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

2011 Korea Junior Math Olympiad, 8
Tags: combinatorics , combination , subset , number theory
There are $n$ students each having $r$ positive integers. Their $nr$ positive integers are all different. Prove that we can divide the students into $k$ classes satisfying the following conditions: (a) $k \le 4r$ (b) If a student $A$ has the number $m$, then the student $B$ in the same class can't have a number $\ell$ such that $(m - 1)! < \ell < (m + 1)! + 1$

2013 Dutch IMO TST, 2
Tags: perfect square , number theory , rational
Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.

2011 Finnish National High School Mathematics Competition, 4
Tags: number theory
Show that there is a perfect square (a number which is a square of an integer) such that sum of its digits is $2011.$

PEN H Problems, 69
Tags: diophantine equation , number theory , relatively prime
Determine all positive rational numbers $r \neq 1$ such that $\sqrt[r-1]{r}$ is rational.

2018 Belarusian National Olympiad, 9.1
Tags: number theory
Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair one number is divisible by another.

2023 Iran MO (3rd Round), 2
Tags: number theory , prime number
Let $N$ be the number of ordered pairs $(x,y)$ st $1 \leq x,y \leq p(p-1)$ and : $$x^{y} \equiv y^{x} \equiv 1 \pmod{p}$$ where $p$ is a fixed prime number. Show that : $$(\phi {(p-1)}d(p-1))^2 \leq N \leq ((p-1)d(p-1))^2$$ where $d(n)$ is the number of divisors of $n$

2020 South East Mathematical Olympiad, 7
Tags: number theory
Arrange all square-free positive integers in ascending order $a_1,a_2,a_3,\ldots,a_n,\ldots$. Prove that there are infinitely many positive integers $n$, such that $a_{n+1}-a_n=2020$.

2021 EGMO, 6
Tags: number theory , asymptotics , algebra , equation , counting
Does there exist a nonnegative integer $a$ for which the equation \[\left\lfloor\frac{m}{1}\right\rfloor + \left\lfloor\frac{m}{2}\right\rfloor + \left\lfloor\frac{m}{3}\right\rfloor + \cdots + \left\lfloor\frac{m}{m}\right\rfloor = n^2 + a\] has more than one million different solutions $(m, n)$ where $m$ and $n$ are positive integers? [i]The expression $\lfloor x\rfloor$ denotes the integer part (or floor) of the real number $x$. Thus $\lfloor\sqrt{2}\rfloor = 1, \lfloor\pi\rfloor =\lfloor 22/7 \rfloor = 3, \lfloor 42\rfloor = 42,$ and $\lfloor 0 \rfloor = 0$.[/i]

2010 China Second Round Olympiad, 2
Tags: modular arithmetic , number theory
Given a fixed integer $k>0,r=k+0.5$,define $f^1(r)=f(r)=r[r],f^l(r)=f(f^{l-1}(r))(l>1)$ where $[x]$ denotes the smallest integer not less than $x$. prove that there exists integer $m$ such that $f^m(r)$ is an integer.

2019 Nigeria Senior MO Round 2, 6
Tags: number theory
Let $N=4^KL$ where $L\equiv\ 7\pmod 8$. Prove that $N$ cannot be written as a sum of 3 squares

2015 JBMO Shortlist, NT3
Tags: number theory , product , difference , divisible
a) Show that the product of all differences of possible couples of six given positive integers is divisible by $960$ b) Show that the product of all differences of possible couples of six given positive integers is divisible by $34560$ PS. a) original from Albania b) modified by problem selecting committee

2017 China Western Mathematical Olympiad, 7
Tags: number theory
Let $n=2^{\alpha} \cdot q$ be a positive integer, where $\alpha$ is a nonnegative integer and $q$ is an odd number. Show that for any positive integer $m$, the number of integer solutions to the equation $x_1^2+x_2^2+\cdots +x_n^2=m$ is divisible by $2^{\alpha +1}$.

2023 All-Russian Olympiad, 5
Tags: number theory
Initially, $10$ ones are written on a blackboard. Grisha and Gleb are playing game, by taking turns; Grisha goes first. On one move Grisha squares some $5$ numbers on the board. On his move, Gleb picks a few (perhaps none) numbers on the board and increases each of them by $1$. If in $10,000$ moves on the board a number divisible by $2023$ appears, Gleb wins, otherwise Grisha wins. Which of the players has a winning strategy?

2006 Costa Rica - Final Round, 2
Tags: floor function , modular arithmetic , number theory
Let $n$ be a positive integer, and let $p$ be a prime, such that $n>p$. Prove that : \[ \displaystyle \binom np \equiv \left\lfloor\frac{n}{p}\right\rfloor \ \pmod p. \]

2022 China Team Selection Test, 2
Tags: floor function , number theory , algebra
Two positive real numbers $\alpha, \beta$ satisfies that for any positive integers $k_1,k_2$, it holds that $\lfloor k_1 \alpha \rfloor \neq \lfloor k_2 \beta \rfloor$, where $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$. Prove that there exist positive integers $m_1,m_2$ such that $\frac{m_1}{\alpha}+\frac{m_2}{\beta}=1$.

2011 Dutch IMO TST, 1
Tags: diophantine equation , number theory
Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\sqrt{x - y}$.

2021 LMT Spring, B4
Tags: number theory
Set $S$ contains exactly $36$ elements in the form of $2^m \cdot 5^n$ for integers $ 0 \le m,n \le 5$. Two distinct elements of $S$ are randomly chosen. Given that the probability that their product is divisible by $10^7$ is $a/b$, where $a$ and $b$ are relatively prime positive integers, find $a +b$. [i]Proposed by Ada Tsui[/i]

2003 Iran MO (3rd Round), 3
Tags: prime number , number theory
assume that A is a finite subset of prime numbers, and a is an positive integer. prove that there are only finitely many positive integers m s.t: prime divisors of a^m-1 are contained in A.

2023 Bangladesh Mathematical Olympiad, P3
Tags: function , number theory
For any positive integer $n$, define $f(n)$ to be the smallest positive integer that does not divide $n$. For example, $f(1)=2$, $f(6)=4$. Prove that for any positive integer $n$, either $f(f(n))$ or $f(f(f(n)))$ must be equal to $2$.

1995 Belarus Team Selection Test, 3
Tags: sequence , number theory , function , functional equation
Show that there is no infinite sequence an of natural numbers such that \[a_{a_n}=a_{n+1}a_{n-1}-a_{n}^2\] for all $n\geq 2$

2015 Switzerland - Final Round, 2
Tags: number theory
Find all pairs $(m,p)$ of natural numbers , such that $p$ is a prime and \[2^mp^2+27\] is the third power of a natural numbers

2007 Hong kong National Olympiad, 4
Tags: floor function , number theory
find all positive integer pairs $(m,n)$,satisfies: (1)$gcd(m,n)=1$,and $m\le\ 2007$ (2)for any $k=1,2,...2007$,we have $[\frac{nk}{m}]=[\sqrt{2}k]$

2006 Tournament of Towns, 5
Tags: combinatorics , number theory
Prove that one can find infinite number of distinct pairs of integers such that every digit of each number is no less than $7$ and the product of two numbers in each pair is also a number with all its digits being no less than $7$. (6)

2020 Turkey Team Selection Test, 1
Tags: number theory
Find all pairs of $(a,b)$ positive integers satisfying the equation: $$\frac {a^3+b^3}{ab+4}=2020$$

2003 Finnish National High School Mathematics Competition, 4
Tags: modular arithmetic , number theory
Find pairs of positive integers $(n, k)$ satisfying \[(n + 1)^k - 1 = n!\]