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: 1766

1997 Pre-Preparation Course Examination, 4
Tags: function , modular arithmetic , number theory proposed , number theory , algebra
Let $f : \mathbb N \to \mathbb N$ be an injective function such that there exists a positive integer $k$ for which $f(n) \leq n^k$. Prove that there exist infinitely many primes $q$ such that the equation $f(x) \equiv 0 \pmod q$ has a solution in prime numbers.

1994 Romania TST for IMO, 2:
Tags: modular arithmetic , number theory proposed , number theory
Let $ n$ be an odd positive integer. Prove that $((n-1)^n+1)^2$ divides $ n(n-1)^{(n-1)^n+1}+n$.

2014 CentroAmerican, 3
Tags: modular arithmetic , number theory proposed , number theory
A positive integer $n$ is [i]funny[/i] if for all positive divisors $d$ of $n$, $d+2$ is a prime number. Find all funny numbers with the largest possible number of divisors.

2011 China Second Round Olympiad, 2
Tags: algebra , polynomial , geometry , geometric transformation , number theory proposed , number theory
For any integer $n\ge 4$, prove that there exists a $n$-degree polynomial $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ satisfying the two following properties: [b](1)[/b] $a_i$ is a positive integer for any $i=0,1,\ldots,n-1$, and [b](2)[/b] For any two positive integers $m$ and $k$ ($k\ge 2$) there exist distinct positive integers $r_1,r_2,...,r_k$, such that $f(m)\ne\prod_{i=1}^{k}f(r_i)$.

2009 India IMO Training Camp, 2
Tags: induction , number theory proposed , number theory
Let us consider a simle graph with vertex set $ V$. All ordered pair $ (a,b)$ of integers with $ gcd(a,b) \equal{} 1$, are elements of V. $ (a,b)$ is connected to $ (a,b \plus{} kab)$ by an edge and to $ (a \plus{} kab,b)$ by another edge for all integer k. Prove that for all $ (a,b)\in V$, there exists a path fromm $ (1,1)$ to $ (a,b)$.

2009 China Team Selection Test, 2
Tags: algebra , polynomial , modular arithmetic , number theory , prime numbers , number theory proposed
Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$

2010 Contests, 1
Tags: function , logarithms , number theory , prime numbers , arithmetic sequence , number theory proposed
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that \[f(a+1), f(a+2), \dots, f(a+n)\] form an arithmetic progression.

2011 Brazil Team Selection Test, 1
Tags: inequalities , quadratics , number theory proposed , number theory
Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.

2001 Vietnam National Olympiad, 2
Tags: number theory , relatively prime , number theory proposed
Let $N = 6^{n}$, where $n$ is a positive integer, and let $M = a^{N}+b^{N}$, where $a$ and $b$ are relatively prime integers greater than $1. M$ has at least two odd divisors greater than $1$ are $p,q$. Find the residue of $p^{N}+q^{N}\mod 6\cdot 12^{n}$.

2000 JBMO ShortLists, 6
Tags: number theory proposed , number theory
Find all four-digit numbers such that when decomposed into prime factors, each number has the sum of its prime factors equal to the sum of the exponents.

2011 China Team Selection Test, 2
Tags: modular arithmetic , number theory proposed , number theory
Let $n>1$ be an integer, and let $k$ be the number of distinct prime divisors of $n$. Prove that there exists an integer $a$, $1<a<\frac{n}{k}+1$, such that $n \mid a^2-a$.

2010 Polish MO Finals, 2
Tags: function , algebra , polynomial , number theory proposed , number theory
Positive rational number $a$ and $b$ satisfy the equality \[a^3 + 4a^2b = 4a^2 + b^4.\] Prove that the number $\sqrt{a}-1$ is a square of a rational number.

2000 JBMO ShortLists, 8
Tags: number theory proposed , number theory
Find all positive integers $a,b$ for which $a^4+4b^4$ is a prime number.

2005 Taiwan TST Round 3, 2
Tags: number theory proposed , number theory
Find all primes $p$ such that the number of distinct positive factors of $p^2+2543$ is less than 16.

1991 IberoAmerican, 5
Tags: algebra , polynomial , quadratics , number theory proposed , number theory
Let $P(x,\, y)=2x^{2}-6xy+5y^{2}$. Let us say an integer number $a$ is a value of $P$ if there exist integer numbers $b$, $c$ such that $P(b,\, c)=a$. a) Find all values of $P$ lying between 1 and 100. b) Show that if $r$ and $s$ are values of $P$, then so is $rs$.

2014 ELMO Shortlist, 4
Tags: function , number theory proposed , number theory
Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]

2021 Kazakhstan National Olympiad, 5
Tags: Kazakhstan , Diophantine equation , number theory proposed , algebra , number theory
Let $a$ be a positive integer. Prove that for any pair $(x,y)$ of integer solutions of equation $$x(y^2-2x^2)+x+y+a=0$$ we have: $$|x| \leqslant a+\sqrt{2a^2+2}$$

1996 All-Russian Olympiad, 3
Tags: number theory , relatively prime , number theory proposed
Find all natural numbers $n$, such that there exist relatively prime integers $x$ and $y$ and an integer $k > 1$ satisfying the equation $3^n =x^k + y^k$. [i]A. Kovaldji, V. Senderov[/i]

2009 Ukraine National Mathematical Olympiad, 4
Tags: number theory proposed , number theory
[b]а)[/b] Prove that for any positive integer $n$ there exist a pair of positive integers $(m, k)$ such that \[{k + m^k + n^{m^k}} = 2009^n.\] [b]b)[/b] Prove that there are infinitely many positive integers $n$ for which there is only one such pair.

2014 Argentina Cono Sur TST, 1
Tags: number theory proposed , number theory
A positive integer $N$ is written on a board. In a step, the last digit $c$ of the number on the board is erased, and after this, the remaining number $m$ is erased and replaced with $|m-3c|$ (for example, if the number $1204$ is on the board, after one step, we will replace it with $120 - 3 \cdot 4 = 108$). We repeat this until the number on the board has only one digit. Find all positive integers $N$ such that after a finite number of steps, the remaining one-digit number is $0$.

2007 Bulgaria Team Selection Test, 2
Tags: inequalities , number theory proposed , number theory
Let $n,k$ be positive integers such that $n\geq2k>3$ and $A= \{1,2,...,n\}.$ Find all $n$ and $k$ such that the number of $k$-element subsets of $A$ is $2n-k$ times bigger than the number of $2$-element subsets of $A.$

2012 ELMO Shortlist, 9
Tags: number theory proposed , number theory
Are there positive integers $m,n$ such that there exist at least $2012$ positive integers $x$ such that both $m-x^2$ and $n-x^2$ are perfect squares? [i]David Yang.[/i]

2005 Flanders Math Olympiad, 3
Tags: geometry , complex numbers , number theory , prime numbers , number theory proposed
Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.

2009 All-Russian Olympiad Regional Round, 11.7
Tags: number theory , greatest common divisor , number theory proposed
$a$, $b$ and $c$ are positive integers with $\textrm{gcd}(a,b,c)=1$. Is it true that there exist a positive integer $n$ such that $a^k+b^k+c^k$ is not divisible by $2^n$ for all $k$?

2011 ELMO Shortlist, 2
Tags: modular arithmetic , number theory proposed , number theory
Let $p\ge5$ be a prime. Show that \[\sum_{k=0}^{(p-1)/2}\binom{p}{k}3^k\equiv 2^p - 1\pmod{p^2}.\] [i]Victor Wang.[/i]