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: 1362

1970 IMO Longlists, 12
Tags: algebra , polynomial , number theory unsolved , number theory
Let $\{x_i\}, 1\le i\le 6$ be a given set of six integers, none of which are divisible by $7$. $(a)$ Prove that at least one of the expressions of the form $x_1\pm x_2\pm x_3\pm x_4\pm x_5\pm x_6$ is divisible by $7$, where the $\pm$ signs are independent of each other. $(b)$ Generalize the result to every prime number.

2010 Argentina Team Selection Test, 5
Tags: algebra , polynomial , number theory unsolved , number theory
Let $p$ and $q$ be prime numbers. The sequence $(x_n)$ is defined by $x_1 = 1$, $x_2 = p$ and $x_{n+1} = px_n - qx_{n-1}$ for all $n \geq 2$. Given that there is some $k$ such that $x_{3k} = -3$, find $p$ and $q$.

2005 Germany Team Selection Test, 3
Tags: inequalities , modular arithmetic , algorithm , induction , number theory , number theory unsolved
We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.

1970 Canada National Olympiad, 7
Tags: number theory unsolved , number theory
Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by 3.

2017 Thailand TSTST, 2
Tags: Euler , function , modular arithmetic , quadratics , number theory , prime numbers , number theory unsolved
Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.

2004 China Team Selection Test, 2
Tags: LaTeX , number theory unsolved , number theory
Let $p_1, p_2, \ldots, p_{25}$ are primes which don’t exceed 2004. Find the largest integer $T$ such that every positive integer $\leq T$ can be expressed as sums of distinct divisors of $(p_1\cdot p_2 \cdot \ldots \cdot p_{25})^{2004}.$

2005 MOP Homework, 5
Tags: number theory unsolved , number theory
Find all ordered triples $(a,b,c)$ of positive integers such that the value of the expression \[\left (b-\frac{1}{a}\right )\left (c-\frac{1}{b}\right )\left (a-\frac{1}{c}\right )\] is an integer.

2007 Kurschak Competition, 2
Tags: arithmetic sequence , number theory , relatively prime , number theory unsolved
Prove that if from any $2007$ consecutive terms of an infinite arithmetic progression of integers starting with $2$, one can choose a term relatively prime to all the $2006$ other terms, then there is also a term amongst any $2008$ consecutive terms relatively prime to the rest.

2007 Germany Team Selection Test, 1
Tags: algebra , polynomial , number theory unsolved , number theory
Let $ k \in \mathbb{N}$. A polynomial is called [i]$ k$-valid[/i] if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.) [b]a.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often. [b]b.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .

2007 Pre-Preparation Course Examination, 21
Tags: number theory , number theory unsolved
Find all primes $p,q$ such that \[p^q-q^p=pq^2-19\]

1986 IMO Longlists, 64
Tags: induction , number theory unsolved , number theory
Let $(a_n)_{n\in \mathbb N}$ be the sequence of integers defined recursively by $a_1 = a_2 = 1, a_{n+2} = 7a_{n+1} - a_n - 2$ for $n \geq 1$. Prove that $a_n$ is a perfect square for every $n.$

2006 China Girls Math Olympiad, 8
Tags: modular arithmetic , number theory unsolved , number theory
Let $p$ be a prime number that is greater than $3$. Show that there exist some integers $a_{1}, a_{2}, \cdots a_{k}$ that satisfy: \[-\frac{p}{2}< a_{1}< a_{2}< \cdots <a_{k}< \frac{p}{2}\] making the product: \[\frac{p-a_{1}}{|a_{1}|}\cdot \frac{p-a_{2}}{|a_{2}|}\cdots \frac{p-a_{k}}{|a_{k}|}\] equals to $3^{m}$ where $m$ is a positive integer.

1976 IMO Longlists, 4
Tags: number theory unsolved , number theory
Find all pairs of natural numbers $(m, n)$ for which $2^m3^n +1$ is the square of some integer.

2008 China National Olympiad, 3
Tags: modular arithmetic , algebra , polynomial , Vieta , number theory unsolved , number theory
Find all triples $(p,q,n)$ that satisfy \[q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)\] where $p,q$ are odd primes and $n$ is an positive integer.

2005 South africa National Olympiad, 2
Tags: number theory unsolved , number theory
Let $F$ be the set of all fractions $m/n$ where $m$ and $n$ are positive integers with $m+n\le 2005$. Find the largest number $a$ in $F$ such that $a < 16/23$.

2010 Malaysia National Olympiad, 5
Tags: number theory unsolved , number theory
Let $n$ be an integer greater than 1. If all digits of $97n$ are odd, find the smallest possible value of $n$.

1991 Federal Competition For Advanced Students, 3
Tags: induction , modular arithmetic , quadratics , number theory unsolved , number theory
Find the number of squares in the sequence given by $ a_0\equal{}91$ and $ a_{n\plus{}1}\equal{}10a_n\plus{}(\minus{}1)^n$ for $ n \ge 0.$

1984 Canada National Olympiad, 3
Tags: induction , number theory unsolved , number theory
An integer is digitally divisible if both of the following conditions are fulfilled: $(a)$ None of its digits is zero; $(b)$ It is divisible by the sum of its digits e.g. $322$ is digitally divisible. Show that there are infinitely many digitally divisible integers.

2014 Romania Team Selection Test, 4
Tags: number theory , greatest common divisor , function , number theory unsolved
Let $n$ be a positive integer and let $A_n$ respectively $B_n$ be the set of nonnegative integers $k<n$ such that the number of distinct prime factors of $\gcd(n,k)$ is even (respectively odd). Show that $|A_n|=|B_n|$ if $n$ is even and $|A_n|>|B_n|$ if $n$ is odd. Example: $A_{10} = \left\{ 0,1,3,7,9 \right\}$, $B_{10} = \left\{ 2,4,5,6,8 \right\}$.

2001 USA Team Selection Test, 8
Tags: number theory unsolved , number theory
Find all pairs of nonnegative integers $(m,n)$ such that \[(m+n-5)^2=9mn.\]

2001 Cono Sur Olympiad, 2
Tags: number theory unsolved , number theory
Find all positive integers $m$ for which $2001\cdot S (m) = m$ where $S(m)$ denotes the sum of the digits of $m$.

1999 Taiwan National Olympiad, 4
Tags: number theory unsolved , number theory
Let $P^{*}$ be the set of primes less than $10000$. Find all possible primes $p\in P^{*}$ such that for each subset $S=\{p_{1},p_{2},...,p_{k}\}$ of $P^{*}$ with $k\geq 2$ and each $p\not\in S$, there is a $q\in P^{*}-S$ such that $q+1$ divides $(p_{1}+1)(p_{2}+1)...(p_{k}+1)$.

2010 Postal Coaching, 6
Tags: floor function , number theory unsolved , number theory
Solve the equation for positive integers $m, n$: \[\left \lfloor \frac{m^2}n \right \rfloor + \left \lfloor \frac{n^2}m \right \rfloor = \left \lfloor \frac mn + \frac nm \right \rfloor +mn\]

2014 Contests, 1
Tags: number theory unsolved , number theory , euler totient function
Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$

2003 Iran MO (3rd Round), 7
Tags: algebra , polynomial , number theory unsolved , number theory
$f_{1},f_{2},\dots,f_{n}$ are polynomials with integer coefficients. Prove there exist a reducible $g(x)$ with integer coefficients that $f_{1}+g,f_{2}+g,\dots,f_{n}+g$ are irreducible.