Olympiad problems

2009 Germany Team Selection Test, 3

Initially, on a board there a positive integer. If board contains the number $x,$ then we may additionally write the numbers $2x+1$ and $\frac{x}{x+2}.$ At some point 2008 is written on the board. Prove, that this number was there from the beginning.

Russian TST 2021, P1

Tags: number theory

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

2010 Contests, 2

Tags: number theory

Fifteen pairwise coprime positive integers chosen so that each of them less than 2010. Show that at least one of them is prime.

1988 Mexico National Olympiad, 5

Tags: number theory , coprime , greatest common divisor

If $a$ and $b$ are coprime positive integers and $n$ an integer, prove that the greatest common divisor of $a^2+b^2-nab$ and $a+b$ divides $n+2$.

2005 Germany Team Selection Test, 3

Tags: number theory

Let $a$, $b$, $c$, $d$ and $n$ be positive integers such that $7\cdot 4^n = a^2+b^2+c^2+d^2$. Prove that the numbers $a$, $b$, $c$, $d$ are all $\geq 2^{n-1}$.

2003 Italy TST, 1

Tags: modular arithmetic , number theory

Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$

2023 LMT Spring, 6

Tags: number theory

Find the least positive integer $m$ such that $105| 9^{(p^2)} -29^p +m$ for all prime numbers $p > 3$.

Kvant 2023, M2734

Tags: number theory , combinatorics

Real numbers are placed at the vertices of an $n{}$-gon. On each side, we write the sum of the numbers on its endpoints. For which $n{}$ is it possible that the numbers on the sides form a permutation of $1, 2, 3,\ldots , n$? [i]From the folklore[/i]

2024 CIIM, 3

Tags: number theory , euler totient function , prime number , summation

Given a positive integer $n$, let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Find all possible positive integers $k$ for which there exist positive integers $1 \leq a_1 < a_2 < \dots < a_k$ such that: \[ \left\lfloor \frac{\phi(a_1)}{a_1} + \frac{\phi(a_2)}{a_2} + \dots + \frac{\phi(a_k)}{a_k} \right\rfloor = 2024 \]

1991 Tournament Of Towns, (312) 2

Tags: algebra , number theory

$11$ girls and $n$ boys went for mushrooms. They have found $n^2+9n -2$ in total, and each child has found the same quantity. Which is greater: the number of girls or the number of boys? (A. Tolpygo, Kiev)

2021 Saint Petersburg Mathematical Olympiad, 7

Tags: number theory , relatively prime

Kolya found several pairwise relatively prime integers, each of which is less than the square of any other. Prove that the sum of reciprocals of these numbers is less than $2$.

1989 India National Olympiad, 5

Tags: number theory

For positive integers $ n$, define $ A(n)$ to be $ \frac {(2n)!}{(n!)^{2}}$. Determine the sets of positive integers $ n$ for which (a) $ A(n)$ is an even number, (b) $ A(n)$ is a multiple of $ 4$.

2016 Azerbaijan Team Selection Test, 1

Tags: floor function , number theory , sequence

Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.

2010 Contests, 1

Tags: number theory , modular arithmetic

a) Replace each letter in the following sum by a digit from $0$ to $9$, in such a way that the sum is correct. $\tab$ $\tab$ $ABC$ $\tab$ $\tab$ $DEF$ [u]$+GHI$[/u] $\tab$ $\tab$ $\tab$ $J J J$ Different letters must be replaced by different digits, and equal letters must be replaced by equal digits. Numbers $ABC$, $DEF$, $GHI$ and $JJJ$ cannot begin by $0$. b) Determine how many triples of numbers $(ABC,DEF,GHI)$ can be formed under the conditions given in a).

1971 Polish MO Finals, 4

Tags: diophantine , number theory

Prove that if positive integers $x,y,z$ satisfy the equation $$x^n + y^n = z^n,$$ then $\min\, (x,y) \ge n$.

2023 Mexico National Olympiad, 6

Tags: number theory

Find all functions $f: \mathbb{N} \rightarrow \mathbb {N}$ such that for all positive integers $m, n$, $f(m+n)\mid f(m)+f(n)$ and $f(m)f(n) \mid f(mn)$.

1975 Polish MO Finals, 4

Tags: divisible , number theory

All decimal digits of some natural number are $1,3,7$, and $9$. Prove that one can rearrange its digits so as to obtain a number divisible by $7$.

2003 Paraguay Mathematical Olympiad, 1

Tags: product of digits , digit , number theory

How many numbers greater than $1.000$ but less than $10.000$ have as a product of their digits $256$?

2018 IMAR Test, 4

Tags: perfect square , number theory

Prove that every non-negative integer $n$ is expressible in the form $n=t^2+u^2+v^2+w^2$, where $t,u,v,w$ are integers such that $t+u+v+w$ is a perfect square. [i]* * *[/i]

1993 Greece National Olympiad, 6

Tags: least common multiple , number theory

What is the smallest positive integer than can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?

2013 Saudi Arabia Pre-TST, 4.2

Tags: number theory , divide , divisible

Let $x, y$ be two integers. Prove that if $2013$ divides $x^{1433} + y^{1433}$ then $2013$ divides $x^7 + y^7$.

2019 Lusophon Mathematical Olympiad, 1

Tags: number theory , digit , divisible

Find a way to write all the digits of $1$ to $9$ in a sequence and without repetition, so that the numbers determined by any two consecutive digits of the sequence are divisible by $7$ or $13$.

1951 Miklós Schweitzer, 8

Tags: number theory , floor function , least common multiple

Given a positive integer $ n>3$, prove that the least common multiple of the products $ x_1x_2\cdots x_k$ ($ k\geq 1$) whose factors $ x_i$ are positive integers with $ x_1\plus{}x_2\plus{}\cdots\plus{}x_k\le n$, is less than $ n!$.

2017 Balkan MO Shortlist, N2

Tags: functional equation , perfect square , functional equation in n , number theory

Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.

2012 BMT Spring, 1

Tags: number theory

Let $S$ be the set of all rational numbers $x \in [0, 1]$ with repeating base $6$ expansion $$x = 0.\overline{a_1a_2 ... a_k} = 0.a_1a_2...a_ka_1a_2...a_k...$$ for some finite sequence $\{a_i\}^{k}_{i=1}$ of distinct nonnegative integers less than $6$. What is the sum of all numbers that can be written in this form? (Put your answer in base $10$.)