Olympiad problems

2017 Bulgaria National Olympiad, 2

Let $m>1$ be a natural number and $N=m^{2017}+1$. On a blackboard, left to right, are written the following numbers: \[N, N-m, N-2m,\dots, 2m+1,m+1, 1.\] On each move, we erase the most left number, written on the board, and all its divisors (if any). This procces continues till all numbers are deleted. Which numbers will be deleted on the last move.

2024 Belarus - Iran Friendly Competition, 2.1

Tags: number theory

Prove that the equation $2+x^3y+y^2+z^2=0$ has no solutions in integers.

1995 Mexico National Olympiad, 1

Tags: array , combinatorics , number theory

$N$ students are seated at desks in an $m \times n$ array, where $m, n \ge 3$. Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are $1020 $handshakes, what is $N$?

2001 All-Russian Olympiad Regional Round, 8.2

Tags: number theory , algebra

$N$ numbers - ones and twos - are arranged in a circle. We mean a number formed by several digits arranged in a row (clockwise or counterclockwise). For what is the smallest value of $N$, all four-digit numbers whose writing contains only numbers $1$ and $2$, could they be among those shown?

2013 Estonia Team Selection Test, 1

Tags: diophantine equation , diophantine , system of equations , number theory

Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$

1979 IMO Longlists, 23

Tags: modular arithmetic , number theory

Consider the set $E$ consisting of pairs of integers $(a, b)$, with $a \geq 1$ and $b \geq 1$, that satisfy in the decimal system the following properties: [b](i)[/b] $b$ is written with three digits, as $\overline{\alpha_2\alpha_1\alpha_0}$, $\alpha_2 \neq 0$; [b](ii)[/b] $a$ is written as $\overline{\beta_p \ldots \beta_1\beta_0}$ for some $p$; [b](iii)[/b] $(a + b)^2$ is written as $\overline{\beta_p\ldots \beta_1 \beta_0 \alpha_2 \alpha_1 \alpha_0}.$ Find the elements of $E$.

2014 Contests, 3

Tags: number theory

Let $a_1 \leq a_2 \leq \cdots$ be a non-decreasing sequence of positive integers. A positive integer $n$ is called [i]good[/i] if there is an index $i$ such that $n=\dfrac{i}{a_i}$. Prove that if $2013$ is [i]good[/i], then so is $20$.

2002 Turkey MO (2nd round), 1

Tags: quadratic , prime number , modular arithmetic , number theory

Find all prime numbers $p$ for which the number of ordered pairs of integers $(x, y)$ with $0\leq x, y < p$ satisfying the condition \[y^2 \equiv x^3 - x \pmod p\] is exactly $p.$

2005 Austria Beginners' Competition, 2

Tags: inequalities , diophantine , number theory

Determine the number of integer pairs $(x, y)$ such that $(|x| - 2)^2 + (|y| - 2)^2 < 5$ .

2009 Stanford Mathematics Tournament, 13

Tags: number theory

A number $N$ has $2009$ positive factors. What is the maximum number of positive factors that $N^2$ could have?

2019 Abels Math Contest (Norwegian MO) Final, 2

Tags: number theory

$find$ all non negative integers $m$, $n$ such that $mn-1$ divides $n^3-1$

2023 SAFEST Olympiad, 1

Tags: number theory

Find all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(1) \neq f(-1)$ and $$f(m+n)^2 \mid f(m)-f(n)$$ for all integers $m, n$. [i]Proposed by Liam Baker, South Africa[/i]

2024/2025 TOURNAMENT OF TOWNS, P1

Tags: number theory

Find the minimum positive integer such that some four of its natural divisors sum up to $2025$.

2011 Estonia Team Selection Test, 2

Tags: number theory

Let $n$ be a positive integer. Prove that for each factor $m$ of the number $1+2+\cdots+n$ such that $m\ge n$, the set $\{1,2,\ldots,n\}$ can be partitioned into disjoint subsets, the sum of the elements of each being equal to $m$. [b]Edit[/b]:Typographical error fixed.

2024 JHMT HS, 3

Tags: number theory

Let $N_2$ be the answer to problem 2. On a number line, Tanya circles the first $\ell$ positive integers. Then, starting with the greatest number in the most recent circle, she circles the next $\ell$ positive integers, so that the two circles have exactly one number in common; she repeats this until $N_2$ is in a circle. Compute the sum of all possible values of $\ell$ for which $N_2$ is the greatest number in a circle.

1997 Tuymaada Olympiad, 6

Tags: divisor , consecutive , number theory

Are there $14$ consecutive positive integers, each of which has a divisor other than $1$ and not exceeding $11$?

2012 NIMO Problems, 10

Tags: expected value , probability , relatively prime , number theory

A [i]triangulation[/i] of a polygon is a subdivision of the polygon into triangles meeting edge to edge, with the property that the set of triangle vertices coincides with the set of vertices of the polygon. Adam randomly selects a triangulation of a regular $180$-gon. Then, Bob selects one of the $178$ triangles in this triangulation. The expected number of $1^\circ$ angles in this triangle can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$. [i]Proposed by Lewis Chen[/i]

2012 Romania Team Selection Test, 1

Tags: quadratic , number theory , algebra , polynomial

Find all triples $(a,b,c)$ of positive integers with the following property: for every prime $p$, if $n$ is a quadratic residue $\mod p$, then $an^2+bn+c$ is a quadratic residue $\mod p$.

2000 Junior Balkan Team Selection Tests - Romania, 2

Tags: modular arithmetic , algebra , number theory

Let be a natural power of two. Find the number of numbers equivalent with $ 1 $ modulo $ 3 $ that divide it. [i]Dan Brânzei[/i]

2008 Indonesia TST, 4

Tags: number theory , greatest common divisor

Let $ a $ and $ b $ be natural numbers with property $ gcd(a,b)=1 $ . Find the least natural number $ k $ such that for every natural number $ r \ge k $ , there exist natural numbers $ m,n >1 $ in such a way that the number $ m^a n^b $ has exactly $ r+1 $ positive divisors.

2020 Tournament Of Towns, 2

Tags: lcm , least common multiple , number theory

Alice had picked positive integers $a, b, c$ and then tried to find positive integers $x, y, z$ such that $a = lcm (x, y)$, $b = lcm(x, z)$, $c = lcm(y, z)$. It so happened that such $x, y, z$ existed and were unique. Alice told this fact to Bob and also told him the numbers $a$ and $b$. Prove that Bob can find $c$. (Note: lcm = least common multiple.) Boris Frenkin

2021 Kyiv City MO Round 1, 8.1

Tags: number theory , fraction , algebra

Find all positive integers $n$ that can be subtracted from both the numerator and denominator of the fraction $\frac{1234}{6789}$, to get, after the reduction, the fraction of form $\frac{a}{b}$, where $a, b$ are single digit numbers. [i]Proposed by Bogdan Rublov[/i]

2011 Princeton University Math Competition, A3 / B5

Tags: number theory

What is the sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43?

2010 Germany Team Selection Test, 3

Tags: size , polynomial , number theory

A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

2012 Tournament of Towns, 4

Tags: prime , divisor , number theory

Let $C(n)$ be the number of prime divisors of a positive integer $n$. (a) Consider set $S$ of all pairs of positive integers $(a, b)$ such that $a \ne b$ and $C(a + b) = C(a) + C(b)$. Is $S$ finite or infinite? (b) Define $S'$ as a subset of S consisting of the pairs $(a, b)$ such that $C(a+b) > 1000$. Is $S'$ finite or infinite?