Found problems: 1766
2023 Bundeswettbewerb Mathematik, 1
Determine the greatest common divisor of the numbers $p^6-7p^2+6$ where $p$ runs through the prime numbers $p \ge 11$.
2020 Bundeswettbewerb Mathematik, 1
Show that there are infinitely many perfect squares of the form $50^m-50^n$, but no perfect square of the form $2020^m+2020^n$, where $m$ and $n$ are positive integers.
1985 Iran MO (2nd round), 4
Let $x$ and $y$ be two real numbers. Prove that the equations
\[\lfloor x \rfloor + \lfloor y \rfloor =\lfloor x +y \rfloor , \quad \lfloor -x \rfloor + \lfloor -y \rfloor =\lfloor -x-y \rfloor\]
Holds if and only if at least one of $x$ or $y$ be integer.
2007 USA Team Selection Test, 4
Determine whether or not there exist positive integers $ a$ and $ b$ such that $ a$ does not divide $ b^n \minus{} n$ for all positive integers $ n$.
2020 Kazakhstan National Olympiad, 3
Let $p$ be a prime number and $k,r$ are positive integers such that $p>r$. If $pk+r$ divides $p^p+1$ then prove that $r$ divides $k$.
1950 Miklós Schweitzer, 2
Show that there exists a positive constant $ c$ with the following property: To every positive irrational $ \alpha$, there can be found infinitely many fractions $ \frac{p}{q}$ with $ (p,q)\equal{}1$ satisfying
$ \left|\alpha\minus{}\frac{p}{q}\right|\le \frac{c}{q^2}$
2004 Junior Balkan MO, 3
If the positive integers $x$ and $y$ are such that $3x + 4y$ and $4x + 3y$ are both perfect squares, prove that both $x$ and $y$ are both divisible with $7$.
2013 Federal Competition For Advanced Students, Part 1, 1
Show that if for non-negative integers $m$, $n$, $N$, $k$ the equation \[(n^2+1)^{2^k}\cdot(44n^3+11n^2+10n+2)=N^m\] holds, then $m = 1$.
1964 IMO Shortlist, 1
(a) Find all positive integers $ n$ for which $ 2^n\minus{}1$ is divisible by $ 7$.
(b) Prove that there is no positive integer $ n$ for which $ 2^n\plus{}1$ is divisible by $ 7$.
2013 Canadian Mathematical Olympiad Qualification Repechage, 3
A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$, $\text{lcm}(x, z) = 900$, and $\text{lcm}(y, z) = n$, where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property.
2005 Taiwan National Olympiad, 1
Find all integer solutions $(x,y)$ to the equation $\displaystyle \frac{x+y}{x^2-xy+y^2}=\frac{3}{7}$.
2010 Stars Of Mathematics, 4
Let $a,b,c$ be given positive integers. Prove that there exists some positive integer $N$ such that
\[ a\mid Nbc+b+c,\ b\mid Nca+c+a,\ c\mid Nab+a+b \]
if and only if, denoting $d=\gcd(a,b,c)$ and $a=dx$, $b=dy$, $c=dz$, the positive integers $x,y,z$ are pairwise coprime, and also $\gcd(d,xyz) \mid x+y+z$.
(Dan Schwarz)
2012 Canada National Olympiad, 2
For any positive integers $n$ and $k$, let $L(n,k)$ be the least common multiple of the $k$ consecutive integers $n,n+1,\ldots ,n+k-1$. Show that for any integer $b$, there exist integers $n$ and $k$ such that $L(n,k)>bL(n+1,k)$.
2014 Kyiv Mathematical Festival, 4a
a) Prove that for every positive integer $y$ the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=x(x+1)$ holds for infinitely many positive integers $x.$
b) Prove that there exists positive integer $y$ such that the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=y(y+1)$ holds for at least 2014 positive integers $x.$
2002 Poland - Second Round, 1
Find all numbers $p\le q\le r$ such that all the numbers
\[pq+r,pq+r^2,qr+p,qr+p^2,rp+q,rp+q^2 \]
are prime.
1977 IMO Longlists, 19
Given any integer $m>1$ prove that there exist infinitely many positive integers $n$ such that the last $m$ digits of $5^n$ are a sequence $a_m,a_{m-1},\ldots ,a_1=5\ (0\le a_j<10)$ in which each digit except the last is of opposite parity to its successor (i.e., if $a_i$ is even, then $a_{i-1}$ is odd, and if $a_i$ is odd, then $a_{i-1}$ is even).
1997 Irish Math Olympiad, 1
Find all pairs of integers $ (x,y)$ satisfying $ 1\plus{}1996x\plus{}1998y\equal{}xy.$
2024 Czech-Polish-Slovak Junior Match, 4
How many positive integers $n<2024$ are divisible by $\lfloor \sqrt{n}\rfloor-1$?
1999 Italy TST, 1
Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.
2005 Poland - Second Round, 1
The polynomial $W(x)=x^2+ax+b$ with integer coefficients has the following property: for every prime number $p$ there is an integer $k$ such that both $W(k)$ and $W(k+1)$ are divisible by $p$. Show that there is an integer $m$ such that $W(m)=W(m+1)=0$.
2014 District Olympiad, 4
Determine all positive integers $a$ for which there exist exactly $2014$ positive integers $b$ such that $\displaystyle2\leq\frac{a}{b}\leq5$.
2014 China Team Selection Test, 6
For positive integer $k>1$, let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$, as $12$ can be factored in these $4$ ways: $12,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3$.
Prove: If $n$ is a positive integer greater than $1$, $p$ is a prime factor of $n$, then $f(n)\leq \frac{n}{p}$
1979 IMO Longlists, 23
Consider the set $E$ consisting of pairs of integers $(a, b)$, with $a \geq 1$ and $b \geq 1$, that satisfy in the decimal system the following properties:
[b](i)[/b] $b$ is written with three digits, as $\overline{\alpha_2\alpha_1\alpha_0}$, $\alpha_2 \neq 0$;
[b](ii)[/b] $a$ is written as $\overline{\beta_p \ldots \beta_1\beta_0}$ for some $p$;
[b](iii)[/b] $(a + b)^2$ is written as $\overline{\beta_p\ldots \beta_1 \beta_0 \alpha_2 \alpha_1 \alpha_0}.$
Find the elements of $E$.
2024 German National Olympiad, 4
Let $k>2$ be a positive integer such that the $k$-digit number $n_k=133\dots 3$, consisting of a digit $1$ followed by $k-1$ digits $3$ is prime. Show that $24 \mid k(k+2)$.
2002 Tuymaada Olympiad, 1
A positive integer $c$ is given. The sequence $\{p_{k}\}$ is constructed by the following rule: $p_{1}$ is arbitrary prime and for $k\geq 1$ the number $p_{k+1}$ is any prime divisor of $p_{k}+c$ not present among the numbers $p_{1}$, $p_{2}$, $\dots$, $p_{k}$. Prove that the sequence $\{p_{k}\}$ cannot be infinite.
[i]Proposed by A. Golovanov[/i]