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: 1766

2011 Bulgaria National Olympiad, 2
Tags: number theory , relatively prime , number theory proposed
For each natural number $a$ we denote $\tau (a)$ and $\phi (a)$ the number of natural numbers dividing $a$ and the number of natural numbers less than $a$ that are relatively prime to $a$. Find all natural numbers $n$ for which $n$ has exactly two different prime divisors and $n$ satisfies $\tau (\phi (n))=\phi (\tau (n))$.

2009 Indonesia TST, 2
Tags: inequalities , logarithms , number theory , relatively prime , number theory proposed
For every positive integer $ n$, let $ \phi(n)$ denotes the number of positive integers less than $ n$ that is relatively prime to $ n$ and $ \tau(n)$ denote the sum of all positive divisors of $ n$. Let $ n$ be a positive integer such that $ \phi(n)|n\minus{}1$ and that $ n$ is not a prime number. Prove that $ \tau(n)>2009$.

2020 Turkey MO (2nd round), 4
Tags: number theory , prime numbers , number theory proposed
Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of number of divisors of $2p^2+2p+1$.

2006 JBMO ShortLists, 6
Tags: factorial , floor function , number theory proposed , number theory
Prove that for every composite number $ n>4$, numbers $ kn$ divides $ (n\minus{}1)!$ for every integer $ k$ such that $ 1\le k\le \lfloor \sqrt{n\minus{}1} \rfloor$.

2021 JBMO TST - Turkey, 6
Tags: number theory , modular arithmetic , number theory proposed
Integers $a_1, a_2, \dots a_n$ are different at $\text{mod n}$. If $a_1, a_2-a_1, a_3-a_2, \dots a_n-a_{n-1}$ are also different at $\text{mod n}$, we call the ordered $n$-tuple $(a_1, a_2, \dots a_n)$ [i]lucky[/i]. For which positive integers $n$, one can find a lucky $n$-tuple?

1995 Vietnam National Olympiad, 2
Tags: number theory proposed , number theory
The sequence (a_n) is defined as follows: $ a_0\equal{}1, a_1\equal{}3$ For $ n\ge 2$, $ a_{n\plus{}2}\equal{}a_{n\plus{}1}\plus{}9a_n$ if n is even, $ a_{n\plus{}2}\equal{}9a_{n\plus{}1}\plus{}5a_n$ if n is odd. Prove that 1) $ (a_{1995})^2\plus{}(a_{1996})^2\plus{}...\plus{}(a_{2000})^2$ is divisible by 20 2) $ a_{2n\plus{}1}$ is not a perfect square for every natural numbers $ n$.

2011 Bulgaria National Olympiad, 1
Tags: number theory proposed , number theory
Prove whether or not there exist natural numbers $n,k$ where $1\le k\le n-2$ such that \[\binom{n}{k}^2+\binom{n}{k+1}^2=\binom{n}{k+2}^4 \]

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)$.

1991 Federal Competition For Advanced Students, 1
Tags: number theory proposed , number theory
Suppose that $ a,b,$ and $ \sqrt[3]{a}\plus{}\sqrt[3]{b}$ are rational numbers. Prove that $ \sqrt[3]{a}$ and $ \sqrt[3]{b}$ are also rational.

2013 Middle European Mathematical Olympiad, 7
Tags: quadratics , ceiling function , number theory proposed , number theory
The numbers from 1 to $ 2013^2 $ are written row by row into a table consisting of $ 2013 \times 2013 $ cells. Afterwards, all columns and all rows containing at least one of the perfect squares $ 1, 4, 9, \cdots, 2013^2 $ are simultaneously deleted. How many cells remain?

1977 Canada National Olympiad, 3
Tags: number theory proposed , number theory
$N$ is an integer whose representation in base $b$ is $777$. Find the smallest integer $b$ for which $N$ is the fourth power of an integer.

2007 All-Russian Olympiad Regional Round, 9.7
Tags: geometry , 3D geometry , algebra , polynomial , arithmetic sequence , number theory proposed , number theory
An infinite increasing arithmetical progression consists of positive integers and contains a perfect cube. Prove that this progression also contains a term which is a perfect cube but not a perfect square.

2003 CentroAmerican, 6
Tags: number theory proposed , number theory
Say a number is [i]tico[/i] if the sum of it's digits is a multiple of $2003$. $\text{(i)}$ Show that there exists a positive integer $N$ such that the first $2003$ multiples, $N,2N,3N,\ldots 2003N$ are all tico. $\text{(ii)}$ Does there exist a positive integer $N$ such that all it's multiples are tico?

1992 Balkan MO, 4
Tags: number theory proposed , number theory
For each integer $n\geq 3$, find the least natural number $f(n)$ having the property $\star$ For every $A \subset \{1, 2, \ldots, n\}$ with $f(n)$ elements, there exist elements $x, y, z \in A$ that are pairwise coprime.

2023 Dutch BxMO TST, 5
Tags: number theory , modular arithmetic , number theory proposed , Diophantine equation
Find all pairs of prime numbers $(p,q)$ for which \[2^p = 2^{q-2} + q!.\]

2009 China Western Mathematical Olympiad, 1
Tags: modular arithmetic , number theory proposed , number theory
Define a sequence $(x_{n})_{n\geq 1}$ by taking $x_{1}\in\left\{5,7\right\}$; when $k\ge 1$, $x_{k+1}\in\left\{5^{x_{k}},7^{x_{k}}\right\}$. Determine all possible last two digits of $x_{2009}$.

2008 Singapore Team Selection Test, 3
Tags: number theory proposed , number theory
Find all odd primes $ p$, if any, such that $ p$ divides $ \sum_{n\equal{}1}^{103}n^{p\minus{}1}$

2012 Canadian Mathematical Olympiad Qualification Repechage, 2
Tags: modular arithmetic , number theory proposed , number theory
Given a positive integer $m$, let $d(m)$ be the number of positive divisors of $m$. Determine all positive integers $n$ such that $d(n) +d(n+ 1) = 5$.

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]

2007 Ukraine Team Selection Test, 12
Tags: polynomial , Diophantine equation , number theory proposed , number theory
Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{2}\plus{}n\plus{}1$ are not more then $ \sqrt{n}$. [hide] Stronger one. Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{3}\minus{}1$ are not more then $ \sqrt{n}$.[/hide]

2010 Contests, 2
Tags: number theory proposed , number theory
Determine the number of positive integers $n$ for which $(n+15)(n+2010)$ is a perfect square.

2005 Romania Team Selection Test, 4
Tags: modular arithmetic , number theory proposed , number theory
a) Prove that there exists a sequence of digits $\{c_n\}_{n\geq 1}$ such that or each $n\geq 1$ no matter how we interlace $k_n$ digits, $1\leq k_n\leq 9$, between $c_n$ and $c_{n+1}$, the infinite sequence thus obtained does not represent the fractional part of a rational number. b) Prove that for $1\leq k_n\leq 10$ there is no such sequence $\{c_n\}_{n\geq 1}$. [i]Dan Schwartz[/i]

1990 IMO Longlists, 26
Tags: modular arithmetic , search , Diophantine equation , number theory proposed , number theory
Prove that there exist infinitely many positive integers $n$ such that the number $\frac{1^2+2^2+\cdots+n^2}{n}$ is a perfect square. Obviously, $1$ is the least integer having this property. Find the next two least integers having this property.

2000 Romania Team Selection Test, 3
Tags: number theory proposed , number theory
Prove that for any positive integers $n$ and $k$ there exist positive integers $a>b>c>d>e>k$ such that \[n=\binom{a}{3}\pm\binom{b}{3}\pm\binom{c}{3}\pm\binom{d}{3}\pm\binom{e}{3}\] [i]Radu Ignat[/i]

2008 German National Olympiad, 6
Tags: number theory proposed , number theory
Find all real numbers $ x$ such that $ 4x^5 \minus{} 7$ and $ 4x^{13} \minus{} 7$ are both perfect squares.