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: 71

1995 India National Olympiad, 6
Tags: number theory , greatest common divisor , number theory solved , primes
Find all primes $p$ for which the quotient \[ \dfrac{2^{p-1} - 1 }{p} \] is a square.

2005 IberoAmerican, 4
Tags: number theory solved , number theory
Denote by $a \bmod b$ the remainder of the euclidean division of $a$ by $b$. Determine all pairs of positive integers $(a,p)$ such that $p$ is prime and \[ a \bmod p + a\bmod 2p + a\bmod 3p + a\bmod 4p = a + p. \]

2004 All-Russian Olympiad, 1
Tags: geometry , 3D geometry , number theory solved , number theory
Are there such pairwise distinct natural numbers $ m, n, p, q$ satisfying $ m \plus{} n \equal{} p \plus{} q$ and $ \sqrt{m} \plus{} \sqrt[3]{n} \equal{} \sqrt{p} \plus{} \sqrt[3]{q} > 2004$ ?

2016 Israel Team Selection Test, 4
Tags: greatest common divisor , number theory , number theory solved
Find the greatest common divisor of all numbers of the form $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a,b,c$ are integers.

2008 Argentina Iberoamerican TST, 2
Tags: number theory , prime numbers , relatively prime , number theory solved
Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.

2005 China Western Mathematical Olympiad, 3
Tags: number theory , prime numbers , relatively prime , number theory solved
Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.

2005 Morocco TST, 2
Tags: induction , number theory solved , number theory
Let $A$ be a set of positive integers such that a) if $a\in A$, the all the positive divisors of $a$ are also in $A$; b) if $a,b\in A$, with $1<a<b$, then $1+ab \in A$. Prove that if $A$ has at least 3 elements, then $A$ is the set of all positive integers.

2004 France Team Selection Test, 1
Tags: quadratics , modular arithmetic , number theory , number theory solved
If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$. Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?

2005 Postal Coaching, 8
Tags: number theory solved , number theory
Prove that For all positive integers $m$ and $n$ , one has $| n \sqrt{2005} - m | > \frac{1}{90n}$

1997 India National Olympiad, 2
Tags: quadratics , modular arithmetic , Diophantine equation , special factorizations , number theory solved , number theory
Show that there do not exist positive integers $m$ and $n$ such that \[ \dfrac{m}{n} + \dfrac{n+1}{m} = 4 . \]

1998 Junior Balkan MO, 4
Tags: number theory solved , number theory , residue
Do there exist 16 three digit numbers, using only three different digits in all, so that the all numbers give different residues when divided by 16? [i]Bulgaria[/i]

2004 Baltic Way, 10
Tags: modular arithmetic , induction , number theory , prime numbers , number theory solved
Is there an infinite sequence of prime numbers $p_1$, $p_2$, $\ldots$, $p_n$, $p_{n+1}$, $\ldots$ such that $|p_{n+1}-2p_n|=1$ for each $n \in \mathbb{N}$?

2002 India National Olympiad, 2
Tags: number theory solved , number theory
Find the smallest positive value taken by $a^3 + b^3 + c^3 - 3abc$ for positive integers $a$, $b$, $c$ . Find all $a$, $b$, $c$ which give the smallest value

1980 Brazil National Olympiad, 2
Tags: number theory , reciprocal sum , number theory solved , construction
Show that for any positive integer $n > 2$ we can find $n$ distinct positive integers such that the sum of their reciprocals is $1$.

1998 India National Olympiad, 3
Tags: number theory solved , number theory
Let $p , q, r , s$ be four integers such that $s$ is not divisible by $5$. If there is an integer $a$ such that $pa^3 + qa^2+ ra +s$ is divisible be 5, prove that there is an integer $b$ such that $sb^3 + rb^2 + qb + p$ is also divisible by 5.

1999 Baltic Way, 19
Tags: number theory solved , number theory
Prove that there exist infinitely many even positive integers $k$ such that for every prime $p$ the number $p^2+k$ is composite.

1992 Polish MO Finals, 3
Tags: number theory solved , number theory
Show that $(k^3)!$ is divisible by $(k!)^{k^2+k+1}$.

2006 Moldova Team Selection Test, 1
Tags: number theory solved , number theory
Let $(a_n)$ be the Lucas sequence: $a_0=2,a_1=1, a_{n+1}=a_n+a_{n-1}$ for $n\geq 1$. Show that $a_{59}$ divides $(a_{30})^{59}-1$.

2004 France Team Selection Test, 3
Tags: induction , number theory , prime numbers , number theory solved
Let $P$ be the set of prime numbers. Consider a subset $M$ of $P$ with at least three elements. We assume that, for each non empty and finite subset $A$ of $M$, with $A \neq M$, the prime divisors of the integer $( \prod_{p \in A} ) - 1$ belong to $M$. Prove that $M = P$.

2003 Bundeswettbewerb Mathematik, 4
Tags: number theory solved , number theory
Let $p$ and $q$ be two positive integers that have no common divisor. The set of integers shall be partioned into three subsets $A$, $B$, $C$ such that for each integer $z$ in each of the sets $A$, $B$, $C$ there is exactly one of the numbers $z$, $z+p$ and $z+q$. a) Prove that such a decomposition is possible if and only if $p+q$ is divisible by $3$. b) In the case we omit the restriction that $p$, $q$ may not have a common divisor, prove that for $p \neq q$ the number $\frac{p+q}{\gcd(p,q)}$ is divisible by 3.

1999 India National Olympiad, 6
Tags: number theory solved , number theory
For which positive integer values of $n$ can the set $\{ 1, 2, 3, \ldots, 4n \}$ be split into $n$ disjoint $4$-element subsets $\{ a,b,c,d \}$ such that in each of these sets $a = \dfrac{b +c +d} {3}$.

2004 India IMO Training Camp, 2
Tags: floor function , number theory solved , number theory
Find all primes $p \geq 3$ with the following property: for any prime $q<p$, the number \[ p - \Big\lfloor \frac{p}{q} \Big\rfloor q \] is squarefree (i.e. is not divisible by the square of a prime).

1999 Romania Team Selection Test, 1
Tags: number theory solved , number theory
a) Prove that it is possible to choose one number out of any 39 consecutive positive integers, having the sum of its digits divisible by 11; b) Find the first 38 consecutive positive integers none of which have the sum of its digits divisible by 11.

2016 Israel Team Selection Test, 2
Tags: number theory , number theory solved
Rothschild the benefactor has a certain number of coins. A man comes, and Rothschild wants to share his coins with him. If he has an even number of coins, he gives half of them to the man and goes away. If he has an odd number of coins, he donates one coin to charity so he can have an even number of coins, but meanwhile another man comes. So now he has to share his coins with two other people. If it is possible to do so evenly, he does so and goes away. Otherwise, he again donates a few coins to charity (no more than 3). Meanwhile, yet another man comes. This goes on until Rothschild is able to divide his coins evenly or until he runs out of money. Does there exist a natural number $N$ such that if Rothschild has at least $N$ coins in the beginning, he will end with at least one coin?

1989 USAMO, 1
Tags: number theory solved , number theory
For each positive integer $n$, let \begin{eqnarray*} S_n &=& 1 + \frac 12 + \frac 13 + \cdots + \frac 1n, \\ T_n &=& S_1 + S_2 + S_3 + \cdots + S_n, \\ U_n &=& \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}. \end{eqnarray*} Find, with proof, integers $0 < a, b,c, d < 1000000$ such that $T_{1988} = a S_{1989} - b$ and $U_{1988} = c S_{1989} - d$.