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

2002 India IMO Training Camp, 16
Tags: sum , modular arithmetic , number theory , additive number theory
Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

2007 Moldova Team Selection Test, 4
Tags: prime number , number theory
Show that there are infinitely many prime numbers $p$ having the following property: there exists a natural number $n$, not dividing $p-1$, such that $p|n!+1$.

2001 India IMO Training Camp, 2
Tags: modular arithmetic , number theory
Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$, define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv 1 \pmod{p}$ and set $kx_k = 1+ pn_k$. Prove that : \[\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}\]

2018 BMT Spring, 2
Tags: number theory
Suppose for some positive integers, that $\frac{p+\frac{1}{q}}{q+\frac{1}{p}}= 17$. What is the greatest integer $n$ such that $\frac{p+q}{n}$ is always an integer?

2014 Junior Regional Olympiad - FBH, 4
Tags: number theory , prime number
Find all prime numbers $p$ and $q$ such that $$(2p-q)^2=17p-10q$$

2024 Caucasus Mathematical Olympiad, 3
Tags: number theory
Let $n$ be a $d$-digit (i.e., having $d$ digits in its decimal representation) positive integer not divisible by $10$. Writing all the digits of $n$ in reverse order, we obtain the number $n'$. Determine if it is possible that the decimal representation of the product $n\cdot n'$ consists of digits $8$ only, if (a) $d = 9998$; (b) $d = 9999?$

2021 AMC 12/AHSME Fall, 16
Tags: number theory , greatest common divisor
Let $a, b,$ and $c$ be positive integers such that $a+b+c=23$ and \[\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.\] What is the sum of all possible distinct values of $a^{2}+b^{2}+c^{2}$? $\textbf{(A)} ~259\qquad\textbf{(B)} ~438\qquad\textbf{(C)} ~516\qquad\textbf{(D)} ~625\qquad\textbf{(E)} ~687$ Proposed by [b]djmathman[/b]

2019 Jozsef Wildt International Math Competition, W. 64
Tags: number theory
Prove that exist different natural numbers $x$, $y$, $z$, $t$ for which $$256\times 2019^{180n+1}=2x^9-2y^6+z^5-t^4$$for all $n\in \mathbb{N}^*$

2016 Hanoi Open Mathematics Competitions, 1
Tags: digit , number theory
How many are there $10$-digit numbers composed from the digits $1, 2, 3$ only and in which, two neighbouring digits differ by $1$ : (A): $48$ (B): $64$ (C): $72$ (D): $128$ (E): None of the above.

2013 NIMO Summer Contest, 2
Tags: number theory , relatively prime
If $\frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6} = \frac{m}{n}$ for relatively prime integers $m$ and $n$, compute $100m+n$. [i]Proposed by Evan Chen[/i]

2020 Iranian Our MO, 3
Tags: number theory
Does there exist a non-constant infinite sequence of distinct natural numbers such that for all large enough $k$ we have that sum of any $k$-sized sub-sequence is square-free if and only if $k$ is square-free. [i]Proposed by Alireza Haqi, Amir Rezaie Moghadam [/i] [b]Rated 3[/b]

2007 Pre-Preparation Course Examination, 1
Tags: function , number theory
a) Find all multiplicative functions $f: \mathbb Z_{p}^{*}\longrightarrow\mathbb Z_{p}^{*}$ (i.e. that $\forall x,y\in\mathbb Z_{p}^{*}$, $f(xy)=f(x)f(y)$.) b) How many bijective multiplicative does exist on $\mathbb Z_{p}^{*}$ c) Let $A$ be set of all multiplicative functions on $\mathbb Z_{p}^{*}$, and $VB$ be set of all bijective multiplicative functions on $\mathbb Z_{p}^{*}$. For each $x\in \mathbb Z_{p}^{*}$, calculate the following sums :\[\sum_{f\in A}f(x),\ \ \sum_{f\in B}f(x)\]

1998 Croatia National Olympiad, Problem 2
Tags: number theory , diophantine equation
Find all positive integer solutions of the equation $10(m+n)=mn$.

2012 Baltic Way, 7
Tags: modular arithmetic , analytic geometry , number theory
On a $2012 \times 2012$ board, some cells on the top-right to bottom-left diagonal are marked. None of the marked cells is in a corner. Integers are written in each cell of this board in the following way. All the numbers in the cells along the upper and the left sides of the board are 1's. All the numbers in the marked cells are 0's. Each of the other cells contains a number that is equal to the sum of its upper neighbour and its left neighbour. Prove that the number in the bottom right corner is not divisible by 2011.

1970 All Soviet Union Mathematical Olympiad, 142
Tags: number theory , digit
All natural numbers containing not more than $n$ digits are divided onto two groups. The first contains the numbers with the even sum of the digits, the second -- with the odd sum. Prove that if $0<k<n$ than the sum of the $k$-th powers of the numbers in the first group equals to the sum of the $k$-th powers of the numbers in the second group.

2019 AIME Problems, 14
Tags: number theory
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5$, $n$, and $n + 1$ cents, $91$ cents is the greatest postage that cannot be formed.

2003 Gheorghe Vranceanu, 3
Tags: modular arithmetic , number theory
Show that $ n\equiv 0\pmod 9 $ if $ 2^n\equiv -1\pmod n, $ where $ n $ is a natural number greater than $ 3. $

VI Soros Olympiad 1999 - 2000 (Russia), 11.7
Tags: number theory , coprime
Prove that there are arithmetic progressions of arbitrary length, consisting of different pairwise coprime natural numbers.

1974 IMO Shortlist, 7
Tags: least common multiple , greatest common divisor , number theory
Let $a_i, b_i$ be coprime positive integers for $i = 1, 2, \ldots , k$, and $m$ the least common multiple of $b_1, \ldots , b_k$. Prove that the greatest common divisor of $a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}$ equals the greatest common divisor of $a_1, \ldots , a_k.$

2007 May Olympiad, 1
Tags: number theory , mutliple
Determine the largest natural number that has all its digits different and is a multiple of $5$, $8$ and $11$.

2018 India PRMO, 20
Tags: product , digit , number theory
Determine the sum of all possible positive integers $n, $ the product of whose digits equals $n^2 -15n -27$.

2016 Gulf Math Olympiad, 2
Tags: number theory
Let $x$ be a real number that satisfies $x^1 + x^{-1} = 3$ Prove that $x^n + x^{-n}$ is an positive integer , then prove that the positive integer $x^{3^{1437}}+x^{3^{-1437}}$ is divisible by at least $1439 \times 2^{1437}$ positive integers

Kvant 2021, M2681
Tags: algebra , number theory
Find all pairs of distinct rational numbers $(a,b)$ such that $a^a=b^b$. [i]Proposed by I. Dorofeev[/i]

2009 QEDMO 6th, 1
Tags: diophantine , diophantine equation , number theory
Solve $y^5 - x^2 = 4$ in integers numbers $x,y$.

2016 Saudi Arabia BMO TST, 3
Tags: number theory , divide , divisible
Show that there are infinitely many positive integers $n$ such that $n$ has at least two prime divisors and $20^n + 16^n$ is divisible by $n^2$.