Found problems: 1766
1999 Baltic Way, 2
Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits.
2001 Junior Balkan Team Selection Tests - Romania, 4
Determine all positive integers in the form $a<b<c<d$ with the property that each of them divides the sum of the other three.
2000 JBMO ShortLists, 1
Prove that there are at least $666$ positive composite numbers with $2006$ digits, having a digit equal to $7$ and all the rest equal to $1$.
2006 Peru IMO TST, 1
[color=blue][size=150]PERU TST IMO - 2006[/size]
Saturday, may 20.[/color]
[b]Question 01[/b]
Find all $(x,y,z)$ positive integers, such that:
$\sqrt{\frac{2006}{x+y}} + \sqrt{\frac{2006}{y+z}} + \sqrt{\frac{2006}{z+x}},$
is an integer.
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[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88509]Spanish version[/url]
$\text{\LaTeX}{}$ed by carlosbr
2013 Tuymaada Olympiad, 6
Solve the equation $p^2-pq-q^3=1$ in prime numbers.
[i]A. Golovanov[/i]
2005 Romania National Olympiad, 3
Prove that for all positive integers $n$ there exists a single positive integer divisible with $5^n$ which in decimal base is written using $n$ digits from the set $\{1,2,3,4,5\}$.
2011 Lusophon Mathematical Olympiad, 2
A non-negative integer $n$ is said to be [i]squaredigital[/i] if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital.
2024 Polish MO Finals, 3
Determine all pairs $(p,q)$ of prime numbers with the following property: There are positive integers $a,b,c$ satisfying
\[\frac{p}{a}+\frac{p}{b}+\frac{p}{c}=1 \quad \text{and} \quad \frac{a}{p}+\frac{b}{p}+\frac{c}{p}=q+1.\]
2000 Brazil National Olympiad, 2
Let $s(n)$ be the sum of all positive divisors of $n$, so $s(6) = 12$. We say $n$ is almost perfect if $s(n) = 2n - 1$. Let $\mod(n, k)$ denote the residue of $n$ modulo $k$ (in other words, the remainder of dividing $n$ by $k$). Put $t(n) = \mod(n, 1) + \mod(n, 2) + \cdots + \mod(n, n)$.
Show that $n$ is almost perfect if and only if $t(n) = t(n-1)$.
2010 Iran Team Selection Test, 1
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that
\[f(a+1), f(a+2), \dots, f(a+n)\]
form an arithmetic progression.
2007 Korea National Olympiad, 4
For all positive integer $ n\geq 2$, prove that product of all prime numbers less or equal than $ n$ is smaller than $ 4^{n}$.
2003 China National Olympiad, 1
Find all integer triples $(a,m,n)$ such that $a^m+1|a^n+203$ where $a,m>1$.
[i]Chen Yonggao[/i]
2002 Tournament Of Towns, 6
Define a sequence $\{a_n\}_{n\ge 1}$ such that $a_1=1,a_2=2$ and $a_{n+1}$ is the smallest positive integer $m$ such that $m$ hasn't yet occurred in the sequence and also $\text{gcd}(m,a_n)\neq 1$. Show all positive integers occur in the sequence.
2005 Cono Sur Olympiad, 1
Let $a_n$ be the last digit of the sum of the digits of $20052005...2005$, where the $2005$ block occurs $n$ times. Find $a_1 +a_2 + \dots +a_{2005}$.
2009 Indonesia TST, 4
Given positive integer $ n > 1$ and define
\[ S \equal{} \{1,2,\dots,n\}.
\]
Suppose
\[ T \equal{} \{t \in S: \gcd(t,n) \equal{} 1\}.
\]
Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)
2009 Czech and Slovak Olympiad III A, 4
A positive integer $n$ is called [i]good[/i] if and only if there exist exactly $4$ positive integers $k_1, k_2, k_3, k_4$ such that $n+k_i|n+k_i^2$ ($1 \leq k \leq 4$). Prove that:
[list]
[*]$58$ is [i]good[/i];
[*]$2p$ is [i]good[/i] if and only if $p$ and $2p+1$ are both primes ($p>2$).[/list]
2009 Indonesia TST, 4
Given positive integer $ n > 1$ and define
\[ S \equal{} \{1,2,\dots,n\}.
\]
Suppose
\[ T \equal{} \{t \in S: \gcd(t,n) \equal{} 1\}.
\]
Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)
2001 Turkey Team Selection Test, 3
For all integers $x,y,z$, let \[S(x,y,z) = (xy - xz, yz-yx, zx - zy).\] Prove that for all integers $a$, $b$ and $c$ with $abc>1$, and for every integer $n\geq n_0$, there exists integers $n_0$ and $k$ with $0<k\leq abc$ such that \[S^{n+k}(a,b,c) \equiv S^n(a,b,c) \pmod {abc}.\] ($S^1 = S$ and for every integer $m\geq 1$, $S^{m+1} = S \circ S^m.$
$(u_1, u_2, u_3) \equiv (v_1, v_2, v_3) \pmod M \Longleftrightarrow u_i \equiv v_i \pmod M (i=1,2,3).$)
2010 Indonesia TST, 4
Prove that for all integers $ m$ and $ n$, the inequality
\[ \dfrac{\phi(\gcd(2^m \plus{} 1,2^n \plus{} 1))}{\gcd(\phi(2^m \plus{} 1),\phi(2^n \plus{} 1))} \ge \dfrac{2\gcd(m,n)}{2^{\gcd(m,n)}}\]
holds.
[i]Nanang Susyanto, Jogjakarta [/i]
2010 Spain Mathematical Olympiad, 3
Let $p$ be a prime number and $A$ an infinite subset of the natural numbers. Let $f_A(n)$ be the number of different solutions of $x_1+x_2+\ldots +x_p=n$, with $x_1,x_2,\ldots x_p\in A$. Does there exist a number $N$ for which $f_A(n)$ is constant for all $n<N$?
2009 Indonesia TST, 1
Prove that for all odd $ n > 1$, we have $ 8n \plus{} 4|C^{4n}_{2n}$.
2004 Canada National Olympiad, 4
Let $p$ be an odd prime. Prove that:
\[\displaystyle\sum_{k\equal{}1}^{p\minus{}1}k^{2p\minus{}1} \equiv \frac{p(p\plus{}1)}{2} \pmod{p^2}\]
1972 IMO Longlists, 42
The decimal number $13^{101}$ is given. It is instead written as a ternary number. What are the two last digits of this ternary number?
2005 Croatia National Olympiad, 3
Show that there is a unique positive integer which consists of the digits $2$ and $5$, having $2005$ digits and divisible by $2^{2005}$.
2010 CentroAmerican, 1
Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation
$n(S(n)-1)=2010.$