Found problems: 1766
KoMaL A Problems 2020/2021, A. 787
Let $p_n$ denote the $n^{\text{th}}$ prime number and define $a_n=\lfloor p_n\nu\rfloor$ for all positive integers $n$ where $\nu$ is a positive irrational number. Is it possible that there exist only finitely many $k$ such that $\binom{2a_k}{a_k}$ is divisible by $p_i^{10}$ for all $i=1,2,\ldots,2020?$
[i]Proposed by Superguy and ayan.nmath[/i]
2010 Pan African, 2
A sequence $a_0,a_1,a_2,\ldots ,a_n,\ldots$ of positive integers is constructed as follows:
[list][*]if the last digit of $a_n$ is less than or equal to $5$ then this digit is deleted and $a_{n+1}$ is the number consisting of the remaining digits. (If $a_{n+1}$ contains no digits the process stops.)
[*]otherwise $a_{n+1}=9a_n$.[/list]
Can one choose $a_0$ so that an infinite sequence is obtained?
2016 Bulgaria JBMO TST, 3
On the board the number 1 is written. If on the board there is n, write $ n^2 $ or $ (n+1)^2 $ or $ (n+2)^2 $. Can we arrive at a number, devisible by 2015?
1999 Flanders Math Olympiad, 4
Let $a,b,m,n$ integers greater than 1. If $a^n-1$ and $b^m+1$ are both primes, give as much info as possible on $a,b,m,n$.
2024 Francophone Mathematical Olympiad, 4
Let $p$ be a fixed prime number. Find all integers $n \ge 1$ with the following property: One can partition the positive divisors of $n$ in pairs $(d,d')$ satisfying $d<d'$ and $p \mid \left\lfloor \frac{d'}{d}\right\rfloor$.
1996 Austrian-Polish Competition, 7
Prove there are no such integers $ k, m $ which satisfy $ k \ge 0, m \ge 0 $ and $ k!+48=48(k+1)^m $.
2000 Baltic Way, 18
Determine all positive real numbers $x$ and $y$ satisfying the equation
\[x+y+\frac{1}{x}+\frac{1}{y}+4=2\cdot (\sqrt{2x+1}+\sqrt{2y+1})\]
1997 Pre-Preparation Course Examination, 4
Let $n$ and $k$ be two positive integers. Prove that there exist infinitely many perfect squares of the form $n \cdot 2^k - 7$.
2011 Iran MO (3rd Round), 3
Let $k$ be a natural number such that $k\ge 7$. How many $(x,y)$ such that $0\le x,y<2^k$ satisfy the equation $73^{73^x}\equiv 9^{9^y} \pmod {2^k}$?
[i]Proposed by Mahyar Sefidgaran[/i]
2012 Iran MO (2nd Round), 3
Prove that if $t$ is a natural number then there exists a natural number $n>1$ such that $(n,t)=1$ and none of the numbers $n+t,n^2+t,n^3+t,....$ are perfect powers.
2008 District Olympiad, 2
Determine $ x$ irrational so that $ x^2\plus{}2x$ and $ x^3\minus{}6x$ are both rational.
1983 IMO Longlists, 7
Find all numbers $x \in \mathbb Z$ for which the number
\[x^4 + x^3 + x^2 + x + 1\]
is a perfect square.
2008 Junior Balkan MO, 3
Find all prime numbers $ p,q,r$, such that $ \frac{p}{q}\minus{}\frac{4}{r\plus{}1}\equal{}1$
2010 Greece National Olympiad, 1
Solve in the integers the diophantine equation
$$x^4-6x^2+1 = 7 \cdot 2^y.$$
2017 Benelux, 4
A [i]Benelux n-square[/i] (with $n\geq 2$) is an $n\times n$ grid consisting of $n^2$ cells, each of them containing a positive integer, satisfying the following conditions:
$\bullet$ the $n^2$ positive integers are pairwise distinct.
$\bullet$ if for each row and each column we compute the greatest common divisor of the $n$ numbers in that row/column, then we obtain $2n$ different outcomes.
(a) Prove that, in each Benelux n-square (with $n \geq 2$), there exists a cell containing a number which is at least $2n^2.$
(b) Call a Benelux n-square [i]minimal[/i] if all $n^2$ numbers in the cells are at most $2n^2.$ Determine all $n\geq 2$ for which there exists a minimal Benelux n-square.
2005 Brazil National Olympiad, 6
Given positive integers $a,c$ and integer $b$, prove that there exists a positive integer $x$ such that
\[ a^x + x \equiv b \pmod c, \]
that is, there exists a positive integer $x$ such that $c$ is a divisor of $a^x + x - b$.
2013 IberoAmerican, 1
A set $S$ of positive integers is said to be [i]channeler[/i] if for any three distinct numbers $a,b,c \in S$, we have $a\mid bc$, $b\mid ca$, $c\mid ab$.
a) Prove that for any finite set of positive integers $ \{ c_1, c_2, \ldots, c_n \} $ there exist infinitely many positive integers $k$, such that the set $ \{ kc_1, kc_2, \ldots, kc_n \} $ is a channeler set.
b) Prove that for any integer $n \ge 3$ there is a channeler set who has exactly $n$ elements, and such that no integer greater than $1$ divides all of its elements.
2011 Singapore Senior Math Olympiad, 2
Determine if there is a set $S$ of 2011 positive integers so that for every pair $m,n$ of distinct elements of $S$, $|m-n|=(m,n)$. Here $(m,n)$ denotes the greatest common divisor of $m$ and $n$.
2002 Romania Team Selection Test, 3
Let $a,b$ be positive real numbers. For any positive integer $n$, denote by $x_n$ the sum of digits of the number $[an+b]$ in it's decimal representation. Show that the sequence $(x_n)_{n\ge 1}$ contains a constant subsequence.
[i]Laurentiu Panaitopol[/i]
2000 Junior Balkan MO, 2
Find all positive integers $n\geq 1$ such that $n^2+3^n$ is the square of an integer.
[i]Bulgaria[/i]
2005 Korea - Final Round, 1
Find all natural numbers that can be expressed in a unique way as a sum of five or less perfect squares.
2006 ISI B.Stat Entrance Exam, 9
Find a four digit number $M$ such that the number $N=4\times M$ has the following properties.
(a) $N$ is also a four digit number
(b) $N$ has the same digits as in $M$ but in reverse order.
2002 Baltic Way, 18
Find all integers $n>1$ such that any prime divisor of $n^6-1$ is a divisor of $(n^3-1)(n^2-1)$.
2009 India IMO Training Camp, 11
Find all integers $ n\ge 2$ with the following property:
There exists three distinct primes $p,q,r$ such that
whenever $ a_1,a_2,a_3,\cdots,a_n$ are $ n$ distinct positive integers with the property that at least one of $ p,q,r$ divides $ a_j - a_k \ \forall 1\le j\le k\le n$,
one of $ p,q,r$ divides all of these differences.
2013 Vietnam National Olympiad, 3
Find all ordered 6-tuples satisfy following system of modular equation:
$ab+a'b' \equiv 1 $(mod 15)
$bc+b'c' \equiv 1 $(mod 15)
$ca+c'a' \equiv 1 $(mod 15)
Given that $a,b,c,a',b',c' \epsilon (0;1;2;...;14)$