Found problems: 1362
2004 South africa National Olympiad, 3
Find all real numbers $x$ such that $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor=88$. The notation $\lfloor x\rfloor$ means the greatest integer less than or equal to $x$.
2002 IberoAmerican, 1
The integer numbers from $1$ to $2002$ are written in a blackboard in increasing order $1,2,\ldots, 2001,2002$. After that, somebody erases the numbers in the $ (3k+1)-th$ places i.e. $(1,4,7,\dots)$. After that, the same person erases the numbers in the $(3k+1)-th$ positions of the new list (in this case, $2,5,9,\ldots$). This process is repeated until one number remains. What is this number?
2010 Serbia National Math Olympiad, 3
Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$,
\[a^n + a^{n-1} + \cdots + a^1 + 1 \mid a^{n!} + a^{(n-1)!} + \cdots + a^{1!} + 1.\]
[i]Proposed by Milos Milosavljevic[/i]
2009 Singapore Senior Math Olympiad, 3
Suppose $ A $ is a subset of $ n $-elements taken from $ 1,2,3,4,...,2009 $ such that the difference of any two numbers in $ A $ is not a prime number. Find the largest value of $ n $ and the set $ A $ with this number of elements.
2010 Contests, 2
Determine the least $n\in\mathbb{N}$ such that $n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n$ has at least $2010$ positive factors.
1998 Korea - Final Round, 3
Denote by $\phi(n)$ for all $n\in\mathbb{N}$ the number of positive integer smaller than $n$ and relatively prime to $n$. Also, denote by $\omega(n)$ for all $n\in\mathbb{N}$ the number of prime divisors of $n$. Given that $\phi(n)|n-1$ and $\omega(n)\leq 3$. Prove that $n$ is a prime number.
2014 South East Mathematical Olympiad, 7
Show that there are infinitely many triples of positive integers $(a_i,b_i,c_i)$, $i=1,2,3,\ldots$, satisfying the equation $a^2+b^2=c^4$, such that $c_n$ and $c_{n+1}$ are coprime for any positive integer $n$.
2014 Tuymaada Olympiad, 1
Given are three different primes. What maximum number of these primes can divide their sum?
[i](A. Golovanov)[/i]
1988 China Team Selection Test, 1
Let $f(x) = 3x + 2.$ Prove that there exists $m \in \mathbb{N}$ such that $f^{100}(m)$ is divisible by $1988$.
1997 Polish MO Finals, 1
The positive integers $x_1, x_2, ... , x_7$ satisfy $x_6 = 144$, $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n = 1, 2, 3, 4$. Find $x_7$.
2008 Argentina National Olympiad, 1
$ 101$ positive integers are written on a line. Prove that we can write signs $ \plus{}$, signs $ \times$ and parenthesis between them, without changing the order of the numbers, in such a way that the resulting expression makes sense and the result is divisible by $ 16!$.
1993 APMO, 5
Let $P_1$, $P_2$, $\ldots$, $P_{1993} = P_0$ be distinct points in the $xy$-plane
with the following properties:
(i) both coordinates of $P_i$ are integers, for $i = 1, 2, \ldots, 1993$;
(ii) there is no point other than $P_i$ and $P_{i+1}$ on the line segment joining $P_i$ with $P_{i+1}$ whose coordinates are both integers, for $i = 0, 1, \ldots, 1992$.
Prove that for some $i$, $0 \leq i \leq 1992$, there exists a point $Q$ with coordinates $(q_x, q_y)$ on the line segment joining $P_i$ with $P_{i+1}$ such that both $2q_x$ and $2q_y$ are odd integers.
2012 Postal Coaching, 2
Let $a_1, a_2,\cdots ,a_n$ be positive integers and let $a$ be an integer greater than $1$ and divisible
by the product $a_1a_2\cdots a_n$. Prove that $a^{n+1} + a-1$ is not divisible by the product
$(a + a_1 - 1)(a + a_2 - 1) \cdots (a + a_n - 1)$.
2013 India Regional Mathematical Olympiad, 1
Prove that there do not exist natural numbers $x$ and $y$ with $x>1$ such that ,
\[ \frac{x^7-1}{x-1}=y^5+1 \]
2004 Postal Coaching, 17
In a system of numeration with base $B$ , there are $n$ one-digit numbers less than $B$ whose cubes have $B-1$ in the units-digits place. Determine the relation between $n$ and $B$
2011 All-Russian Olympiad, 1
Two natural numbers $d$ and $d'$, where $d'>d$, are both divisors of $n$. Prove that $d'>d+\frac{d^2}{n}$.
1999 All-Russian Olympiad, 1
Do there exist $19$ distinct natural numbers with equal sums of digits, whose sum equals $1999$?
1984 IMO Longlists, 28
A “number triangle” $(t_{n, k}) (0 \le k \le n)$ is defined by $t_{n,0} = t_{n,n} = 1 (n \ge 0),$
\[t_{n+1,m} =(2 -\sqrt{3})^mt_{n,m} +(2 +\sqrt{3})^{n-m+1}t_{n,m-1} \quad (1 \le m \le n)\]
Prove that all $t_{n,m}$ are integers.
1997 IberoAmerican, 1
Let $r\geq1$ be areal number that holds with the property that for each pair of positive integer numbers $m$ and $n$, with $n$ a multiple of $m$, it is true that $\lfloor{nr}\rfloor$ is multiple of $\lfloor{mr}\rfloor$. Show that $r$ has to be an integer number.
[b]Note: [/b][i]If $x$ is a real number, $\lfloor{x}\rfloor$ is the greatest integer lower than or equal to $x$}.[/i]
2014 Vietnam National Olympiad, 3
Find all sets of not necessary distinct 2014 rationals such that:if we remove an arbitrary number in the set, we can divide remaining 2013 numbers into three sets such that each set has exactly 671 elements and the product of all elements in each set are the same.
2006 Estonia Math Open Junior Contests, 4
Does there exist a natural number with the sum of digits of its $ kth$ power being
equal to $ k$, if a) $ k \equal{} 2004$; b) $ k \equal{} 2006?$
2003 Hungary-Israel Binational, 3
Let $n$ be a positive integer. Show that there exist three distinct integers
between $n^{2}$ and $n^{2}+n+3\sqrt{n}$, such that one of them divides the product of the other two.
2014 ITAMO, 3
For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers.
(a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$.
(b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.
2007 Cono Sur Olympiad, 2
Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$s that may occur among the $100$ numbers.
1983 Federal Competition For Advanced Students, P2, 5
Given positive integers $ a,b,$ find all positive integers $ x,y$ satisfying the equation: $ x^{a\plus{}b}\plus{}y\equal{}x^a y^b$.