Found problems: 15460
2020 Moldova Team Selection Test, 1
All members of geometrical progression $(b_n)_{n\geq1}$ are members of some arithmetical progression. It is known that $b_1$ is an integer. Prove that all members of this geometrical progression are integers. (progression is infinite)
2007 District Olympiad, 4
Let $n$ be a positive integer which is not prime. Prove that there exist $k, a_{1},a_{2},...a_{k}>1$ positive integers such that $a_{1}+a_{2}+\cdots+a_{k}=n(\frac1{a_{1}}+\frac1{a_{2}}+\cdots+\frac1{a_{k}})$
Edit: the $a_{i}'s$ have to be grater than 1. Sorry, my mistake :blush:
2004 China Team Selection Test, 3
$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying:
(1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$.
(2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$.
Find the largest possible value of $ |S|$.
1993 Tournament Of Towns, (381) 3
A natural number $A$ is given. One may add to it one of its divisors $d$ ($1 < d < A$). One may then repeat this operation with the new number $A + d$ and so on. Prove that starting from $A = 4$ one can get any composite number by these operations.
(M Vyalyi)
KoMaL A Problems 2023/2024, A. 876
Find all non negative integers $a{}$ and $b{}$ satisfying $5^a+6=31^b$
[i]Proposed by Erik Füredi, Budapest[/i]
2009 South africa National Olympiad, 1
Determine the smallest integer $n > 1$ with the property that $n^2(n - 1)$ is divisible by 2009.
1998 Akdeniz University MO, 1
Prove that, for $k \in {\mathbb Z^+}$
$$k(k+1)(k+2)(k+3)$$
is not a perfect square.
1983 IMO Shortlist, 2
Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is [i]superabundant[/i] (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$
Prove that there exists an infinity of [i]superabundant[/i] numbers.
2016 Brazil Team Selection Test, 3
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
2002 IMO Shortlist, 2
Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.
2022 Stars of Mathematics, 4
Let $a{}$ be an even positive integer which is not a power of two. Prove that at least one of $2^{2^n}+1$ and $a^{2^n}+1$ is composite, for infinitely many positive integers $n$.
[i]Bojan Bašić[/i]
Sri Lankan Mathematics Challenge Competition 2022, P1
[b]Problem 1[/b] : Find the smallest positive integer $n$, such that $\sqrt[5]{5n}$, $\sqrt[6]{6n}$ , $\sqrt[7]{7n}$ are integers.
2021 Saudi Arabia BMO TST, 3
Let $x$, $y$ and $z$ be odd positive integers such that $\gcd \ (x, y, z) = 1$ and the sum $x^2 +y^2 +z^2$ is divisible by $x+y+z$. Prove that $x+y+z- 2$ is not divisible by $3$.
2013 Balkan MO Shortlist, N6
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^3$
2012 Indonesia TST, 4
Determine all integer $n > 1$ such that
\[\gcd \left( n, \dfrac{n-m}{\gcd(n,m)} \right) = 1\]
for all integer $1 \le m < n$.
2021 Moldova Team Selection Test, 2
Prove that if $p$ and $q$ are two prime numbers, such that
$$p+p^2+p^3+...+p^q=q+q^2+q^3+...+q^p,$$
then $p=q$.
2024 Dutch BxMO/EGMO TST, IMO TSTST, 1
Find all pairs of prime numbers $p, q$ for which there exist positive integers $(m, n)$ such that $$(p+q)^m=(p-q)^n$$.
2015 Canadian Mathematical Olympiad Qualification, 3
Let $N$ be a 3-digit number with three distinct non-zero digits. We say that $N$ is [i]mediocre[/i] if it has the property that when all six 3-digit permutations of $N$ are written down, the average is $N$. For example, $N = 481$ is mediocre, since it is the average of $\{418, 481, 148, 184, 814, 841\}$.
Determine the largest mediocre number.
1997 Tournament Of Towns, (524) 1
How many integers from $1$ to $1997$ have the sum of their digits divisible by $5$?
(AI Galochkin)
2011 Spain Mathematical Olympiad, 3
The sequence $S_0,S_1,S_2,\ldots$ is defined by[list][*]$S_n=1$ for $0\le n\le 2011$, and
[*]$S_{n+2012}=S_{n+2011}+S_n$ for $n\ge 0$.[/list]Prove that $S_{2011a}-S_a$ is a multiple of $2011$ for all nonnegative integers $a$.
2019 BMT Spring, 8
Let $(k_i)$ be a sequence of unique nonzero integers such that $x^2- 5x + k_i$ has rational solutions. Find the minimum possible value of $$\frac15 \sum_{i=1}^{\infty} \frac{1}{k_i}$$
1988 IMO Longlists, 14
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.
1976 IMO Longlists, 41
Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$
2020 BMT Fall, 21
Let $P$ be the probability that the product of $2020$ real numbers chosen independently and uniformly at random from the interval $[-1, 2]$ is positive. The value of $2P - 1$ can be written in the form $\left(\frac{m}{n}\right)^b$ , where $m, n$ and $b$ are positive integers such that $m$ and $n$ are relatively prime and $b$ is as large as possible. Compute $m + n + b$.
1979 Brazil National Olympiad, 4
Show that the number of positive integer solutions to $x_1 + 2^3x_2 + 3^3x_3 + \ldots + 10^3x_{10} = 3025$ (*) equals the number of non-negative integer solutions to the equation $y_1 + 2^3y_2 + 3^3y_3 + \ldots + 10^3y_{10} = 0$. Hence show that (*) has a unique solution in positive integers and find it.