Olympiad problems

2011 Federal Competition For Advanced Students, Part 2, 1

Determine all pairs $(a,b)$ of non-negative integers, such that $a^b+b$ divides $a^{2b}+2b$. (Remark: $0^0=1$.)

2011 Iran MO (3rd Round), 4

Tags: algebra , polynomial , number theory proposed , number theory

Suppose that $n$ is a natural number and $n$ is not divisible by $3$. Prove that $(n^{2n}+n^n+n+1)^{2n}+(n^{2n}+n^n+n+1)^n+1$ has at least $2d(n)$ distinct prime factors where $d(n)$ is the number of positive divisors of $n$. [i]proposed by Mahyar Sefidgaran[/i]

2003 Tuymaada Olympiad, 4

Tags: algebra , polynomial , calculus , integration , number theory proposed , number theory

Given are polynomial $f(x)$ with non-negative integral coefficients and positive integer $a.$ The sequence $\{a_{n}\}$ is defined by $a_{1}=a,$ $a_{n+1}=f(a_{n}).$ It is known that the set of primes dividing at least one of the terms of this sequence is finite. Prove that $f(x)=cx^{k}$ for some non-negative integral $c$ and $k.$ [i]Proposed by F. Petrov[/i] [hide="For those of you who liked this problem."] Check [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62259]this thread[/url] out.[/hide]

2011 Indonesia TST, 4

Tags: modular arithmetic , number theory proposed , number theory

Prove that there exists infinitely many positive integers $n$ such that $n^2+1$ has a prime divisor greater than $2n+\sqrt{5n+2011}$.

2008 Iran MO (2nd Round), 1

Tags: number theory proposed , number theory

$\mathbb{N}$ is the set of positive integers and $a\in\mathbb{N}$. We know that for every $n\in\mathbb{N}$, $4(a^n+1)$ is a perfect cube. Prove that $a=1$.

2004 Baltic Way, 7

Tags: ratio , number theory proposed , number theory

Find all sets $X$ consisting of at least two positive integers such that for every two elements $m,n\in X$, where $n>m$, there exists an element $k\in X$ such that $n=mk^2$.

2014 Korea - Final Round, 5

Tags: Euler , abstract algebra , number theory proposed , number theory , Thues Lemma

Let $p>5$ be a prime. Suppose that there exist integer $k$ such that $ k^2 + 5 $ is divisible by $p$. Prove that there exist two positive integers $m,n$ satisfying $ p^2 = m^2 + 5n^2 $.

2012 International Zhautykov Olympiad, 3

Tags: modular arithmetic , number theory , greatest common divisor , Diophantine equation , number theory proposed

Find all integer solutions of the equation the equation $2x^2-y^{14}=1$.

1993 Baltic Way, 3

Tags: number theory proposed , number theory

Let’s call a positive integer [i]interesting[/i] if it is a product of two (distinct or equal) prime numbers. What is the greatest number of consecutive positive integers all of which are interesting?

2006 Iran MO (3rd Round), 5

Tags: abstract algebra , group theory , number theory proposed , number theory

For each $n$, define $L(n)$ to be the number of natural numbers $1\leq a\leq n$ such that $n\mid a^{n}-1$. If $p_{1},p_{2},\ldots,p_{k}$ are the prime divisors of $n$, define $T(n)$ as $(p_{1}-1)(p_{2}-1)\cdots(p_{k}-1)$. a) Prove that for each $n\in\mathbb N$ we have $n\mid L(n)T(n)$. b) Prove that if $\gcd(n,T(n))=1$ then $\varphi(n) | L(n)T(n)$.

1989 Federal Competition For Advanced Students, 1

Tags: number theory proposed , number theory

Natural numbers $ a \le b \le c \le d$ satisfy $ a\plus{}b\plus{}c\plus{}d\equal{}30$. Find the maximum value of the product $ P\equal{}abcd.$

2005 Georgia Team Selection Test, 7

Tags: number theory proposed , number theory

Determine all positive integers $ n$, for which $ 2^{n\minus{}1}n\plus{}1$ is a perfect square.

1993 Korea - Final Round, 4

Tags: geometry , number theory proposed , number theory

An integer which is the area of a right-angled triangle with integer sides is called [i]Pythagorean[/i]. Prove that for every positive integer $n > 12$ there exists a Pythagorean number between $n$ and $2n.$

Oliforum Contest IV 2013, 4

Tags: algebra , polynomial , quadratics , number theory proposed , number theory , Contest 8

Let $p,q$ be integers such that the polynomial $x^2+px+q+1$ has two positive integer roots. Show that $p^2+q^2$ is composite.

2024 Bundeswettbewerb Mathematik, 2

Tags: number theory , number theory proposed , Digits , Perfect Squares , Perfect Square

Can a number of the form $44\dots 41$, with an odd number of decimal digits $4$ followed by a digit $1$, be a perfect square?

2008 Czech-Polish-Slovak Match, 1

Tags: quadratics , number theory proposed , number theory

Prove that there exists a positive integer $n$, such that the number $k^2+k+n$ does not have a prime divisor less than $2008$ for any integer $k$.

2007 Middle European Mathematical Olympiad, 4

Tags: modular arithmetic , algebra , polynomial , number theory , divisibility tests , number theory proposed

Find all positive integers $ k$ with the following property: There exists an integer $ a$ so that $ (a\plus{}k)^{3}\minus{}a^{3}$ is a multiple of $ 2007$.

2006 Baltic Way, 18

Tags: modular arithmetic , number theory proposed , number theory

For a positive integer $n$ let $a_n$ denote the last digit of $n^{(n^n)}$. Prove that the sequence $(a_n)$ is periodic and determine the length of the minimal period.

2009 Indonesia TST, 3

Tags: modular arithmetic , floor function , ceiling function , inequalities , number theory proposed , number theory

Let $ n \ge 2009$ be an integer and define the set: \[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}. \] Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that \[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}. \]

2020 Bundeswettbewerb Mathematik, 4

Tags: number theory , prime factorization , number theory proposed

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 $a_{19702020}$.

2000 Baltic Way, 14

Tags: inequalities , function , induction , number theory , prime factorization , number theory proposed

Find all positive integers $n$ such that $n$ is equal to $100$ times the number of positive divisors of $n$.

2024 Dutch IMO TST, 4

Tags: number theory , number theory proposed , blackboard , Digits

Initially, a positive integer $N$ is written on a blackboard. We repeatedly replace the number according to the following rules: 1) replace the number by a positive multiple of itself 2) replace the number by a number with the same digits in a different order. (The new number is allowed to have leading digits, which are then deleted.) [i]A possible sequence of moves is given by $5 \to 20 \to 140 \to 041=41$.[/i] Determine for which values of $N$ it is possible to obtain $1$ after a finite number of such moves.

1996 All-Russian Olympiad, 5

Tags: geometry , 3D geometry , greatest common divisor , number theory proposed , number theory

At the vertices of a cube are written eight pairwise distinct natural numbers, and on each of its edges is written the greatest common divisor of the numbers at the endpoints of the edge. Can the sum of the numbers written at the vertices be the same as the sum of the numbers written at the edges? [i]A. Shapovalov[/i]

2000 Irish Math Olympiad, 3

Tags: number theory proposed , number theory

For each positive integer $ n$ find all positive integers $ m$ for which there exist positive integers $ x_1<x_2<...<x_n$ with: $ \frac{1}{x_1}\plus{}\frac{2}{x_2}\plus{}...\plus{}\frac{n}{x_n}\equal{}m.$

2014 Iran MO (3rd Round), 6

Tags: modular arithmetic , arithmetic series , number theory proposed , number theory

Prove that there are 100 natural number $a_1 < a_2 < ... < a_{99} < a_{100}$ ( $ a_i < 10^6$) such that A , A+A , 2A , A+2A , 2A + 2A are five sets apart ? $A = \{a_1 , a_2 ,... , a_{99} ,a_{100}\}$ $2A = \{2a_i \vert 1\leq i\leq 100\}$ $A+A = \{a_i + a_j \vert 1\leq i<j\leq 100\}$ $A + 2A = \{a_i + 2a_j \vert 1\leq i,j\leq 100\}$ $2A + 2A = \{2a_i + 2a_j \vert 1\leq i<j\leq 100\}$ (20 ponits )