This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 15460

2019 Saudi Arabia JBMO TST, 4
Tags: number theory
Find all positive integers $k>1$, such that there exist positive integer $n$, such that the number $A=17^{18n}+4.17^{2n}+7.19^{5n}$ is product of $k$ consecutive positive integers.

2020 Bulgaria Team Selection Test, 2
Tags: algebra , number theory
Given two odd natural numbers $ a,b$ prove that for each $ n\in\mathbb{N}$ there exists $ m\in\mathbb{N}$ such that either $ a^mb^2-1$ or $ b^ma^2-1$ is multiple of $ 2^n.$

2019 IMO Shortlist, N2
Tags: number theory
Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.

2012 Kyrgyzstan National Olympiad, 5
Tags: number theory , modular arithmetic
The sequence of natural numbers is defined as follows: for any $ k\geq 1 $,$ a_{k+2}= a_{k+1}\cdot a_k+1 $. Prove that for $ k\geq 9 $ the number $ a_k-22 $ is composite.

2023 Simon Marais Mathematical Competition, B4
Tags: number theory
[i](The following problem is open in the sense that the answer to part (b) is not currently known.)[/i] [list=a] [*] Let $n$ be a positive integer that is not a perfect square. Find all pairs $(a,b)$ of positive integers for which there exists a positive real number $r$, such that $$r^a+\sqrt{n} \ \ \text{and} \ \ r^b+\sqrt{n}$$ are both rational numbers. [*] Let $n$ be a positive integer that is not a perfect square. Find all pairs $(a,b)$ of positive integers for which there exists a real number $r$, such that $$r^a+\sqrt{n} \ \ \text{and} \ \ r^b+\sqrt{n}$$ are both rational numbers. [/list]

2008 USA Team Selection Test, 4
Tags: number theory
Prove that for no integer $ n$ is $ n^7 \plus{} 7$ a perfect square.

2023 Switzerland - Final Round, 3
Tags: greatest common divisor , number theory
Let $x,y$ and $a_0, a_1, a_2, \cdots $ be integers satisfying $a_0 = a_1 = 0$, and $$a_{n+2} = xa_{n+1}+ya_n+1$$for all integers $n \geq 0$. Let $p$ be any prime number. Show that $\gcd(a_p,a_{p+1})$ is either equal to $1$ or greater than $\sqrt{p}$.

2008 Thailand Mathematical Olympiad, 8
Tags: perfect square , number theory
Prove that $2551 \cdot 543^n -2008\cdot 7^n$ is never a perfect square, where $n$ varies over the set of positive integers

2018 Polish Junior MO First Round, 6
Tags: equation , number theory , algebra
Positive integers $k, m, n$ satisfy the equation $m^2 + n = k^2 + k$. Show that $m \le n$.

2024 Caucasus Mathematical Olympiad, 3
Tags: number theory
Given $10$ positive integers with a sum equal to $1000$. The product of their factorials is a $10$-th power of an integer. Prove that all these numbers are equal.

2001 AIME Problems, 9
Tags: probability , rectangle , geometry , relatively prime , number theory
Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2014 Argentina Cono Sur TST, 4
Tags: number theory
Find all pairs of positive prime numbers $(p,q)$ such that $p^5+p^3+2=q^2-q$

2006 Belarusian National Olympiad, 8
Tags: perfect square , number theory
a) Do there exist positive integers $a$ and $b$ such that for any positive,integer $n$ the number $a \cdot 2^n+ b\cdot 5^n$ is a perfect square ? b) Do there exist positive integers $a, b$ and $c$, such that for any positive integer $n$ the number $a\cdot 2^n+ b\cdot 5^n + c$ is a perfect square? (M . Blotski)

1991 Chile National Olympiad, 4
Tags: divide , divisible , number theory
Show that the expressions $2x + 3y$, $9x + 5y$ are both divisible by $17$, for the same values of $x$ and $y$.

2004 Finnish National High School Mathematics Competition, 4
Tags: prime number , number theory
The numbers $2005! + 2, 2005! + 3, ... , 2005! + 2005$ form a sequence of $2004$ consequtive integers, none of which is a prime number. Does there exist a sequence of $2004$ consequtive integers containing exactly $12$ prime numbers?

2012 Puerto Rico Team Selection Test, 6
Tags: number theory
The increasing sequence $1; 3; 4; 9; 10; 12; 13; 27; 28; 30; 31, \ldots$ is formed with positive integers which are powers of $3$ or sums of different powers of $3$. Which number is in the $100^{th}$ position?

2025 Macedonian TST, Problem 6
Tags: number theory
Let $n>2$ be an even integer, and let $V$ be an arbitrary set of $8$ distinct integers. Define \[ E(V,n) \;=\; \bigl\{(u,v)\in V\times V : u < v,\ u+v = n^k\text{ for some }k\in\mathbb{N}\bigr\}. \] For each even $n>2$, determine the maximum possible size of the set $E(V,n)$.

2002 Moldova Team Selection Test, 4
Tags: arc , integer , sum , projection , distance , geometry , number theory
Let $C$ be the circle with center $O(0,0)$ and radius $1$, and $A(1,0), B(0,1)$ be points on the circle. Distinct points $A_1,A_2, ....,A_{n-1}$ on $C$ divide the smaller arc $AB$ into $n$ equal parts ($n \ge 2$). If $P_i$ is the orthogonal projection of $A_i$ on $OA$ ($i =1, ... ,n-1$), find all values of $n$ such that $P_1A^{2p}_1 +P_2A^{2p}_2 +...+P_{n-1}A^{2p}_{n-1}$ is an integer for every positive integer $p$.

1999 German National Olympiad, 1
Tags: diophantine equation , diophantine , number theory
Find all $x,y$ which satisfy the equality $x^2 +xy+y^2 = 97$, when $x,y$ are a) natural numbers, b) integers

1971 All Soviet Union Mathematical Olympiad, 144
Tags: number theory , power of 2 , digit
Prove that for every natural $n$ there exists a number, containing only digits "$1$" and "$2$" in its decimal notation, that is divisible by $2^n$ ( $n$-th power of two ).

2007 Bulgaria Team Selection Test, 2
Tags: number theory , inequalities
Let $n,k$ be positive integers such that $n\geq2k>3$ and $A= \{1,2,...,n\}.$ Find all $n$ and $k$ such that the number of $k$-element subsets of $A$ is $2n-k$ times bigger than the number of $2$-element subsets of $A.$

2007 QEDMO 4th, 11
Tags: set , multiple , number theory
Let $S_{1},$ $S_{2},$ $...,$ $S_{n}$ be finitely many subsets of $\mathbb{N}$ such that $S_{1}\cup S_{2}\cup...\cup S_{n}=\mathbb{N}.$ Prove that there exists some $k\in\left\{ 1,2,...,n\right\} $ such that for each positive integer $m,$ the set $S_{k}$ contains infinitely many multiples of $m.$

2013 Iran MO (3rd Round), 4
Tags: polynomial , vieta , diophantine equation , pell equation , number theory , algebra
Prime $p=n^2 +1$ is given. Find the sets of solutions to the below equation: \[x^2 - (n^2 +1)y^2 = n^2.\] (25 points)

2011 Vietnam National Olympiad, 1
Tags: algebra , quadratic , modular arithmetic , polynomial , number theory
Define the sequence of integers $\langle a_n\rangle$ as; \[a_0=1, \quad a_1=-1, \quad \text{ and } \quad a_n=6a_{n-1}+5a_{n-2} \quad \forall n\geq 2.\] Prove that $a_{2012}-2010$ is divisible by $2011.$

2015 NZMOC Camp Selection Problems, 4
Tags: diophantine , diophantine equation , number theory
For which positive integers $m$ does the equation: $$(ab)^{2015} = (a^2 + b^2)^m$$ have positive integer solutions?