Found problems: 15460
2021 Saudi Arabia JBMO TST, 1
Let $(a_n)_{n\ge 1}$ be a sequence given by $a_1 = 45$ and $$a_n = a^2_{n-1} + 15a_{n-1}$$ for $n > 1$. Prove that the sequence contains no perfect squares.
2015 China Team Selection Test, 6
Prove that there exist infinitely many integers $n$ such that $n^2+1$ is squarefree.
2022 Baltic Way, 19
Find all triples $(x, y, z)$ of nonnegative integers such that
$$ x^5+x^4+1=3^y7^z $$
2014 AIME Problems, 3
Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and denominator have a sum of $1000$.
1989 Swedish Mathematical Competition, 3
Find all positive integers $n$ such that $n^3 - 18n^2 + 115n - 391$ is the cube of a positive intege
2007 National Olympiad First Round, 6
How many positive integers $n$ are there such that $n!(2n+1)$ and $221$ are relatively prime?
$
\textbf{(A)}\ 10
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 12
\qquad\textbf{(D)}\ 13
\qquad\textbf{(E)}\ \text{None of the above}
$
Mathematical Minds 2023, P1
Determine all positive integers $n{}$ which can be expressed as $d_1+d_2+d_3$ where $d_1,d_2,d_3$ are distinct positive divisors of $n{}$.
2019 India PRMO, 24
For $n \geq 1$, let $a_n$ be the number beginning with $n$ $9$'s followed by $744$; eg., $a_4=9999744$. Define $$f(n)=\text{max}\{m\in \mathbb{N} \mid2^m ~ \text{divides} ~ a_n \}$$, for $n\geq 1$. Find $f(1)+f(2)+f(3)+ \cdots + f(10)$.
2010 Estonia Team Selection Test, 1
For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$
Let $n$ be a positive integer. Prove that the following conditions are equivalent:
(i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$,
(ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.
2023 BMT, 16
Let $n$ be the smallest positive integer such that there exist integers, $a$, $b$, and $c$, satisfying:
$$\frac{n}{2}= a^2 \,\,\, \,\,\, \frac{n}{3}= b^3 \,\,\ , \,\,\ \frac{n}{5}= c^5.$$
Find the number of positive integer factors of $n$.
2005 International Zhautykov Olympiad, 3
Find all prime numbers $ p,q < 2005$ such that $ q | p^{2} \plus{} 8$ and $ p|q^{2} \plus{} 8.$
2016 Romania Team Selection Tests, 4
Determine the integers $k\geq 2$ for which the sequence $\Big\{ \binom{2n}{n} \pmod{k}\Big\}_{n\in \mathbb{Z}_{\geq 0}}$ is eventually periodic.
2012 ELMO Shortlist, 6
Consider a directed graph $G$ with $n$ vertices, where $1$-cycles and $2$-cycles are permitted. For any set $S$ of vertices, let $N^{+}(S)$ denote the out-neighborhood of $S$ (i.e. set of successors of $S$), and define $(N^{+})^k(S)=N^{+}((N^{+})^{k-1}(S))$ for $k\ge2$.
For fixed $n$, let $f(n)$ denote the maximum possible number of distinct sets of vertices in $\{(N^{+})^k(X)\}_{k=1}^{\infty}$, where $X$ is some subset of $V(G)$. Show that there exists $n>2012$ such that $f(n)<1.0001^n$.
[i]Linus Hamilton.[/i]
2023 ELMO Shortlist, N5
An ordered pair \((k,n)\) of positive integers is [i]good[/i] if there exists an ordered quadruple \((a,b,c,d)\) of positive integers such that \(a^3+b^k=c^3+d^k\) and \(abcd=n\). Prove that there exist infinitely many positive integers \(n\) such that \((2022,n)\) is not good but \((2023,n)\) is good.
[i]Proposed by Luke Robitaille[/i]
2015 Argentina National Olympiad Level 2, 4
Let $N$ be the number of ordered lists of $9$ positive integers $(a,b,c,d,e,f,g,h,i)$ such that
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f}+\frac{1}{g}+\frac{1}{h}+\frac{1}{i}=1.$$
Determine whether $N$ is even or odd.
2013 Austria Beginners' Competition, 1
Find all natural numbers $n> 1$ for which the following applies:
The sum of the number $n$ and its second largest divisor is $2013$.
(R. Henner, Vienna)
1998 Bosnia and Herzegovina Team Selection Test, 6
Sequence of integers $\{u_n\}_{n \in \mathbb{N}_0}$ is given as: $u_0=0$, $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for all $n \in \mathbb{N}_0$
$a)$ Find $u_{1998}$
$b)$ If $p$ is a positive integer and $m=(2^p-1)^2$, find $u_m$
2023 Brazil National Olympiad, 6
For a positive integer $k$, let $p(k)$ be the smallest prime that does not divide $k$. Given a positive integer $a$, define the infinite sequence $a_0, a_1, \ldots$ by $a_0 = a$ and, for $n > 0$, $a_n$ is the smallest positive integer with the following properties:
• $a_n$ has not yet appeared in the sequence, that is, $a_n \neq a_i$ for $0 \leq i < n$;
• $(a_{n-1})^{a_n} - 1$ is a multiple of $p(a_{n-1})$.
Prove that every positive integer appears as a term in the sequence, that is, for every positive integer $m$ there is $n$ such that $a_n = m$.
India EGMO 2024 TST, 4
Let $N \geq 3$ be an integer, and let $a_0, \dots, a_{N-1}$ be pairwise distinct reals so that $a_i \geq a_{2i}$ for all $i$ (indices are taken $\bmod~ N$). Find all possible $N$ for which this is possible.
[i]Proposed by Sutanay Bhattacharya[/i]
2022 Germany Team Selection Test, 1
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?
2005 South East Mathematical Olympiad, 4
Find all positive integer solutions $(a, b, c)$ to the function $a^{2} + b^{2} + c^{2} = 2005$, where $a \leq b \leq c$.
2011 LMT, 18
Let $x$ and $y$ be distinct positive integers below $15$. For any two distinct numbers $a, b$ from the set $\{2, x,y\}$, $ab + 1$ is always a positive square. Find all possible values of the square $xy + 1$.
2020 LIMIT Category 1, 2
Prove that any integer has a multiple consisting of all ten digits $\{0,1,2,3,4,5,6,7,8,9\}$.
\\
[i]Note: Any digit can be repeated any number of times[/i]
2012 Middle European Mathematical Olympiad, 7
Find all triplets $ (x,y,z) $ of positive integers such that
\[ x^y + y^x = z^y \]\[ x^y + 2012 = y^{z+1} \]
2010 India IMO Training Camp, 12
Prove that there are infinitely many positive integers $m$ for which there exists consecutive odd positive integers $p_m<q_m$ such that $p_m^2+p_mq_m+q_m^2$ and $p_m^2+m\cdot p_mq_m+q_m^2$ are both perfect squares. If $m_1, m_2$ are two positive integers satisfying this condition, then we have $p_{m_1}\neq p_{m_2}$