Found problems: 15460
2015 Kazakhstan National Olympiad, 4
$P_k(n) $ is the product of all positive divisors of $n$ that are divisible by $k$ (the empty product is equal to $1$). Show that $P_1(n)P_2(n)\cdots P_n(n)$ is a perfect square, for any positive integer $n$.
2010 ELMO Shortlist, 5
Find the set $S$ of primes such that $p \in S$ if and only if there exists an integer $x$ such that $x^{2010} + x^{2009} + \cdots + 1 \equiv p^{2010} \pmod{p^{2011}}$.
[i]Brian Hamrick.[/i]
2009 Silk Road, 4
Prove that for any prime number $p$ there are infinitely many fours $(x, y, z, t)$ pairwise distinct natural numbers such that the number $(x^2+p t^2)(y^2+p t^2)(z^2+p t^2)$ is a perfect square.
2009 VTRMC, Problem 2
Given that $40!=\overline{abcdef283247897734345611269596115894272pqrstuvwx}$, find $a,b,c,d,e,f,p,q,r,s,t,u,v,w,x$.
2013 Czech-Polish-Slovak Junior Match, 4
Determine the largest two-digit number $d$ with the following property:
for any six-digit number $\overline{aabbcc}$ number $d$ is a divisor of the number $\overline{aabbcc}$ if and only if the number $d$ is a divisor of the corresponding three-digit number $\overline{abc}$.
Note The numbers $a \ne 0, b$ and $c$ need not be different.
2023 BMT, 7
For an integer $n > 0$, let $p(n)$ be the product of the digits of $n$. Compute the sum of all integers $n$ such that $n - p(n) = 52$.
2018 USA Team Selection Test, 1
Let $n \ge 2$ be a positive integer, and let $\sigma(n)$ denote the sum of the positive divisors of $n$. Prove that the $n^{\text{th}}$ smallest positive integer relatively prime to $n$ is at least $\sigma(n)$, and determine for which $n$ equality holds.
[i]Proposed by Ashwin Sah[/i]
2005 Junior Balkan MO, 4
Find all 3-digit positive integers $\overline{abc}$ such that \[ \overline{abc} = abc(a+b+c) , \] where $\overline{abc}$ is the decimal representation of the number.
1988 Balkan MO, 4
Let $(a_{n})_{n\geq 1}$ be a sequence defined by $a_{n}=2^{n}+49$. Find all values of $n$ such that $a_{n}=pg, a_{n+1}=rs$, where $p,q,r,s$ are prime numbers with $p<q, r<s$ and $q-p=s-r$.
2003 Iran MO (3rd Round), 8
A positive integer $n$ is said to be a [i]perfect power[/i] if $n=a^b$ for some integers $a,b$ with $b>1$.
$(\text{a})$ Find $2004$ perfect powers in arithmetic progression.
$(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.
2006 Germany Team Selection Test, 1
For any positive integer $n$, let $w\left(n\right)$ denote the number of different prime divisors of the number $n$. (For instance, $w\left(12\right)=2$.) Show that there exist infinitely many positive integers $n$ such that $w\left(n\right)<w\left(n+1\right)<w\left(n+2\right)$.
2006 All-Russian Olympiad Regional Round, 8.1
Find some nine-digit number $N$, consisting of different digits, such that among all the numbers obtained from $N$ by crossing out seven digits, there would be no more than one prime. Prove that the number found is correct. (If the number obtained by crossing out the digits starts at zero, then the zero is crossed out.)
2007 Peru MO (ONEM), 3
We say that a natural number of at least two digits $E$ is [i]special [/i] if each time two adjacent digits of $E$ are added, a divisor of $E$ is obtained. For example, $2124$ is special, since the numbers $2 + 1$, $1 + 2$ and $2 + 4$ are all divisors of $2124$. Find the largest value of $n$ for which there exist $n$ consecutive natural numbers such that they are all special.
2003 Korea - Final Round, 2
For a positive integer, $m$, answer the following questions.
1) Show that $2^{m+1}+1$ is a prime number, when $2^{m+1}+1$ is a factor of $3^{2^m}+1$.
2) Is converse of 1) true?
2011 Bosnia And Herzegovina - Regional Olympiad, 2
If $p>2$ is prime number and $m$ and $n$ are positive integers such that $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}$$ Prove that $p$ divides $m$
2017 Irish Math Olympiad, 5
The sequence $a = (a_0, a_1,a_2,...)$ is defined by $a_0 = 0, a_1 =2$ and $$a_{n+2} = 2a_{n+1} + 41a_n$$Prove that $a_{2016}$ is divisible by $2017.$
2020 BMT Fall, 20
Compute the number of positive integers $n \le 1890$ such that n leaves an odd remainder when divided by all of $2, 3, 5$, and $7$.
2022 Regional Competition For Advanced Students, 4
We are given the set $$M = \{-2^{2022}, -2^{2021}, . . . , -2^{2}, -2, -1, 1, 2, 2^2, . . . , 2^{2021}, 2^{2022}\}.$$
Let $T$ be a subset of $M$, such that neighbouring numbers have the same difference when the elements are ordered by size.
(a) Determine the maximum number of elements that such a set $T$ can contain.
(b) Determine all sets $T$ with the maximum number of elements.
[i](Walther Janous)[/i]
2005 iTest, 4
How many multiples of $2005$ are factors of $(2005)^2$?
2015 Brazil Team Selection Test, 2
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/i]
2023 AMC 12/AHSME, 24
Integers $a, b, c, d$ satisfy the following:
$abcd=2^6\cdot 3^9\cdot 5^7$
$\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3$
$\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3$
$\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3$
$\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2$
$\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2$
$\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2$
Find $\text{gcd}(a,b,c,d)$
$\textbf{(A)}~30\qquad\textbf{(B)}~45\qquad\textbf{(C)}~3\qquad\textbf{(D)}~15\qquad\textbf{(E)}~6$
2021 Kyiv City MO Round 1, 8.1
Find all positive integers $n$ that can be subtracted from both the numerator and denominator of the fraction $\frac{1234}{6789}$, to get, after the reduction, the fraction of form $\frac{a}{b}$, where $a, b$ are single digit numbers.
[i]Proposed by Bogdan Rublov[/i]
1975 Spain Mathematical Olympiad, 6
Let $\{x_n\}$ and $\{y_n\}$ be two sequences of natural numbers defined as follow:
$x_1 = 1, \,\,\, x_2 = 1, \,\,\, x_{n+2} = x_{n+1} + 2x_n$ for $n = 1, 2, 3, ...$
$y_1 = 1, \,\,\, y_2 = 7, \,\,\, y_{n+2} = 2y_{n+1} + 3y_n$ for $n = 1, 2, 3, ...$
Prove that, except for the case $x_1 = y_1 = 1$, there is no natural value that occurs in the two sequences.
2021 JBMO TST - Turkey, 2
For which positive integers $n$, one can find a non-integer rational number $x$ such that $$x^n+(x+1)^n$$ is an integer?
2007 Indonesia TST, 3
Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that:
(i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and
(ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.