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 JBMO Shortlist, 5
Tags: number theory , sum of digits
Find the least positive integer such that the sum of its digits is $2011$ and the product of its digits is a power of $6$.

2022 Belarusian National Olympiad, 11.1
Tags: sequence , number theory
A sequence of positive integer numbers $a_1,a_2,\ldots$ for $i \geq 3$ satisfies $$a_{i+1}=a_i+gcd(a_{i-1},a_{i-2})$$ Prove that there exist two positive integer numbers $N, M$, such that $a_{n+1}-a_n=M$ for all $n \geq N$

1977 IMO Longlists, 26
Tags: number theory
Let $p$ be a prime number greater than $5.$ Let $V$ be the collection of all positive integers $n$ that can be written in the form $n = kp + 1$ or $n = kp - 1 \ (k = 1, 2, \ldots).$ A number $n \in V$ is called [i]indecomposable[/i] in $V$ if it is impossible to find $k, l \in V$ such that $n = kl.$ Prove that there exists a number $N \in V$ that can be factorized into indecomposable factors in $V$ in more than one way.

2009 IMO Shortlist, 5
Tags: number theory , permutation , algebra , polynomial
Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$. [i]Proposed by Jozsef Pelikan, Hungary[/i]

2022 Pan-American Girls' Math Olympiad, 2
Tags: diophantine equation , number theory
Find all ordered triplets $(p,q,r)$ of positive integers such that $p$ and $q$ are two (not necessarily distinct) primes, $r$ is even, and \[p^3+q^2=4r^2+45r+103.\]

2013 Finnish National High School Mathematics Competition, 5
Tags: search , quadratic , finland , modular arithmetic , number theory
Find all integer triples $(m,p,q)$ satisfying \[2^mp^2+1=q^5\] where $m>0$ and both $p$ and $q$ are prime numbers.

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 $1$.

2014 NIMO Problems, 2
Tags: symmetry , number theory , perimeter , relatively prime , probability , analytic geometry , geometry
Two points $A$ and $B$ are selected independently and uniformly at random along the perimeter of a unit square with vertices at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. The probability that the $y$-coordinate of $A$ is strictly greater than the $y$-coordinate of $B$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. [i]Proposed by Rajiv Movva[/i]

2012 Stars of Mathematics, 2
Tags: number theory
Prove the value of the expression $$\displaystyle \dfrac {\sqrt{n + \sqrt{0}} + \sqrt{n + \sqrt{1}} + \sqrt{n + \sqrt{2}} + \cdots + \sqrt{n + \sqrt{n^2-1}} + \sqrt{n + \sqrt{n^2}}} {\sqrt{n - \sqrt{0}} + \sqrt{n - \sqrt{1}} + \sqrt{n - \sqrt{2}} + \cdots + \sqrt{n - \sqrt{n^2-1}} + \sqrt{n - \sqrt{n^2}}}$$ is constant over all positive integers $n$. ([i]Folklore (also Philippines Olympiad)[/i])

2017 IFYM, Sozopol, 4
Tags: number theory
Find all pairs of natural numbers $(a,n)$, $a\geq n \geq 2,$ for which $a^n+a-2$ is a power of $2$.

2016 Taiwan TST Round 2, 2
Tags: divisibility , number theory
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

2019 Silk Road, 3
Tags: number theory , eulers function , diophantine equation , relatively prime
Find all pairs of $ (a, n) $ natural numbers such that $ \varphi (a ^ n + n) = 2 ^ n. $ ($ \varphi (n) $ is the Euler function, that is, the number of integers from $1$ up to $ n $, relative prime to $ n $)

1978 Swedish Mathematical Competition, 2
Tags: algebra , number theory , sum , digit
Let $s_m$ be the number $66\cdots 6$ with $m$ digits $6$. Find \[ s_1 + s_2 + \cdots + s_n \]

Russian TST 2018, P1
Tags: number theory
Let $k>1$ be the given natural number and $p\in \mathbb{P}$ such that $n=kp+1$ is composite number. Given that $n\mid 2^{n-1}-1.$ Prove that $n<2^k.$

2005 Rioplatense Mathematical Olympiad, Level 3, 3
Tags: number theory
Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$. (Note: $n$ is written in the usual base ten notation.)

2020 May Olympiad, 2
Tags: number theory
Paul wrote the list of all four-digit numbers such that the hundreds digit is $5$ and the tens digit is $7$. For example, $1573$ and $7570$ are on Paul's list, but $2754$ and $571$ are not. Find the sum of all the numbers on Pablo's list. $Note$. The numbers on Pablo's list cannot start with zero.

1996 Yugoslav Team Selection Test, Problem 1
Tags: combinatorics , set , number theory
Let $\mathfrak F=\{A_1,A_2,\ldots,A_n\}$ be a collection of subsets of the set $S=\{1,2,\ldots,n\}$ satisfying the following conditions: (a) Any two distinct sets from $\mathfrak F$ have exactly one element in common; (b) each element of $S$ is contained in exactly $k$ of the sets in $\mathfrak F$. Can $n$ be equal to $1996$?

2010 Saudi Arabia Pre-TST, 1.3
Tags: divide , digit , divisible , number theory
1) Let $a$ and $b$ be relatively prime positive integers. Prove that there is a positive integer $n$ such that $1 \le n \le b$ and $b$ divides $a^n - 1$. 2) Prove that there is a multiple of $7^{2010}$ of the form $99... 9$ ($n$ nines), for some positive integer $n$ not exceeding $7^{2010}$.

2011 Tuymaada Olympiad, 4
Tags: number theory , euler , quadratic
Prove that, among $100000$ consecutive $100$-digit positive integers, there is an integer $n$ such that the length of the period of the decimal expansion of $\frac1n$ is greater than $2011$.

2022 Tuymaada Olympiad, 2
Tags: number theory , polynomial
Given are integers $a, b, c$ and an odd prime $p.$ Prove that $p$ divides $x^2 + y^2 + ax + by + c$ for some integers $x$ and $y.$ [i](A. Golovanov )[/i]

2012 International Zhautykov Olympiad, 1
Tags: induction , number theory , function
Do there exist integers $m, n$ and a function $f\colon \mathbb R \to \mathbb R$ satisfying simultaneously the following two conditions? $\bullet$ i) $f(f(x))=2f(x)-x-2$ for any $x \in \mathbb R$; $\bullet$ ii) $m \leq n$ and $f(m)=n$.

2017 CHMMC (Fall), 5
Tags: perfect square , number theory
Find the number of primes $p$ such that $p! + 25p$ is a perfect square.

2017 Singapore Senior Math Olympiad, 5
Tags: algebra , number theory , combinatorics , arithmetic progression , sequence , integer
Given $7$ distinct positive integers, prove that there is an infinite arithmetic progression of positive integers $a, a + d, a + 2d,..$ with $a < d$, that contains exactly $3$ or $4$ of the $7$ given integers.

1957 Poland - Second Round, 1
Tags: integer , number theory
Prove that if $ n $ is an integer, then $$ \frac{n^5}{120} - \frac{n^3}{24} + \frac{n}{30}$$ is also an integer.

2005 Estonia Team Selection Test, 3
Tags: number theory , diophantine equation , diophantine
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.