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

2009 Indonesia TST, 1
Tags: algebra , polynomial , quadratics , number theory proposed , number theory
Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.

2011 APMO, 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.

2004 Iran MO (3rd Round), 20
Tags: algebra , polynomial , number theory proposed , number theory
$ p(x)$ is a polynomial in $ \mathbb{Z}[x]$ such that for each $ m,n\in \mathbb{N}$ there is an integer $ a$ such that $ n\mid p(a^m)$. Prove that $0$ or $1$ is a root of $ p(x)$.

2013 North Korea Team Selection Test, 6
Tags: induction , number theory proposed , number theory
Show that $ x^3 + x+ a^2 = y^2 $ has at least one pair of positive integer solution $ (x,y) $ for each positive integer $ a $.

2004 Nordic, 2
Tags: arithmetic sequence , number theory proposed , number theory
Show that there exist strictly increasing infinite arithmetic sequence of integers which has no numbers in common with the Fibonacci sequence.

2001 JBMO ShortLists, 4
Tags: quadratics , algebra , number theory proposed , number theory
The discriminant of the equation $x^2-ax+b=0$ is the square of a rational number and $a$ and $b$ are integers. Prove that the roots of the equation are integers.

2009 Tournament Of Towns, 3
Tags: symmetry , number theory proposed , number theory
Find all positive integers $a$ and $b$ such that $(a + b^2)(b + a^2) = 2^m$ for some integer $m.$ [i](6 points)[/i]

1973 Bundeswettbewerb Mathematik, 2
Tags: number theory proposed , number theory
We work in the decimal system and the following operations are allowed to be done with a positive integer: a) append $4$ at the end of the number. b) append $0$ at the end of the number. c) divide the number by $2$ if it's even. Show that starting with $4$, we can reach every positive integer by a finite number of these operations

2000 Spain Mathematical Olympiad, 1
Tags: modular arithmetic , floor function , number theory proposed , number theory
Find the largest integer $N$ satisfying the following two conditions: [b](i)[/b] $\left[ \frac N3 \right]$ consists of three equal digits; [b](ii)[/b] $\left[ \frac N3 \right] = 1 + 2 + 3 +\cdots + n$ for some positive integer $n.$

2004 Turkey Team Selection Test, 3
Tags: number theory proposed , number theory
Let $n$ be a positive integer. Determine integers, $n+1 \leq r \leq 3n+2$ such that for all integers $a_1,a_2,\dots,a_m,b_1,b_2,\dots,b_m$ satisfying the equations \[ a_1b_1^k+a_2b_2^k+\dots + a_mb_m^k=0 \] for every $1 \leq k \leq n$, the condition \[ r \mid a_1b_1^r+a_2b_2^r+\dots + a_mb_m^r=0 \] also holds.

2024 All-Russian Olympiad, 8
Tags: number theory , number theory proposed
Prove that there exists $c>0$ such that for any odd prime $p=2k+1$, the numbers $1^0, 2^1,3^2,\dots,k^{k-1}$ give at least $c\sqrt{p}$ distinct residues modulo $p$. [i]Proposed by M. Turevsky, I. Bogdanov[/i]

2003 Federal Competition For Advanced Students, Part 1, 1
Tags: number theory proposed , number theory
Find all triples of prime numbers $(p, q, r)$ such that $p^q + p^r$ is a perfect square.

2012 China Team Selection Test, 2
Tags: number theory proposed , number theory
For a positive integer $n$, denote by $\tau (n)$ the number of its positive divisors. For a positive integer $n$, if $\tau (m) < \tau (n)$ for all $m < n$, we call $n$ a good number. Prove that for any positive integer $k$, there are only finitely many good numbers not divisible by $k$.

2024 Baltic Way, 20
Tags: number theory , number theory proposed , Diophantine equation , Discriminant
Positive integers $a$, $b$ and $c$ satisfy the system of equations \begin{align*} (ab-1)^2&=c(a^2+b^2)+ab+1,\\ a^2+b^2&=c^2+ab. \end{align*} a) Prove that $c+1$ is a perfect square. b) Find all such triples $(a,b,c)$.

2004 Turkey MO (2nd round), 3
Tags: number theory proposed , number theory
[b](a)[/b] Determine if exist an integer $n$ such that $n^2 -k$ has exactly $10$ positive divisors for each $k = 1, 2, 3.$ [b](b) [/b]Show that the number of positive divisors of $n^2 -4$ is not $10$ for any integer $n.$

1981 Bundeswettbewerb Mathematik, 4
Tags: modular arithmetic , number theory proposed , number theory
Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.

2006 All-Russian Olympiad, 5
Tags: number theory proposed , number theory
Let $a_1$, $a_2$, ..., $a_{10}$ be positive integers such that $a_1<a_2<...<a_{10}$. For every $k$, denote by $b_k$ the greatest divisor of $a_k$ such that $b_k<a_k$. Assume that $b_1>b_2>...>b_{10}$. Show that $a_{10}>500$.

2014 Turkey Junior National Olympiad, 2
Tags: number theory proposed , number theory
Determine the minimum possible amount of distinct prime divisors of $19^{4n}+4$, for a positive integer $n$.

2010 International Zhautykov Olympiad, 1
Tags: number theory proposed , number theory
Find all primes $p,q$ such that $p^3-q^7=p-q$.

1999 Italy TST, 1
Tags: modular arithmetic , number theory proposed , number theory
Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.

2012 JBMO TST - Turkey, 2
Tags: number theory , greatest common divisor , modular arithmetic , number theory proposed , GCD
Find all positive integers $m,n$ and prime numbers $p$ for which $\frac{5^m+2^np}{5^m-2^np}$ is a perfect square.

1985 IMO Longlists, 21
Tags: number theory proposed , number theory
Let $A$ be a set of positive integers such that for any two distinct elements $x, y\in A$ we have $|x-y| \geq \frac{xy}{25}.$ Prove that $A$ contains at most nine elements. Give an example of such a set of nine elements.

2004 Poland - Second Round, 1
Tags: number theory proposed , number theory
Find all positive integers $n$ which have exactly $\sqrt{n}$ positive divisors.

2010 Contests, 2
Tags: number theory proposed , number theory
Find all prime numbers $p, q, r$ such that \[15p+7pq+qr=pqr.\]

2010 Indonesia TST, 1
Tags: algebra , polynomial , number theory proposed , number theory
Let $ f$ be a polynomial with integer coefficients. Assume that there exists integers $ a$ and $ b$ such that $ f(a)\equal{}41$ and $ f(b)\equal{}49$. Prove that there exists an integer $ c$ such that $ 2009$ divides $ f(c)$. [i]Nanang Susyanto, Jogjakarta[/i]