Found problems: 1362
2004 239 Open Mathematical Olympiad, 3
Prove that for any integer $a$ there exist infinitely many positive integers $n$ such that $a^{2^n}+2^n$ is not a prime.
[b]proposed by S. Berlov[/b]
1996 IberoAmerican, 1
Given a natural number $n \geq 2$, consider all the fractions of the form $\frac{1}{ab}$, where $a$ and $b$ are natural numbers, relative primes and such that:
$a < b \leq n$,
$a+b>n$.
Show that for each $n$, the sum of all this fractions are $\frac12$.
2007 China National Olympiad, 2
Show that:
1) If $2n-1$ is a prime number, then for any $n$ pairwise distinct positive integers $a_1, a_2, \ldots , a_n$, there exists $i, j \in \{1, 2, \ldots , n\}$ such that
\[\frac{a_i+a_j}{(a_i,a_j)} \geq 2n-1\]
2) If $2n-1$ is a composite number, then there exists $n$ pairwise distinct positive integers $a_1, a_2, \ldots , a_n$, such that for any $i, j \in \{1, 2, \ldots , n\}$ we have
\[\frac{a_i+a_j}{(a_i,a_j)} < 2n-1\]
Here $(x,y)$ denotes the greatest common divisor of $x,y$.
1991 Vietnam Team Selection Test, 2
For every natural number $n$ we define $f(n)$ by the following rule: $f(1) = 1$ and for $n>1$ then $f(n) = 1 + a_1 \cdot p_1 + \ldots + a_k \cdot p_k$, where $n = p_1^{a_1} \cdots p_k^{a_k}$ is the canonical prime factorisation of $n$ ($p_1, \ldots, p_k$ are distinct primes and $a_1, \ldots, a_k$ are positive integers). For every positive integer $s$, let $f_s(n) = f(f(\ldots f(n))\ldots)$, where on the right hand side there are exactly $s$ symbols $f$. Show that for every given natural number $a$, there is a natural number $s_0$ such that for all $s > s_0$, the sum $f_s(a) + f_{s-1}(a)$ does not depend on $s$.
2009 Czech-Polish-Slovak Match, 5
The $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers satisfies the following:
[list](i) $1\le a_1<a_2<\cdots < a_n\le 50$
(ii) for each $n$-tuple $(b_1,b_2,\ldots,b_n)$ of positive integers, there exist a positive integer $m$ and an $n$-tuple $(c_1,c_2,\ldots,c_n)$ of positive integers such that \[mb_i=c_i^{a_i}\qquad\text{for } i=1,2,\ldots,n. \] [/list]Prove that $n\le 16$ and determine the number of $n$-tuples $(a_1,a_2,\ldots,a_n$) satisfying these conditions for $n=16$.
2015 India National Olympiad, 2
For any natural number $n > 1$ write the finite decimal expansion of $\frac{1}{n}$ (for example we write $\frac{1}{2}=0.4\overline{9}$ as its infinite decimal expansion not $0.5)$. Determine the length of non-periodic part of the (infinite) decimal expansion of $\frac{1}{n}$.
2010 Malaysia National Olympiad, 9
Let $m$ and $n$ be positive integers such that $2^n+3^m$ is divisible by $5$. Prove that $2^m+3^n$ is divisible by $5$.
2010 Rioplatense Mathematical Olympiad, Level 3, 1
Let $r_2, r_3,\ldots, r_{1000}$ denote the remainders when a positive odd integer is divided by $2,3,\ldots,1000$, respectively. It is known that the remainders are pairwise distinct and one of them is $0$. Find all values of $k$ for which it is possible that $r_k = 0$.
1999 China Team Selection Test, 2
Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$.
2013 Indonesia MO, 6
A positive integer $n$ is called "strong" if there exists a positive integer $x$ such that $x^{nx} + 1$ is divisible by $2^n$.
a. Prove that $2013$ is strong.
b. If $m$ is strong, determine the smallest $y$ (in terms of $m$) such that $y^{my} + 1$ is divisible by $2^m$.
2003 Germany Team Selection Test, 3
Let $N$ be a natural number and $x_1, \ldots , x_n$ further natural numbers less than $N$ and such that the least common multiple of any two of these $n$ numbers is greater than $N$. Prove that the sum of the reciprocals of these $n$ numbers is always less than $2$: $\sum^n_{i=1} \frac{1}{x_i} < 2.$
2011 JBMO Shortlist, 2
Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$
2011 Postal Coaching, 2
Let $\tau(n)$ be the number of positive divisors of a natural number $n$, and $\sigma(n)$ be their sum. Find the largest real number $\alpha$ such that
\[\frac{\sigma(n)}{\tau(n)}\ge\alpha \sqrt{n}\]
for all $n \ge 1$.
2007 Pre-Preparation Course Examination, 6
Let $a,b$ be two positive integers and $b^2+a-1|a^2+b-1$. Prove that $b^2+a-1$ has at least two prime divisors.
2013 Romania Team Selection Test, 1
Let $a$ and $b$ be two square-free, distinct natural numbers. Show that there exist $c>0$ such that
\[
\left | \{n\sqrt{a}\}-\{n\sqrt{b}\} \right |>\frac{c}{n^3}\]
for every positive integer $n$.
2003 Baltic Way, 16
Find all pairs of positive integers $(a,b)$ such that $a-b$ is a prime number and $ab$ is a perfect square.
2005 France Team Selection Test, 4
Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.
1997 Irish Math Olympiad, 5
Let $ S$ be the set of odd integers greater than $ 1$. For each $ x \in S$, denote by $ \delta (x)$ the unique integer satisfying the inequality $ 2^{\delta (x)}<x<2^{\delta (x) \plus{}1}$. For $ a,b \in S$, define:
$ a \ast b\equal{}2^{\delta (a)\minus{}1} (b\minus{}3)\plus{}a.$
Prove that if $ a,b,c \in S$, then:
$ (a)$ $ a \ast b \in S$ and
$ (b)$ $ (a \ast b)\ast c\equal{}a \ast (b \ast c)$.
2014 Macedonia National Olympiad, 2
Solve the following equation in $\mathbb{Z}$:
\[3^{2a + 1}b^2 + 1 = 2^c\]
2010 China Team Selection Test, 3
Let $k>1$ be an integer, set $n=2^{k+1}$. Prove that for any positive integers
$a_1<a_2<\cdots<a_n$, the number $\prod_{1\leq i<j\leq n}(a_i+a_j)$ has at least $k+1$ different prime divisors.
1987 IMO Longlists, 8
Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.
[hide="Note"]Note. Here (and generally in MathLinks) natural numbers supposed to be positive.[/hide]
2005 Rioplatense Mathematical Olympiad, Level 3, 3
Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$. (Note: $n$ is written in the usual base ten notation.)
2007 Cono Sur Olympiad, 3
Show that for each positive integer $n$, there is a positive integer $k$ such that the decimal representation of each of the numbers $k, 2k,\ldots, nk$ contains all of the digits $0, 1, 2,\ldots, 9$.
2008 Middle European Mathematical Olympiad, 4
Determine that all $ k \in \mathbb{Z}$ such that $ \forall n$ the numbers $ 4n\plus{}1$ and $ kn\plus{}1$ have no common divisor.
1985 Vietnam National Olympiad, 1
Find all pairs $ (x, y)$ of integers such that $ x^3 \minus{} y^3 \equal{} 2xy \plus{} 8$.