Found problems: 1362
2009 Brazil National Olympiad, 1
Emerald writes $ 2009^2$ integers in a $ 2009\times 2009$ table, one number in each entry of the table. She sums all the numbers in each row and in each column, obtaining $ 4018$ sums. She notices that all sums are distinct. Is it possible that all such sums are perfect squares?
2009 Rioplatense Mathematical Olympiad, Level 3, 2
Find all pairs $(a,b)$ of integers with $a>1$ and $b>1$ such that $a$ divides $b+1$ and $b$ divides $a^3-1$.
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$.
2011 Junior Balkan MO, 2
Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$
1982 USAMO, 2
Let $X_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$, \[(*)\quad\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}\] for $(m,n)=(2,3),(3,2),(2,5)$, or $(5,2)$. Determine [i]all[/i] other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,z$ such that $x+y+z=0$.
2001 Tournament Of Towns, 2
In three piles there are $51, 49$, and $5$ stones, respectively. You can combine any two piles into one pile or divide a pile consisting of an even number of stones into two equal piles. Is it possible to get $105$ piles with one stone in each?
2006 Rioplatense Mathematical Olympiad, Level 3, 3
An infinite sequence $x_1,x_2,\ldots$ of positive integers satisfies \[ x_{n+2}=\gcd(x_{n+1},x_n)+2006 \] for each positive integer $n$. Does there exist such a sequence which contains exactly $10^{2006}$ distinct numbers?
2002 China Team Selection Test, 2
Find all non-negative integers $m$ and $n$, such that $(2^n-1) \cdot (3^n-1)=m^2$.
2011 District Round (Round II), 3
Find all pairs $(m, n)$ of positive integers for which $4 (mn +1)$ is divisible by $(m + n)^2$.
2010 Tournament Of Towns, 1
Alex has a piece of cheese. He chooses a positive number $a\neq 1$ and cuts the piece into several pieces one by one. Every time he chooses a piece and cuts it in the same ratio $1:a.$ His goal is to divide the cheese into two piles of equal masses. Can he do it?
2012 Indonesia TST, 4
The sequence $a_i$ is defined as $a_1 = 1$ and
\[a_n = a_{\left\lfloor \dfrac{n}{2} \right\rfloor} + a_{\left\lfloor \dfrac{n}{3} \right\rfloor} + a_{\left\lfloor \dfrac{n}{4} \right\rfloor} + \cdots + a_{\left\lfloor \dfrac{n}{n} \right\rfloor} + 1\]
for every positive integer $n > 1$. Prove that there are infinitely many values of $n$ such that $a_n \equiv n \mod 2012$.
1987 China Team Selection Test, 2
Find all positive integer $n$ such that the equation $x^3+y^3+z^3=n \cdot x^2 \cdot y^2 \cdot z^2$ has positive integer solutions.
2005 MOP Homework, 6
Let $p$ be a prime number, and let $0 \le a_1<a_2<...<a_m<p$ and $0 \le b_1<b_2<...<b_n<p$ be arbitrary integers. Denote by $k$ the number of different remainders of $a_i+b_j$, $1 \le i \le m$ and $1 \le j \le n$, modulo $p$. Prove that
(i) if $m+n>p$, then $k=p$
(ii) if $m+n \le p$, then $k \ge m+n-1$
2014 Contests, 3
For all integers $n\ge 2$ with the following property:
[list]
[*] for each pair of positive divisors $k,~\ell <n$, at least one of the numbers $2k-\ell$ and $2\ell-k$ is a (not necessarily positive) divisor of $n$ as well.[/list]
1999 IberoAmerican, 1
Let $B$ be an integer greater than 10 such that everyone of its digits belongs to the set $\{1,3,7,9\}$. Show that $B$ has a [b]prime divisor[/b] greater than or equal to 11.
1992 Vietnam Team Selection Test, 1
Let two natural number $n > 1$ and $m$ be given. Find the least positive integer $k$ which has the following property: Among $k$ arbitrary integers $a_1, a_2, \ldots, a_k$ satisfying the condition $a_i - a_j$ ( $1 \leq i < j \leq k$) is not divided by $n$, there exist two numbers $a_p, a_s$ ($p \neq s$) such that $m + a_p - a_s$ is divided by $n$.
2005 India IMO Training Camp, 2
Given real numbers $a,\alpha,\beta, \sigma \ and \ \varrho$ s.t. $\sigma, \varrho > 0$ and $\sigma \varrho = \frac{1}{16}$, prove that there exist integers $x$ and $y$ s.t.
\[ - \sigma \leq (x+\alpha_(ax + y + \beta ) \leq \varrho \]
2002 China Team Selection Test, 3
The positive integers $ \alpha, \beta, \gamma$ are the roots of a polynomial $ f(x)$ with degree $ 4$ and the coefficient of the first term is $ 1$. If there exists an integer such that $ f(\minus{}1)\equal{}f^2(s)$.
Prove that $ \alpha\beta$ is not a perfect square.
2001 Tournament Of Towns, 2
Clara computed the product of the first $n$ positive integers, and Valerie computed the product of the first $m$ even positive integers, where $m\ge2$. They got the same answer. Prove that one of them had made a mistake.
2001 India IMO Training Camp, 2
A strictly increasing sequence $(a_n)$ has the property that $\gcd(a_m,a_n) = a_{\gcd(m,n)}$ for all $m,n\in \mathbb{N}$. Suppose $k$ is the least positive integer for which there exist positive integers $r < k < s$ such that $a_k^2 = a_ra_s$. Prove that $r | k$ and $k | s$.
2010 Albania Team Selection Test, 4
With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?
2014 Postal Coaching, 2
Let $d(n)$ be the number of positive divisors of a natural number $n$.Find all $k\in \mathbb{N}$ such that there exists $n\in \mathbb{N}$ with $d(n^2)/d(n)=k$.
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]
1983 Canada National Olympiad, 1
Find all positive integers $w$, $x$, $y$ and $z$ which satisfy $w! = x! + y! + z!$.
1986 IMO Longlists, 16
Given a positive integer $k$, find the least integer $n_k$ for which there exist five sets $S_1, S_2, S_3, S_4, S_5$ with the following properties:
\[|S_j|=k \text{ for } j=1, \cdots , 5 , \quad |\bigcup_{j=1}^{5} S_j | = n_k ;\]
\[|S_i \cap S_{i+1}| = 0 = |S_5 \cap S_1|, \quad \text{for } i=1,\cdots ,4 \]