Found problems: 1766
2001 JBMO ShortLists, 6
Find all integers $x$ and $y$ such that $x^3\pm y^3 =2001p$, where $p$ is prime.
2009 India IMO Training Camp, 8
Let $ n$ be a natural number $ \ge 2$ which divides $ 3^n\plus{}4^n$.Prove That $ 7\mid n$.
2004 Bundeswettbewerb Mathematik, 1
Let $k$ be a positive integer. A natural number $m$ is called [i]$k$-typical[/i] if each divisor of $m$ leaves the remainder $1$ when being divided by $k$.
Prove:
[b]a)[/b] If the number of all divisors of a positive integer $n$ (including the divisors $1$ and $n$) is $k$-typical, then $n$ is the $k$-th power of an integer.
[b]b)[/b] If $k > 2$, then the converse of the assertion [b]a)[/b] is not true.
2012 ELMO Shortlist, 2
For positive rational $x$, if $x$ is written in the form $p/q$ with $p, q$ positive relatively prime integers, define $f(x)=p+q$. For example, $f(1)=2$.
a) Prove that if $f(x)=f(mx/n)$ for rational $x$ and positive integers $m, n$, then $f(x)$ divides $|m-n|$.
b) Let $n$ be a positive integer. If all $x$ which satisfy $f(x)=f(2^nx)$ also satisfy $f(x)=2^n-1$, find all possible values of $n$.
[i]Anderson Wang.[/i]
2002 Romania Team Selection Test, 2
Let $P(x)$ and $Q(x)$ be integer polynomials of degree $p$ and $q$ respectively. Assume that $P(x)$ divides $Q(x)$ and all their coefficients are either $1$ or $2002$. Show that $p+1$ is a divisor of $q+1$.
[i]Mihai Cipu[/i]
1997 Romania Team Selection Test, 2
Let $a>1$ be a positive integer. Show that the set of integers
\[\{a^2+a-1,a^3+a^2-1,\ldots ,a^{n+1}+a^n-1,\ldots\}\]
contains an infinite subset of pairwise coprime integers.
[i]Mircea Becheanu[/i]
2009 Indonesia TST, 2
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$.
2005 Romania Team Selection Test, 2
Let $m,n$ be co-prime integers, such that $m$ is even and $n$ is odd. Prove that the following expression does not depend on the values of $m$ and $n$:
\[ \frac 1{2n} + \sum^{n-1}_{k=1} (-1)^{\left[ \frac{mk}n \right]} \left\{ \frac {mk}n \right\} . \]
[i]Bogdan Enescu[/i]
1982 Vietnam National Olympiad, 1
Find all positive integers $x, y, z$ such that $2^x + 2^y + 2^z = 2336$.
2011 All-Russian Olympiad Regional Round, 10.3
$a_1,a_2,\dots,a_{14}$ are different positive integers. All 196 numbers of the form $a_k+a_l$ with $1\leq k,l\leq 14$ are written on a board. Is it possible that for any two-digit combination, there exists a number among all 196 that ends with that combination (i.e., there exist numbers ending with $00, 01, \dots, 99$)?
(Author: P. Kozhevnikov)
2014 Dutch BxMO/EGMO TST, 1
Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$
2006 Mexico National Olympiad, 6
Let n be the sum of the digits in a natural number A. The number A it's said to be "surtido" if every number 1,2,3,4....,n can be expressed as a sum of digits in A.
a)Prove that, if 1,2,3,4,5,6,7,8 are sums of digits in A, then A is "Surtido"
b)If 1,2,3,4,5,6,7 are sums of digits in A, does it follow that A is "Surtido"?
2012 Turkey Junior National Olympiad, 1
Let $x, y$ be integers and $p$ be a prime for which
\[ x^2-3xy+p^2y^2=12p \]
Find all triples $(x,y,p)$.
2013 China Western Mathematical Olympiad, 8
Find all positive integers $a$ such that for any positive integer $n\ge 5$ we have $2^n-n^2\mid a^n-n^a$.
2008 Iran MO (3rd Round), 3
a) Prove that there are two polynomials in $ \mathbb Z[x]$ with at least one coefficient larger than 1387 such that coefficients of their product is in the set $ \{\minus{}1,0,1\}$.
b) Does there exist a multiple of $ x^2\minus{}3x\plus{}1$ such that all of its coefficient are in the set $ \{\minus{}1,0,1\}$
1994 Irish Math Olympiad, 1
Let $ x,y$ be positive integers with $ y>3$ and $ x^2\plus{}y^4\equal{}2((x\minus{}6)^2\plus{}(y\plus{}1)^2).$ Prove that: $ x^2\plus{}y^4\equal{}1994.$
2007 Indonesia TST, 3
For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.
2010 Contests, 3
Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$
IMSC 2024, 1
For a positive integer $n$ denote by $P_0(n)$ the product of all non-zero digits of $n$. Let $N_0$ be the set of all positive integers $n$ such that $P_0(n)|n$. Find the largest possible value of $\ell$ such that $N_0$ contains infinitely many strings of $\ell$ consecutive integers.
[i]Proposed by Navid Safaei, Iran[/i]
2005 Bulgaria Team Selection Test, 2
Find the number of the subsets $B$ of the set $\{1,2,\cdots, 2005 \}$ such that the sum of the elements of $B$ is congruent to $2006$ modulo $2048$
2012 All-Russian Olympiad, 4
For a positive integer $n$ define $S_n=1!+2!+\ldots +n!$. Prove that there exists an integer $n$ such that $S_n$ has a prime divisor greater than $10^{2012}$.
2007 Moldova Team Selection Test, 1
Find the least positive integers $m,k$ such that
a) There exist $2m+1$ consecutive natural numbers whose sum of cubes is also a cube.
b) There exist $2k+1$ consecutive natural numbers whose sum of squares is also a square.
The author is Vasile Suceveanu
1995 Baltic Way, 1
Find all triples $(x,y,z)$ of positive integers satisfying the system of equations
\[\begin{cases} x^2=2(y+z)\\ x^6=y^6+z^6+31(y^2+z^2)\end{cases}\]
2009 Indonesia TST, 3
Let $ n \ge 2009$ be an integer and define the set:
\[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}.
\]
Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that
\[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}.
\]
2024 Abelkonkurransen Finale, 1b
Find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that the numbers
\[n, f(n),f(f(n)),\dots,f^{m-1}(n)\]
are distinct modulo $m$ for all integers $n,m$ with $m>1$.
(Here $f^k$ is defined by $f^0(n)=n$ and $f^{k+1}(n)=f(f^{k}(n))$ for $k \ge 0$.)