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

2009 Cuba MO, 9
Tags: number theory , divide
Find all the triples of prime numbers $(p, q, r)$ such that $$p | 2qr + r \,\,\,, \,\,\,q |2pr + p \,\,\, and \,\,\, r | 2pq + q.$$

2022 Harvard-MIT Mathematics Tournament, 2
Tags: number theory
Compute the number of positive integers that divide at least two of the integers in the set $\{1^1,2^2,3^3,4^4,5^5,6^6,7^7,8^8,9^9,10^{10}\}$.

2011 Belarus Team Selection Test, 2
Tags: number theory , divide , divisible
Do they exist natural numbers $m,x,y$ such that $$m^2 +25 \vdots (2011^x-1007^y) ?$$ S. Finskii

2007 Argentina National Olympiad, 1
Tags: number theory , diophantine equation , diophantine
Find all the prime numbers $p$ and $q$ such that $ p^2+q=37q^2+p $. Clarification: $1$ is not a prime number.

2005 ITAMO, 2
Tags: modular arithmetic , euler , function , number theory , totient function , algebra
Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and \[a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }\\\\a_n+h \text{ otherwise }.\end{array}\] For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$?

1984 Poland - Second Round, 1
Tags: number theory , diophantine , diophantine equation
For a given natural number $ n $, find the number of solutions to the equation $ \sqrt{x} + \sqrt{y} = n $ in natural numbers $ x, y $.

2010 Grand Duchy of Lithuania, 2
Tags: number theory , diophantine , diophantine equation
Find all positive integers $n$ for which there are distinct integer numbers $a_1, a_2, ... , a_n$ such that $$\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{n}{a_n}=\frac{a_1 + a_2 + ... + a_n}{2}$$

2013 Romanian Masters In Mathematics, 1
Tags: floor function , modular arithmetic , quadratic , number theory
For a positive integer $a$, define a sequence of integers $x_1,x_2,\ldots$ by letting $x_1=a$ and $x_{n+1}=2x_n+1$ for $n\geq 1$. Let $y_n=2^{x_n}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_1,\ldots,y_k$ are all prime.

2011 Paraguay Mathematical Olympiad, 3
Tags: number theory
If number $\overline{aaaa}$ is divided by $\overline{bb}$, the quotient is a number between $140$ and $160$ inclusively, and the remainder is equal to $\overline{(a-b)(a-b)}$. Find all pairs of positive integers $(a,b)$ that satisfy this.

2018 Pan-African Shortlist, N7
Tags: diophantine equation , number theory
Find all non-negative integers $n$ for which the equation \[ {\left( x^2 + y^2 \right)}^n = {(xy)}^{2018} \] admits positive integral solutions.

2025 239 Open Mathematical Olympiad, 2
Tags: number theory
Let's call a power of two [i]compact[/i] if it can be represented as the sum of no more than $10^9$ not necessarily distinct factorials of positive integer numbers. Prove that the set of compact powers of two is finite.

2011 All-Russian Olympiad Regional Round, 10.5
Tags: number theory
Find all $a$ such that for any positive integer $n$, the number $an(n+2)(n+3)(n+4)$ is an integer. (Author: O. Podlipski) [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=427802](similar to Problem 5 of grade 9)[/url] Same problem for grades 10 and 11

2019 Harvard-MIT Mathematics Tournament, 8
Tags: hmmt , algebra , number theory
There is a unique function $f: \mathbb{N} \to \mathbb{R}$ such that $f(1) > 0$ and such that \[\sum_{d \mid n} f(d) f\left(\frac{n}{d}\right) = 1\] for all $n \ge 1$. What is $f(2018^{2019})$?

2009 VTRMC, Problem 6
Tags: number theory , diophantine equation
Let $n$ be a nonzero integer. Prove that $n^4-7n^2+1$ can never be a perfect square.

PEN P Problems, 18
Tags: modular arithmetic , number theory , additive number theory
Let $p$ be a prime with $p \equiv 1 \pmod{4}$. Let $a$ be the unique integer such that \[p=a^{2}+b^{2}, \; a \equiv-1 \pmod{4}, \; b \equiv 0 \; \pmod{2}\] Prove that \[\sum^{p-1}_{i=0}\left( \frac{i^{3}+6i^{2}+i }{p}\right) = 2 \left( \frac{2}{p}\right),\] where $\left(\frac{k}{p}\right)$ denotes the Legendre Symbol.

1961 Poland - Second Round, 1
Tags: number theory , consecutive , power of 2
Prove that no number of the form $ 2^n $, where $ n $ is a natural number, is the sum of two or more consecutive natural numbers.

1966 IMO Shortlist, 34
Tags: number theory , equation
Find all pairs of positive integers $\left( x;\;y\right) $ satisfying the equation $2^{x}=3^{y}+5.$

2023 Junior Balkan Team Selection Tests - Moldova, 5
Tags: geometry , number theory
The positive integers $ a, b, c $ are the lengths of the sides of a right triangle. Prove that $abc$ is divisible by $60$.

2014 Contests, 4
Tags: limit , real analysis , number theory , analytic number theory
Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that \[n \mid a^{f(n)}-1.\] Prove that $S$ has density $0$; that is, prove that $\lim_{n\rightarrow \infty} \frac{|S\cap \{1,...,n\}|}{n}=0$.

1955 Moscow Mathematical Olympiad, 307
Tags: trinomial , perfect square , number theory , algebra
* The quadratic expression $ax^2 + bx + c$ is a square (of an integer) for any integer $x$. Prove that $ax^2 + bx + c = (dx + e)^2$ for some integers d and e.

1996 Yugoslav Team Selection Test, Problem 1
Tags: number theory , set , combinatorics
Let $\mathfrak F=\{A_1,A_2,\ldots,A_n\}$ be a collection of subsets of the set $S=\{1,2,\ldots,n\}$ satisfying the following conditions: (a) Any two distinct sets from $\mathfrak F$ have exactly one element in common; (b) each element of $S$ is contained in exactly $k$ of the sets in $\mathfrak F$. Can $n$ be equal to $1996$?

LMT Team Rounds 2021+, 4
Tags: number theory
Find the least positive integer ending in $7$ with exactly $12$ positive divisors.

1967 IMO, 3
Tags: binomial coefficient , number theory , divisibility
Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that \[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\] is divisible by the product $c_1c_2\ldots c_n$.

2005 IMO Shortlist, 1
Tags: fermat's little theorem , number theory
Determine all positive integers relatively prime to all the terms of the infinite sequence \[ a_n=2^n+3^n+6^n -1,\ n\geq 1. \]

Russian TST 2015, P1
Tags: number theory , divisibility
Find all pairs of natural numbers $(a,b)$ satisfying the following conditions: [list] [*]$b-1$ is divisible by $a+1$ and [*]$a^2+a+2$ is divisible by $b$. [/list]