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 Stanford Mathematics Tournament, 13
Tags: number theory
A number $N$ has $2009$ positive factors. What is the maximum number of positive factors that $N^2$ could have?

2020 Estonia Team Selection Test, 3
Tags: number theory , remainder
The prime numbers $p$ and $q$ and the integer $a$ are chosen such that $p> 2$ and $a \not\equiv 1$ (mod $q$), but $a^p \equiv 1$ (mod $q$). Prove that $(1 + a^1)(1 + a^2)...(1 + a^{p - 1})\equiv 1$ (mod $q$) .

2004 ITAMO, 5
Tags: number theory
Decide if the following statement is true or false: For every sequence $\{x_n\}_{n\in \mathbb{N}}$ of non-negative real numbers, there exist sequences $\{a_n\}_{n\in\mathbb{N}}$ and $\{b_n\}_{n\in\mathbb{N}}$ of non-negative real numbers such that: (a) $x_n = a_n + b_n$ for all $n$; (b) $a_1 + \cdots + a_n \le n$ for infinitely many values of $n$; (c) $b_1 + \cdots + b_n \le n$ for infinitely many values of $n$.

2022/2023 Tournament of Towns, P6
Tags: number theory , sum of digits
Peter added a positive integer $M{}$ to a positive integer $N{}$ and noticed that the sum of the digits of the resulting integer is the same as the sum of the digits of $N{}$. Then he added $M{}$ to the result again, and so on. Will Peter eventually get a number with the same digit sum as the number $N{}$ again?

2023 Brazil Team Selection Test, 2
Tags: number theory , sequence
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

1999 Bosnia and Herzegovina Team Selection Test, 5
Tags: set , product , sum , number theory , combinatorics
For any nonempty set $S$, we define $\sigma(S)$ and $\pi(S)$ as sum and product of all elements from set $S$, respectively. Prove that $a)$ $\sum \limits_{} \frac{1}{\pi(S)} =n$ $b)$ $\sum \limits_{} \frac{\sigma(S)}{\pi(S)} =(n^2+2n)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(n+1)$ where $\sum$ denotes sum by all nonempty subsets $S$ of set $\{1,2,...,n\}$

2015 Kyiv Math Festival, P3
Tags: positive integer , number theory , sum
Is it true that every positive integer greater than $50$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?

2021 HMNT, 6
Tags: number theory
Let $n$ be the answer to this problem. $a$ and $b$ are positive integers satisfying $$3a + 5b \equiv 19 \,\,\, (mod \,\,\, n + 1)$$ $$4a + 2b \equiv 25 \,\,\, (mod \,\,\, n + 1)$$ Find $ 2a + 6b$.

1979 IMO Longlists, 19
Tags: number theory
For $k = 1, 2, \ldots$ consider the $k$-tuples $(a_1, a_2, \ldots, a_k)$ of positive integers such that \[a_1 + 2a_2 + \cdots + ka_k = 1979.\] Show that there are as many such $k$-tuples with odd $k$ as there are with even $k$.

1989 Tournament Of Towns, (226) 4
Tags: number theory , diophantine , diophantine equation
Find the positive integer solutions of the equation $$ x+ \frac{1}{y+ \frac{1}{z}}= \frac{10}{7}$$ (G. Galperin)

2006 IMO Shortlist, 3
Tags: calculus , floor function , number theory , sequence , summation
We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by \[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right), \] where $\lfloor x\rfloor$ denotes the integer part of $x$. [b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often. [b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often. [i]Proposed by Johan Meyer, South Africa[/i]

2025 Turkey Team Selection Test, 4
Tags: number theory , coprime integers
Let $a,b,c$ be given pairwise coprime positive integers where $a>bc$. Let $m<n$ be positive integers. We call $m$ to be a grandson of $n$ if and only if, for all possible piles of stones whose total mass adds up to $n$ and consist of stones with masses $a,b,c$, it's possible to take some of the stones out from this pile in a way that in the end, we can obtain a new pile of stones with total mass of $m$. Find the greatest possible number that doesn't have any grandsons.

2013 India Regional Mathematical Olympiad, 3
Tags: number theory , easy , nice
Find all primes $p$ and $q$ such that $p$ divides $q^2-4$ and $q$ divides $p^2-1$.

1989 IMO Longlists, 45
Tags: number theory , algebra , logarithm
Let $ (\log_2(x))^2 \minus{} 4 \cdot \log_2(x) \minus{} m^2 \minus{} 2m \minus{} 13 \equal{} 0$ be an equation in $ x.$ Prove: [b](a)[/b] For any real value of $ m$ the equation has two distinct solutions. [b](b)[/b] The product of the solutions of the equation does not depend on $ m.$ [b](c)[/b] One of the solutions of the equation is less than 1, while the other solution is greater than 1. Find the minimum value of the larger solution and the maximum value of the smaller solution.

2015 Balkan MO Shortlist, N5
Tags: number theory , maximum , power of 2 , divide , product
For a positive integer $s$, denote with $v_2(s)$ the maximum power of $2$ that divides $s$. Prove that for any positive integer $m$ that: $$v_2\left(\prod_{n=1}^{2^m}\binom{2n}{n}\right)=m2^{m-1}+1.$$ (FYROM)

2020 Czech-Austrian-Polish-Slovak Match, 5
Tags: number theory , diophantine equation
Let $n$ be a positive integer and let $d(n)$ denote the number of ordered pairs of positive integers $(x,y)$ such that $(x+1)^2-xy(2x-xy+2y)+(y+1)^2=n$. Find the smallest positive integer $n$ satisfying $d(n) = 61$. (Patrik Bak, Slovakia)

2022 IMO Shortlist, N6
Tags: number theory , prime factorization , function , prime number , prime
Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2009 IMAC Arhimede, 5
Tags: number theory , diophantine equation , diophantine , exponential equation , exponential
Find all natural numbers $x$ and $y$ such that $x^y-y^x=1$ .

2015 China Northern MO, 4
Tags: number theory , combinatorics
If the set $S = \{1,2,3,…,16\}$ is partitioned into $n$ subsets, there must be a subset in which elements $a, b, c$ (can be the same) exist, satisfying $a+ b=c$. Find the maximum value of $n$.

2023 Durer Math Competition (First Round), 3
Tags: combinatorics , number theory
Let $n \ge 3$ be an integer and $A$ be a subset of the real numbers of size n. Denote by $B$ the set of real numbers that are of the form $ x \cdot y$, where $x, y \in A$ and $x\ne y$. At most how many distinct positive primes could $B$ contain (depending on $n$)?

2009 All-Russian Olympiad, 3
Tags: number theory
Given are positive integers $ n>1$ and $ a$ so that $ a>n^2$, and among the integers $ a\plus{}1, a\plus{}2, \ldots, a\plus{}n$ one can find a multiple of each of the numbers $ n^2\plus{}1, n^2\plus{}2, \ldots, n^2\plus{}n$. Prove that $ a>n^4\minus{}n^3$.

2021 Middle European Mathematical Olympiad, 8
Tags: perfect square , number theory , digit
Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$.

2019 Moroccan TST, 3
Tags: number theory , diophantine equation
Find all couples $(x,y)$ over the positive integers such that: $7^x+x^4+47=y^2$

2007 Junior Macedonian Mathematical Olympiad, 1
Tags: number theory
Does there exist a positive integer $n$, such that the number $n(n + 1)(n + 2)$ is the square of a positive integer?

1999 Finnish National High School Mathematics Competition, 3
Tags: modular arithmetic , geometric series , algebra , difference of squares , special factorizations , number theory
Determine how many primes are there in the sequence \[101, 10101, 1010101 ....\]