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 Al-Khwarizmi IJMO, 2
Tags: number theory
For how many $x \in \{1,2,3,\dots, 2024\}$ is it possible that [i]Bekhzod[/i] summed $2024$ non-negative consecutive integers, [i]Ozod[/i] summed $2024+x$ non-negative consecutive integers and they got the same result? [i]Proposed by Marek Maruin, Slovakia[/i]

2000 Czech and Slovak Match, 3
Tags: number theory , divisor , power of 2
Let $n$ be a positive integer. Prove that $n$ is a power of two if and only if there exists an integer $m$ such that $2^n-1$ is a divisor of $m^2 +9$.

2021 Science ON Juniors, 1
Tags: number theory
Let $a,p,q\in \mathbb{Z}_{\ge 1}$ be such that $a$ is a perfect square, $a=pq$ and $$2021~|~p^3+q^3+p^2q+pq^2.$$ Prove that $2021$ divides $\sqrt a$.\\ \\ [i](Cosmin Gavrilă)[/i]

2013 Romanian Masters In Mathematics, 2
Tags: floor function , induction , number theory
Given a positive integer $k\geq2$, set $a_1=1$ and, for every integer $n\geq 2$, let $a_n$ be the smallest solution of equation \[x=1+\sum_{i=1}^{n-1}\left\lfloor\sqrt[k]{\frac{x}{a_i}}\right\rfloor\] that exceeds $a_{n-1}$. Prove that all primes are among the terms of the sequence $a_1,a_2,\ldots$

2005 Korea - Final Round, 5
Tags: modular arithmetic , number theory
Find all positive integers $m$ and $n$ such that both $3^{m}+1$ and $3^{n}+1$ are divisible by $mn$.

2017 China Team Selection Test, 5
Tags: number theory , algebra , polynomial
Let $ \varphi(x)$ be a cubic polynomial with integer coefficients. Given that $ \varphi(x)$ has have 3 distinct real roots $u,v,w $ and $u,v,w $ are not rational number. there are integers $ a, b,c$ such that $u=av^2+bv+c$. Prove that $b^2 -2b -4ac - 7$ is a square number .

2002 Kazakhstan National Olympiad, 4
Tags: number theory , remainder
Prove that there is a set $ A $ consisting of $2002$ different natural numbers satisfying the condition: for each $ a \in A $, the product of all numbers from $ A $, except $ a $, when divided by $ a $ gives the remainder $1$.

1957 Miklós Schweitzer, 8
Tags: number theory
[b]8.[/b] Find all integers $a>1$ for which the least (integer) solution $n$ of the congruence $a^{n} \equiv 1 \pmod{p}$ differs from 6 (p is any prime number). [b](N. 9)[/b]

2011 Middle European Mathematical Olympiad, 8
Tags: modular arithmetic , greatest common divisor , arithmetic sequence , relatively prime , algebra , system of equations , number theory
We call a positive integer $n$ [i]amazing[/i] if there exist positive integers $a, b, c$ such that the equality \[n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)\] holds. Prove that there exist $2011$ consecutive positive integers which are [i]amazing[/i]. [b]Note.[/b] By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.

2010 Cono Sur Olympiad, 1
Tags: number theory
Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that: [list] [*]The sum of the fractions is equal to $2$. [*]The sum of the numerators of the fractions is equal to $1000$. [/list] In how many ways can Pedro do this?

2018 Thailand TST, 2
Tags: number theory
Let $(x_1,x_2,\dots,x_{100})$ be a permutation of $(1,2,...,100)$. Define $$S = \{m \mid m\text{ is the median of }\{x_i, x_{i+1}, x_{i+2}\}\text{ for some }i\}.$$ Determine the minimum possible value of the sum of all elements of $S$.

2025 Malaysian IMO Team Selection Test, 1
Tags: number theory
Determine all integers $n\ge 2$ such that for any two infinite sequences of positive integers $a_1<a_2< \cdots $ and $b_1, b_2, \cdots$, such that $a_i\mid a_j$ for all $i<j$, there always exists a real number $c$ such that $$\lfloor{ca_i}\rfloor \equiv b_i \pmod {n}$$ for all $i\ge 1$. [i]Proposed by Wong Jer Ren & Ivan Chan Kai Chin[/i]

2021 CMIMC, 8
Tags: algebra , number theory
Determine the number of functions $f$ from the integers to $\{1,2,\cdots,15\}$ which satisfy $$f(x)=f(x+15)$$ and $$f(x+f(y))=f(x-f(y))$$ for all $x,y$. [i]Proposed by Vijay Srinivasan[/i]

2001 May Olympiad, 4
Tags: number theory , prime number
Using only prime numbers, a set is formed with the following conditions: Any one-digit prime number can be in the set. For a prime number with more than one digit to be in the set, the number that results from deleting only the first digit and also the number that results from deleting only the last digit must be in the set. Write, of the sets that meet these conditions, the one with the greatest number of elements. Justify why there cannot be one with more elements. Remember that the number $1$ is not prime.

2016 Romania Team Selection Tests, 2
Tags: number theory
Given a positive integer $k$ and an integer $a\equiv 3 \pmod{8}$, show that $a^m+a+2$ is divisible by $2^k$ for some positive integer $m$.

2018 Junior Balkan Team Selection Tests - Romania, 1
Tags: sum , algebra , number theory
Determine the positive integers $n \ge 3$ such that, for every integer $m \ge 0$, there exist integers $a_1, a_2,..., a_n$ such that $a_1 + a_2 +...+ a_n = 0$ and $a_1a_2 + a_2a_3 + ...+a_{n-1}a_n + a_na_1 = -m$ Alexandru Mihalcu

2003 Federal Competition For Advanced Students, Part 2, 1
Tags: algebra , polynomial , number theory
Consider the polynomial $P(n) = n^3 -n^2 -5n+ 2$. Determine all integers $n$ for which $P(n)^2$ is a square of a prime. [hide="Remark."]I'm not sure if the statement of this problem is correct, because if $P(n)^2$ be a square of a prime, then $P(n)$ should be that prime, and I don't think the problem means that.[/hide]

2011 Iran MO (3rd Round), 3
Tags: modular arithmetic , number theory
Let $k$ be a natural number such that $k\ge 7$. How many $(x,y)$ such that $0\le x,y<2^k$ satisfy the equation $73^{73^x}\equiv 9^{9^y} \pmod {2^k}$? [i]Proposed by Mahyar Sefidgaran[/i]

1985 Vietnam National Olympiad, 1
Tags: number theory
Find all pairs $ (x, y)$ of integers such that $ x^3 \minus{} y^3 \equal{} 2xy \plus{} 8$.

2015 Princeton University Math Competition, A6
Tags: number theory
For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$. What is the smallest positive integer $n$ such that \[\sum_{t \mid n} d(t)^3\]is divisible by $35$?

2019 Nepal TST, P1
Tags: number theory
Prove that there exist infinitely many pairs of different positive integers $(m, n)$ for which $m!n!$ is a square of an integer. [i]Proposed by Anton Trygub[/i]

2020 APMO, 5
Tags: algebra , number theory , algorithm
Let $n \geq 3$ be a fixed integer. The number $1$ is written $n$ times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers $a$ and $b$, replacing them with the numbers $1$ and $a+b$, then adding one stone to the first bucket and $\gcd(a, b)$ stones to the second bucket. After some finite number of moves, there are $s$ stones in the first bucket and $t$ stones in the second bucket, where $s$ and $t$ are positive integers. Find all possible values of the ratio $\frac{t}{s}$.

1995 Romania Team Selection Test, 1
Tags: number theory
The sequence $ (x_n)$ is defined by $ x_1\equal{}1,x_2\equal{}a$ and $ x_n\equal{}(2n\plus{}1)x_{n\minus{}1}\minus{}(n^2\minus{}1)x_{n\minus{}2}$ $ \forall n \geq 3$, where $ a \in N^*$.For which value of $ a$ does the sequence have the property that $ x_i|x_j$ whenever $ i<j$.

2013 NZMOC Camp Selection Problems, 12
Tags: prime divisor , divisor , prime , number theory , inequalities
For a positive integer $n$, let $p(n)$ denote the largest prime divisor of $n$. Show that there exist infinitely many positive integers m such that $p(m-1) < p(m) < p(m + 1)$.

2000 Switzerland Team Selection Test, 4
Tags: sum , sum of digits , number theory
Let $q(n)$ denote the sum of the digits of a natural number $n$. Determine $q(q(q(2000^{2000})))$.