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.

AND:
OR:
NO:

Found problems: 15460

2024 Vietnam National Olympiad, 6
Tags: number theory
For each positive integer $n$, let $\tau (n)$ be the number of positive divisors of $n$. a) Find all positive integers $n$ such that $\tau(n)+2023=n$. b) Prove that there exist infinitely many positive integers $k$ such that there are exactly two positive integers $n$ satisfying $\tau(kn)+2023=n$.

2024 Polish MO Finals, 3
Tags: prime number , system of equations , diophantine equation , number theory
Determine all pairs $(p,q)$ of prime numbers with the following property: There are positive integers $a,b,c$ satisfying \[\frac{p}{a}+\frac{p}{b}+\frac{p}{c}=1 \quad \text{and} \quad \frac{a}{p}+\frac{b}{p}+\frac{c}{p}=q+1.\]

2004 Thailand Mathematical Olympiad, 17
Tags: number theory , remainder , sum of powers
Compute the remainder when $1^{2547} + 2^{2547} +...+ 2547^{2547}$ is divided by $25$.

1969 IMO Shortlist, 54
Tags: algebra , polynomial , modular arithmetic , number theory , divisibility
$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$

2016 Vietnam Team Selection Test, 1
Tags: number theory
Find all $a,n\in\mathbb{Z}^+$ ($a>2$) such that each prime divisor of $a^n-1$ is also prime divisor of $a^{3^{2016}}-1$

2005 France Team Selection Test, 4
Tags: floor function , ceiling function , induction , inequalities , number theory , logarithm
Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.

2024 India IMOTC, 17
Tags: number theory , algebra , polynomial
Fix a positive integer $a > 1$. Consider triples $(f(x), g(x), h(x))$ of polynomials with integer coefficients, such that 1. $f$ is a monic polynomial with $\deg f \ge 1$. 2. There exists a positive integer $N$ such that $g(x)>0$ for $x \ge N$ and for all positive integers $n \ge N$, we have $f(n) \mid a^{g(n)} + h(n)$. Find all such possible triples. [i]Proposed by Mainak Ghosh and Rijul Saini[/i]

1983 IMO Longlists, 55
Tags: number theory
For every $a \in \mathbb N$ denote by $M(a)$ the number of elements of the set \[ \{ b \in \mathbb N | a + b \text{ is a divisor of } ab \}.\] Find $\max_{a\leq 1983} M(a).$

2016 Federal Competition For Advanced Students, P1, 3
Tags: number theory
Consider 2016 points arranged on a circle. We are allowed to jump ahead by 2 or 3 points in clockwise direction. What is the minimum number of jumps required to visit all points and return to the starting point? (Gerd Baron)

2019 All-Russian Olympiad, 5
Tags: rectangle , number theory , prime number
In a kindergarten, a nurse took $n$ congruent cardboard rectangles and gave them to $n$ kids, one per each. Each kid has cut its rectangle into congruent squares(the squares of different kids could be of different sizes). It turned out that the total number of the obtained squares is a prime number. Prove that all the initial squares were in fact squares.

1973 Chisinau City MO, 70
Tags: diophantine , diophantine equation , number theory
The natural numbers $p, q$ satisfy the relation $p^p + q^q = p^q + q^p$. Prove that $p = q$.

2001 Baltic Way, 19
Tags: number theory
What is the smallest positive odd integer having the same number of positive divisors as $360$?

2015 China Northern MO, 1
Tags: diophantine equation , diophantine , number theory
Find all integer solutions to the equation $$\frac{xyz}{w}+\frac{yzw}{x}+\frac{zwx}{y}+\frac{wxy}{z}=4$$

2015 ISI Entrance Examination, 1
Tags: number theory , algebra
Let $m_1< m_2 < \ldots m_{k-1}< m_k$ be $k$ distinct positive integers such that their reciprocals are in arithmetic progression. 1.Show that $k< m_1 + 2$. 2. Give an example of such a sequence of length $k$ for any positive integer $k$.

2019 Switzerland Team Selection Test, 12
Tags: number theory
Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

2006 IberoAmerican, 1
Tags: number theory
Find all pairs $(a,\, b)$ of positive integers such that $2a-1$ and $2b+1$ are coprime and $a+b$ divides $4ab+1.$

2021 CMIMC, 5
Tags: number theory
Let $N$ be the fifth largest number that can be created by combining $2021$ $1$'s using addition, multiplication, and exponentiation, in any order (parentheses are allowed). If $f(x)=\log_2(x)$, and $k$ is the least positive integer such that $f^k(N)$ is not a power of $2$, what is the value of $f^k(N)$? (Note: $f^k(N)=f(f(\cdots(f(N))))$, where $f$ is applied $k$ times.) [i]Proposed by Adam Bertelli[/i]

1994 IMO Shortlist, 6
Tags: modular arithmetic , number theory , integer sequence , recurrence relation , divisibility
Define the sequence $ a_1, a_2, a_3, ...$ as follows. $ a_1$ and $ a_2$ are coprime positive integers and $ a_{n \plus{} 2} \equal{} a_{n \plus{} 1}a_n \plus{} 1$. Show that for every $ m > 1$ there is an $ n > m$ such that $ a_m^m$ divides $ a_n^n$. Is it true that $ a_1$ must divide $ a_n^n$ for some $ n > 1$?

2011 Singapore MO Open, 4
Tags: algebra , polynomial , calculus , number theory
Find all polynomials $P(x)$ with real coefficients such that \[P(a)\in\mathbb{Z}\ \ \ \text{implies that}\ \ \ a\in\mathbb{Z}.\]

2023 Bulgarian Spring Mathematical Competition, 9.3
Tags: number theory
Given a prime $p$, find $\gcd(\binom{2^pp}{1},\binom{2^pp}{3},\ldots, \binom{2^pp}{2^pp-1}) $.

2008 All-Russian Olympiad, 7
Tags: number theory
For which integers $ n>1$ do there exist natural numbers $ b_1,b_2,\ldots,b_n$ not all equal such that the number $ (b_1\plus{}k)(b_2\plus{}k)\cdots(b_n\plus{}k)$ is a power of an integer for each natural number $ k$? (The exponents may depend on $ k$, but must be greater than $ 1$)

2004 South africa National Olympiad, 1
Tags: greatest common divisor , algorithm , number theory
Let $a=1111\dots1111$ and $b=1111\dots1111$ where $a$ has forty ones and $b$ has twelve ones. Determine the greatest common divisor of $a$ and $b$.

2009 Croatia Team Selection Test, 4
Tags: quadratic , diophantine equation , number theory
Prove that there are infinite many positive integers $ n$ such that $ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.

2017 Regional Olympiad of Mexico Southeast, 4
Tags: number theory , diophantine equation , factorial
Find all couples of positive integers $m$ and $n$ such that $$n!+5=m^3$$

2000 JBMO ShortLists, 1
Tags: number theory
Prove that there are at least $666$ positive composite numbers with $2006$ digits, having a digit equal to $7$ and all the rest equal to $1$.