags: modular arithmetic , quadratic , number theory , divisibility

Let $a$ and $b$ be integers. Prove that $\frac{2a^2-1}{b^2+2}$ is not an integer.

2009 All-Russian Olympiad, 8

Tags: number theory

Let $ x$, $ y$ be two integers with $ 2\le x, y\le 100$. Prove that $ x^{2^n} \plus{} y^{2^n}$ is not a prime for some positive integer $ n$.

2011 IFYM, Sozopol, 1

Tags: number theory , irrational number

Prove that for $\forall n>1$, $n\in \mathbb{N}$ , there exist infinitely many pairs of positive irrational numbers $a$ and $b$, such that $a^n=b$.

2006 AIME Problems, 4

Tags: factorial , number theory , prime factorization

Let $N$ be the number of consecutive 0's at the right end of the decimal representation of the product $1!\times2!\times3!\times4!\cdots99!\times100!.$ Find the remainder when $N$ is divided by 1000.

2020 Bundeswettbewerb Mathematik, 4

Tags: prime factorization , number theory

Define a sequence $(a_n)$ recursively by $a_1=0, a_2=2, a_3=3$ and $a_n=\max_{0<d<n} a_d \cdot a_{n-d}$ for $n \ge 4$. Determine the prime factorization of $a_{19702020}$.

1995 Tournament Of Towns, (441) 1

Tags: algebra , number theory

Sonia has $10$, $15$ and $20$ cent stamps with total face value of $\$5$. She has $30$ stamps altogether. Prove that she has more $20$ cent stamps than $10$ cent stamps.

2020 BMT Fall, 2

Tags: combinatorics , number theory

Haydn picks two different integers between $1$ and $100$, inclusive, uniformly at random. The probability that their product is divisible by $4$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

2018 BMT Spring, 10

Tags: number theory

Evaluate the following $$\prod^{50}_{j=1} \left( 2 cos \left( \frac{4\pi j}{101} \right) + 1\right).$$

2021 Irish Math Olympiad, 1

Tags: number theory , factorial

Let $N = 15! = 15\cdot 14\cdot 13 ... 3\cdot 2\cdot 1$. Prove that $N$ can be written as a product of nine different integers all between $16$ and $30$ inclusive.

2010 IMO Shortlist, 6

Tags: combinatorics , matrix , number theory

The rows and columns of a $2^n \times 2^n$ table are numbered from $0$ to $2^{n}-1.$ The cells of the table have been coloured with the following property being satisfied: for each $0 \leq i,j \leq 2^n - 1,$ the $j$-th cell in the $i$-th row and the $(i+j)$-th cell in the $j$-th row have the same colour. (The indices of the cells in a row are considered modulo $2^n$.) Prove that the maximal possible number of colours is $2^n$. [i]Proposed by Hossein Dabirian, Sepehr Ghazi-nezami, Iran[/i]

1987 Tournament Of Towns, (159) 3

Tags: number theory , divisible , divide

Prove that there are infinitely many pairs of natural numbers $a$ and $b$ such that $a^2 + 1$ is divisible by $b$ and $b^2 + 1$ is divisible by $a$ .

2024 Australian Mathematical Olympiad, P8

Tags: irrational number , number theory

Let $r=0.d_0d_1d_2\ldots$ be a real number. Let $e_n$ denote the number formed by the digits $d_n, d_{n-1}, \ldots, d_0$ written from left to right (leading zeroes are permitted). Given that $d_0=6$ and for each $n \geq 0$, $e_n$ is equal to the number formed by the $n+1$ rightmost digits of $e_n^2$. Show that $r$ is irrational.

2010 Slovenia National Olympiad, 1

Tags: prime number , number theory

Find all prime numbers $p, q$ and $r$ such that $p>q>r$ and the numbers $p-q, p-r$ and $q-r$ are also prime.

1999 Baltic Way, 16

Tags: modular arithmetic , number theory

Find the smallest positive integer $k$ which is representable in the form $k=19^n-5^m$ for some positive integers $m$ and $n$.

2004 Switzerland Team Selection Test, 2

Tags: perfect square , number theory

Find the largest natural number $n$ for which $4^{995} +4^{1500} +4^n$ is a square.

2021 Flanders Math Olympiad, 1

Tags: number theory , combinatorics

Johnny once saw plums hanging, like eggs so big and numbered according to the first natural numbers. He is the first to pick the plum with number $2$. After that, Jantje picks the plum each time with the smallest number $n$ that satisfies the following two conditions: $\bullet$ $n$ is greater than all numbers on the already picked plums, $\bullet$ $n$ is not the product of two equal or different numbers on already picked plums. We call the numbers on the picked plums plum numbers. Is $100 000$ a plum number? Justify your answer.

2007 Pre-Preparation Course Examination, 18

Tags: greatest common divisor , number theory

Prove that the equation $x^3+y^3+z^3=t^4$ has infinitely many solutions in positive integers such that $\gcd(x,y,z,t)=1$. [i]Mihai Pitticari & Sorin Rǎdulescu[/i]

2020-IMOC, N2

Tags: diophantine equation , number theory

Find all positive integers $N$ such that the following holds: There exist pairwise coprime positive integers $a,b,c$ with $$\frac1a+\frac1b+\frac1c=\frac N{a+b+c}.$$