Found problems: 2008
1983 Vietnam National Olympiad, 1
Are there positive integers $a, b$ with $b \ge 2$ such that $2^a + 1$ is divisible by $2^b - 1$?
2009 Singapore Senior Math Olympiad, 2
Find all positive integers $ m,n $ that satisfy the equation \[ 3.2^m +1 = n^2 \]
2012 USA TSTST, 3
Let $\mathbb N$ be the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function satisfying the following two conditions:
(a) $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime.
(b) $n \le f(n) \le n+2012$ for all $n$.
Prove that for any natural number $n$ and any prime $p$, if $p$ divides $f(n)$ then $p$ divides $n$.
2010 Contests, 3
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
2014 National Olympiad First Round, 22
What is remainder when $2014^{2015}$ is divided by $121$?
$
\textbf{(A)}\ 45
\qquad\textbf{(B)}\ 34
\qquad\textbf{(C)}\ 23
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ 1
$
2014 NIMO Problems, 6
Let $\varphi(k)$ denote the numbers of positive integers less than or equal to $k$ and relatively prime to $k$. Prove that for some positive integer $n$, \[ \varphi(2n-1) + \varphi(2n+1) < \frac{1}{1000} \varphi(2n). \][i]Proposed by Evan Chen[/i]
2013 Olympic Revenge, 4
Find all triples $(p,n,k)$ of positive integers, where $p$ is a Fermat's Prime, satisfying \[p^n + n = (n+1)^k\].
[i]Observation: a Fermat's Prime is a prime number of the form $2^{\alpha} + 1$, for $\alpha$ positive integer.[/i]
2015 District Olympiad, 1
[b]a)[/b] Solve the equation $ x^2-x+2\equiv 0\pmod 7. $
[b]b)[/b] Determine the natural numbers $ n\ge 2 $ for which the equation $ x^2-x+2\equiv 0\pmod n $ has an unique solution modulo $ n. $