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

2022 LMT Spring, 5
Tags: number theory
Find the sum $$\sum^{2020}_{n=1} \gcd (n^3 -2n^2 +2021,n^2 -3n +3).$$

2024 Regional Competition For Advanced Students, 4
Tags: number theory , divisible , divide
Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$. [i](Walther Janous)[/i]

2010 Contests, 2
Tags: number theory , power of 2 , consecutive
A number is called polite if it can be written as $ m + (m+1)+...+ n$, for certain positive integers $ m <n$ . For example: $18$ is polite, since $18 =5 + 6 + 7$. A number is called a power of two if it can be written as $2^{\ell}$ for some integer $\ell \ge 0$. (a) Show that no number is both polite and a power of two. (b) Show that every positive integer is polite or a power of two.

2015 Estonia Team Selection Test, 7
Tags: number theory , digit , consecutive , power
Prove that for every prime number $p$ and positive integer $a$, there exists a natural number $n$ such that $p^n$ contains $a$ consecutive equal digits.

2022 Regional Olympiad of Mexico West, 1
Tags: number theory , divide
Find a subset of $\{1,2, ...,2022\}$ with maximum number of elements such that it does not have two elements $a$ and $b$ such that $a = b + d$ for some divisor $d$ of $b$.

2000 Italy TST, 1
Tags: number theory
Determine all triples $(x,y,z)$ of positive integers such that \[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997} \]

2006 MOP Homework, 1
Tags: number theory
Let $n$ be an integer greater than $1$, and let $a_1$, $a_2$, ..., $a_n$ be not all identical positive integers. Prove that there are infinitely many primes $p$ such that $p$ divides $a_1^k+a_2^k+...+a_n^k$ for some positive integer $k$.

2002 France Team Selection Test, 2
Tags: number theory
Consider the set $S$ of integers $k$ which are products of four distinct primes. Such an integer $k=p_1p_2p_3p_4$ has $16$ positive divisors $1=d_1<d_2<\ldots <d_{15}<d_{16}=k$. Find all elements of $S$ less than $2002$ such that $d_9-d_8=22$.

2007 Romania Team Selection Test, 2
Tags: limit , probability , number theory
The world-renowned Marxist theorist [i]Joric[/i] is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer $n$, he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the [i]defect[/i] of the number $n$. Determine the average value of the defect (over all positive integers), that is, if we denote by $\delta(n)$ the defect of $n$, compute \[\lim_{n \rightarrow \infty}\frac{\sum_{k = 1}^{n}\delta(k)}{n}.\] [i]Iurie Boreico[/i]

2007 Indonesia TST, 3
Tags: inequalities , floor function , diophantine equation , number theory
For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.

2019 Thailand TST, 3
Tags: function , number theory
Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.

2019 Estonia Team Selection Test, 9
Tags: number theory , arithmetic function , number of divisors , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2011 NIMO Problems, 1
Tags: probability , relatively prime , number theory
A jar contains 4 blue marbles, 3 green marbles, and 5 red marbles. If Helen reaches in the jar and selects a marble at random, then the probability that she selects a red marble can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2023 IRN-SGP-TWN Friendly Math Competition, 6
Tags: functional equation , polynomial , polynomial with integer coeffi , algebra , number theory
$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\] (We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.) [i]Proposed by the4seasons.[/i]

2021 May Olympiad, 5
Tags: number theory
Bob writes $36$ consecutive positive integers in a white paper(in ascending order), next he computes the sum of digits of each one of $36$ numbers(in the order) and writes the first $16$ results in a red paper and the last $10$ results in a blue paper. Determine if Bob can choose the $36$ integers, such that the sum of the numbers in the red paper is less than or equal to sum of the numbers in the blue paper.

2024 Caucasus Mathematical Olympiad, 6
Tags: number theory
Given is a permutation of $1, 2, \ldots, 2023, 2024$ and two positive integers $a, b$, such that for any two adjacent numbers, at least one of the following conditions hold: 1) their sum is $a$; 2) the absolute value of their difference is $b$. Find all possible values of $b$.

2014 Serbia National Math Olympiad, 4
Tags: number theory
We call natural number $n$ [i]$crazy$[/i] iff there exist natural numbers $a$, $b >1$ such that $n=a^b+b$. Whether there exist $2014$ consecutive natural numbers among which are $2012$ [i]$crazy$[/i] numbers? [i]Proposed by Milos Milosavljevic[/i]

2005 MOP Homework, 3
Tags: relatively prime , number theory
For any positive integer $n$, the sum $1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}$ is written in the lowest form $\frac{p_n}{q_n}$; that is, $p_n$ and $q_n$ are relatively prime positive integers. Find all $n$ such that $p_n$ is divisible by $3$.

1997 Moldova Team Selection Test, 1
Tags: number theory
Let $a$ and $b$ be two odd positive integers. Define the sequence $(x_n)_{n\in\mathbb{N}}$ as such: $x_1=a, x_2=b,$ for every $n\geq3$ the term $x_n{}$ is the greatest odd integer of $x_{n-1}+x_{n-2}$. Show that starting with a term, all the following terms are constant.

2008 Junior Balkan Team Selection Tests - Romania, 4
Tags: floor function , irrational number , number theory
Let $ a,b$ be real nonzero numbers, such that number $ \lfloor an \plus{} b \rfloor$ is an even integer for every $ n \in \mathbb{N}$. Prove that $ a$ is an even integer.

1996 Brazil National Olympiad, 1
Tags: algorithm , number theory
Show that there exists infinite triples $(x,y,z) \in N^3$ such that $x^2+y^2+z^2=3xyz$.

2007 Alexandru Myller, 1
Tags: algebra , modular arithmetic , number theory
[b]a)[/b] Show that $ n^2+2n+2007 $ is squarefree for any natural number $ n. $ [b]b)[/b] Prove that for any natural number $ k\ge 2 $ there is a nonnegative integer $ m $ such that $ m^2+2m+2k $ is a perfect square.

2004 Tournament Of Towns, 4
Tags: modular arithmetic , number theory
A positive integer $a > 1$ is given (in decimal notation). We copy it twice and obtain a number $b = \overline{aa}$ which happens to be a multiple of $a^2$. Find all possible values of $b/a^2$.

2015 India PRMO, 6
Tags: perfect square , number theory , digit , sum of digits
$6.$ How many two digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square $?$

2020 Italy National Olympiad, #2
Tags: number theory
Determine all the pairs $(a,b)$ of positive integers, such that all the following three conditions are satisfied: 1- $b>a$ and $b-a$ is a prime number 2- The last digit of the number $a+b$ is $3$ 3- The number $ab$ is a square of an integer.