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 Dutch Mathematical Olympiad, 2
Tags: diophantine , diophantine equation , number theory
Determine all triplets $(x, y, z)$ of positive integers with $x \le y \le z$ that satisfy $\left(1+\frac1x \right)\left(1+\frac1y \right)\left(1+\frac1z \right) = 3$

2007 Hanoi Open Mathematics Competitions, 4
Tags: number theory
[help me] Let m and n denote the number of digits in $2^{2007}$ and $5^{2007}$ when expressed in base 10. What is the sum m + n?

PEN M Problems, 14
Tags: relatively prime , number theory , recursive sequences
Let $x_{1}$ and $x_{2}$ be relatively prime positive integers. For $n \ge 2$, define $x_{n+1}=x_{n}x_{n-1}+1$.[list=a][*] Prove that for every $i>1$, there exists $j>i$ such that ${x_{i}}^{i}$ divides ${x_{j}}^{j}$. [*] Is it true that $x_{1}$ must divide ${x_{j}}^{j}$ for some $j>1$? [/list]

OMMC POTM, 2024 10
Tags: number theory , combinatorics
There are three positive integers written on a blackboard every minute. You can pick two written numbers $a$ and $b$ and replace them with $a \cdot b$ and $|a-b|$. Prove that it is always possible to make two of the numbers zero.

Russian TST 2021, P2
Tags: lattice points , number theory
The natural numbers $t{}$ and $q{}$ are given. For an integer $s{}$, we denote by $f(s)$ the number of lattice points lying in the triangle with vertices $(0;-t/q), (0; t/q)$ and $(t; ts/q)$. Suppose that $q{}$ divides $rs-1{}$. Prove that $f(r) = f(s)$.

2016 Bosnia and Herzegovina Team Selection Test, 3
Tags: number theory , prime number , integer sequence
For an infinite sequence $a_1<a_2<a_3<...$ of positive integers we say that it is [i]nice[/i] if for every positive integer $n$ holds $a_{2n}=2a_n$. Prove the following statements: $a)$ If there is given a [i]nice[/i] sequence and prime number $p>a_1$, there exist some term of the sequence which is divisible by $p$. $b)$ For every prime number $p>2$, there exist a [i]nice[/i] sequence such that no terms of the sequence are divisible by $p$.

2017 IMO Shortlist, N5
Tags: number theory
Find all pairs $(p,q)$ of prime numbers which $p>q$ and $$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$ is an integer.

2008 Irish Math Olympiad, 3
Tags: number theory
Determine, with proof, all integers $ x$ for which $ x(x\plus{}1)(x\plus{}7)(x\plus{}8)$ is a perfect square.

2022 Saudi Arabia JBMO TST, 4
Tags: number theory , combinatorics
You plan to organize your birthday party, which will be attended either by exactly $m$ persons or by exactly $n$ persons (you are not sure at the moment). You have a big birthday cake and you want to divide it into several parts (not necessarily equal), so that you are able to distribute the whole cake among the people attending the party with everybody getting cake of equal mass (however, one may get one big slice, while others several small slices - the sizes of slices may differ). What is the minimal number of parts you need to divide the cake, so that it is possible, regardless of the number of guests.

2021 International Zhautykov Olympiad, 1
Tags: number theory
Prove that there exists a positive integer $n$, such that the remainder of $3^n$ when divided by $2^n$ is greater than $10^{2021} $.

2010 Contests, 1
Tags: number theory , algebra
a) Factorize $xy - x - y + 1$. b) Prove that if integers $a$ and $b$ satisfy $ |a + b| > |1 + ab|$, then $ab = 0$.

2010 China Northern MO, 3
Tags: diophantine , number theory , diophantine equation
Find all positive integer triples $(x, y, z)$ such that $1 + 2^x \cdot 3^y=5^z$ is true.

2021 JBMO TST - Turkey, 6
Tags: number theory , modular arithmetic
Integers $a_1, a_2, \dots a_n$ are different at $\text{mod n}$. If $a_1, a_2-a_1, a_3-a_2, \dots a_n-a_{n-1}$ are also different at $\text{mod n}$, we call the ordered $n$-tuple $(a_1, a_2, \dots a_n)$ [i]lucky[/i]. For which positive integers $n$, one can find a lucky $n$-tuple?

1971 IMO Longlists, 53
Tags: floor function , number theory , logarithm , limit
Denote by $x_n(p)$ the multiplicity of the prime $p$ in the canonical representation of the number $n!$ as a product of primes. Prove that $\frac{x_n(p)}{n}<\frac{1}{p-1}$ and $\lim_{n \to \infty}\frac{x_n(p)}{n}=\frac{1}{p-1}$.

2025 All-Russian Olympiad Regional Round, 9.5
Tags: number theory
Find all pairs of integer numbers $m$ and $n>2$ such that $((n-1)!-n)(n-2)!=m(m-2)$. [i]A. Kuznetsov[/i]

1992 IMO Longlists, 31
Tags: algebra , number theory , divisibility , coefficient , polynomial
Let $ f(x) \equal{} x^8 \plus{} 4x^6 \plus{} 2x^4 \plus{} 28x^2 \plus{} 1.$ Let $ p > 3$ be a prime and suppose there exists an integer $ z$ such that $ p$ divides $ f(z).$ Prove that there exist integers $ z_1, z_2, \ldots, z_8$ such that if \[ g(x) \equal{} (x \minus{} z_1)(x \minus{} z_2) \cdot \ldots \cdot (x \minus{} z_8),\] then all coefficients of $ f(x) \minus{} g(x)$ are divisible by $ p.$

2015 Indonesia MO Shortlist, N4
Tags: greatest common divisor , exponential equation , exponential , diophantine equation , number theory
Suppose that the natural number $a, b, c, d$ satisfy the equation $a^ab^{a + b} = c^cd^{c + d}$. (a) If gcd $(a, b) = $ gcd $(c, d) = 1$, prove that $a = c$ and $b = d$. (b) Does the conclusion $a = c$ and $b = d$ apply, without the condition gcd $(a, b) = $ gcd $(c, d) = 1$?

2024 Brazil National Olympiad, 4
Tags: number theory , counting , combinatorics
A number is called [i]trilegal[/i] if its digits belong to the set \(\{1, 2, 3\}\) and if it is divisible by \(99\). How many trilegal numbers with \(10\) digits are there?

2021 Moldova EGMO TST, 11
Tags: number theory
Find all solutions for (x,y) , both integers such that: $xy=3(\sqrt{x^2+y^2}-1)$

2024 Argentina Cono Sur TST, 6
Tags: number theory
Find all pairs of positive integers $(n, k)$ that satisfy the equation $$n!+n=n^k$$

2009 Ukraine Team Selection Test, 7
Tags: number theory , sequence , divisibility , modular arithmetic
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

2013 Turkey Team Selection Test, 1
Tags: greatest common divisor , modular arithmetic , number theory , polynomial , algebra
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

1955 Moscow Mathematical Olympiad, 311
Tags: floor function , number theory
Find all numbers $a$ such that (1) all numbers $[a], [2a], . . . , [Na]$ are distinct and (2) all numbers $\left[ \frac{1}{a}\right], \left[ \frac{2}{a}\right], ..., \left[ \frac{M}{a}\right]$ are distinct.

2012 Indonesia TST, 4
Tags: number theory
The Fibonacci sequence $\{F_n\}$ is defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$. Determine all triplets of positive integers $(k,m,n)$ such that $F_n = F_m^k$.

2017 Latvia Baltic Way TST, 6
Tags: combinatorics , number theory , divisible
A natural number is written in each box of the $13 \times 13$ grid area. Prove that you can choose $2$ rows and $4$ columns such that the sum of the numbers written at their $8$ intersections is divisible by $8$.