Found problems: 1362
2005 All-Russian Olympiad, 1
Ten mutually distinct non-zero reals are given such that for any two, either their sum or their product is rational. Prove that squares of all these numbers are rational.
2011 India IMO Training Camp, 1
Find all positive integer $n$ satisfying the conditions
$a)n^2=(a+1)^3-a^3$
$b)2n+119$ is a perfect square.
2009 Hungary-Israel Binational, 1
For a given prime $ p > 2$ and positive integer $ k$ let \[ S_k \equal{} 1^k \plus{} 2^k \plus{} \ldots \plus{} (p \minus{} 1)^k\] Find those values of $ k$ for which $ p \, |\, S_k$.
1987 IMO Longlists, 42
Find the integer solutions of the equation
\[ \left[ \sqrt{2m} \right] = \left[ n(2+\sqrt 2) \right] \]
2006 India IMO Training Camp, 3
Let $A_1,A_2,\cdots , A_n$ be arithmetic progressions of integers, each of $k$ terms, such that any two of these arithmetic progressions have at least two common elements. Suppose $b$ of these arithmetic progressions have common difference $d_1$ and the remaining arithmetic progressions have common difference $d_2$ where $0<b<n$. Prove that
\[b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.\]
2002 China Team Selection Test, 3
Let $ p_i \geq 2$, $ i \equal{} 1,2, \cdots n$ be $ n$ integers such that any two of them are relatively prime. Let:
\[ P \equal{} \{ x \equal{} \sum_{i \equal{} 1}^{n} x_i \prod_{j \equal{} 1, j \neq i}^{n} p_j \mid x_i \text{is a non \minus{} negative integer}, i \equal{} 1,2, \cdots n \}
\]
Prove that the biggest integer $ M$ such that $ M \not\in P$ is greater than $ \displaystyle \frac {n \minus{} 2}{2} \cdot \prod_{i \equal{} 1}^{n} p_i$, and also find $ M$.
2009 Indonesia MO, 3
A pair of integers $ (m,n)$ is called [i]good[/i] if
\[ m\mid n^2 \plus{} n \ \text{and} \ n\mid m^2 \plus{} m\]
Given 2 positive integers $ a,b > 1$ which are relatively prime, prove that there exists a [i]good[/i] pair $ (m,n)$ with $ a\mid m$ and $ b\mid n$, but $ a\nmid n$ and $ b\nmid m$.
2007 Vietnam National Olympiad, 2
Let $x,y$ be integer number with $x,y\neq-1$ so that $\frac{x^{4}-1}{y+1}+\frac{y^{4}-1}{x+1}\in\mathbb{Z}$. Prove that $x^{4}y^{44}-1$ is divisble by $x+1$
2014 Contests, 1
Determine the last two digits of the product of the squares of all positive odd integers less than $2014$.
1993 Bundeswettbewerb Mathematik, 4
Does there exist a non-negative integer n, such that the first four digits of n! is 1993?
2014 Postal Coaching, 1
Let $d(n)$ denote the number of positive divisors of the positive integer $n$.Determine those numbers $n$ for which $d(n^3)=5d(n)$.
1994 Brazil National Olympiad, 5
Call a super-integer an infinite sequence of decimal digits: $\ldots d_n \ldots d_2d_1$.
(Formally speaking, it is the sequence $(d_1,d_2d_1,d_3d_2d_1,\ldots)$ )
Given two such super-integers $\ldots c_n \ldots c_2c_1$ and $\ldots d_n \ldots d_2d_1$, their product $\ldots p_n \ldots p_2p_1$ is formed by taking $p_n \ldots p_2p_1$ to be the last n digits of the product $c_n \ldots c_2c_1$ and $d_n \ldots d_2d_1$.
Can we find two non-zero super-integers with zero product?
(a zero super-integer has all its digits zero)
2004 China Team Selection Test, 3
The largest one of numbers $ p_1^{\alpha_1}, p_2^{\alpha_2}, \cdots, p_t^{\alpha_t}$ is called a $ \textbf{Good Number}$ of positive integer $ n$, if $ \displaystyle n\equal{} p_1^{\alpha_1} \cdot p_2^{\alpha_2} \cdots p_t^{\alpha_t}$, where $ p_1$, $ p_2$, $ \cdots$, $ p_t$ are pairwisely different primes and $ \alpha_1, \alpha_2, \cdots, \alpha_t$ are positive integers. Let $ n_1, n_2, \cdots, n_{10000}$ be $ 10000$ distinct positive integers such that the $ \textbf{Good Numbers}$ of $ n_1, n_2, \cdots, n_{10000}$ are all equal.
Prove that there exist integers $ a_1, a_2, \cdots, a_{10000}$ such that any two of the following $ 10000$ arithmetical progressions $ \{ a_i, a_i \plus{} n_i, a_i \plus{} 2n_i, a_i \plus{} 3n_i, \cdots \}$($ i\equal{}1,2, \cdots 10000$) have no common terms.
1997 Irish Math Olympiad, 1
Given a positive integer $ n$, denote by $ \sigma (n)$ the sum of all positive divisors of $ n$. We say that $ n$ is $ abundant$ if $ \sigma (n)>2n.$ (For example, $ 12$ is abundant since $ \sigma (12)\equal{}28>2 \cdot 12$.) Let $ a,b$ be positive integers and suppose that $ a$ is abundant. Prove that $ ab$ is abundant.
2007 Pre-Preparation Course Examination, 16
Prove that $2^{2^{n}}+2^{2^{{n-1}}}+1$ has at least $n$ distinct prime divisors.
2005 Rioplatense Mathematical Olympiad, Level 3, 1
Find all numbers $n$ that can be expressed in the form $n=k+2\lfloor\sqrt{k}\rfloor+2$ for some nonnegative integer $k$.
2014 Dutch IMO TST, 4
Determine all pairs $(p, q)$ of primes for which $p^{q+1}+q^{p+1}$ is a perfect square.
2012 Brazil National Olympiad, 3
Find the least non-negative integer $n$ such that exists a non-negative integer $k$ such that the last 2012 decimal digits of $n^k$ are all $1$'s.
2007 Pre-Preparation Course Examination, 10
Let $a >1$ be a positive integer. Prove that the set $\{a^2+a-1,a^3+a-1,\cdots\}$ have a subset $S$ with infinite members and for any two members of $S$ like $x,y$ we have $\gcd(x,y)=1$. Then prove that the set of primes has infinite members.
2022 Mexican Girls' Contest, 1
Determine all finite nonempty sets $S$ of positive integers satisfying
\[ {i+j\over (i,j)}\qquad\mbox{is an element of S for all i,j in S}, \]
where $(i,j)$ is the greatest common divisor of $i$ and $j$.
2005 Federal Competition For Advanced Students, Part 1, 1
Prove that there are infinitely many multiples of 2005 that contain all the digits 0, 1, 2,...,9 an equal number of times.
2005 Moldova Team Selection Test, 4
Given functions $f,g:N^*\rightarrow N^*$, $g$ is surjective and $2f(n)^2=n^2+g(n)^2$, $\forall n>0$. Prove that if $|f(n)-n|\le2005\sqrt n$, $\forall n>0$, then $f(n)=n$ for infinitely many $n$.
2009 Germany Team Selection Test, 1
For which $ n \geq 2, n \in \mathbb{N}$ are there positive integers $ A_1, A_2, \ldots, A_n$ which are not the same pairwise and have the property that the product $ \prod^n_{i \equal{} 1} (A_i \plus{} k)$ is a power for each natural number $ k.$
2003 Iran MO (3rd Round), 5
Let $p$ be an odd prime number. Let $S$ be the sum of all primitive roots modulo $p$. Show that if $p-1$ isn't squarefree (i. e., if there exist integers $k$ and $m$ with $k>1$ and $p-1=k^2m$), then $S \equiv 0 \mod p$.
If not, then what is $S$ congruent to $\mod p$ ?
1977 IMO Longlists, 37
Let $A_1,A_2,\ldots ,A_{n+1}$ be positive integers such that $(A_i,A_{n+1})=1$ for every $i=1,2,\ldots ,n$. Show that the equation
\[x_1^{A_1}+x_2^{A_2}+\ldots + x_n^{A_n}=x_{n+1}^{A_{n+1} }\]
has an infinite set of solutions $(x_1,x_2,\ldots , x_{n+1})$ in positive integers.