Found problems: 2008
2013 Online Math Open Problems, 22
Find the sum of all integers $m$ with $1 \le m \le 300$ such that for any integer $n$ with $n \ge 2$, if $2013m$ divides $n^n-1$ then $2013m$ also divides $n-1$.
[i]Proposed by Evan Chen[/i]
1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 3
What is the last digit of $ 17^{1996}$?
A. 1
B. 3
C. 5
D. 7
E. 9
2011 National Olympiad First Round, 6
For how many primes $p$, $|p^4-86|$ is also prime?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$
1998 National Olympiad First Round, 27
For which of the following $ n$, $ n\times n$ chessboard cannot be covered using at most one unit square piece and many L-shaped pieces (an L-shaped piece is a 2x2 piece with one square removed)?
$\textbf{(A)}\ 96 \qquad\textbf{(B)}\ 97 \qquad\textbf{(C)}\ 98 \qquad\textbf{(D)}\ 99 \qquad\textbf{(E)}\ 100$
2005 Flanders Math Olympiad, 1
For all positive integers $n$, find the remainder of $\dfrac{(7n)!}{7^n \cdot n!}$ upon division by 7.
2014 Brazil National Olympiad, 2
Find all integers $n$, $n>1$, with the following property: for all $k$, $0\le k < n$, there exists a multiple of $n$ whose digits sum leaves a remainder of $k$ when divided by $n$.
2012 ITAMO, 4
Let $x_1,x_2,x_3, \cdots$ be a sequence defined by the following recurrence relation:
\[ \begin{cases}x_{1}&= 4\\ x_{n+1}&= x_{1}x_{2}x_{3}\cdots x_{n}+5\text{ for }n\ge 1\end{cases} \]
The first few terms of the sequence are $x_1=4,x_2=9,x_3=41 \cdots$
Find all pairs of positive integers $\{a,b\}$ such that $x_a x_b$ is a perfect square.
2005 IMO Shortlist, 6
Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$.
[i]Proposed by Mohsen Jamali, Iran[/i]
2005 Junior Balkan Team Selection Tests - Moldova, 4
Let the $A$ be the set of all nonenagative integers.
It is given function such that $f:\mathbb{A}\rightarrow\mathbb{A}$ with $f(1) = 1$ and for every element $n$ od set $A$ following holds:
[b]1)[/b] $3 f(n) \cdot f(2n+1) = f(2n) \cdot (1+3 \cdot f(n))$;
[b]2)[/b] $f(2n) < 6f(n)$,
Find all solutions of $f(k)+f(l) = 293$, $k<l$.
PEN M Problems, 17
A sequence of integers, $\{a_{n}\}_{n \ge 1}$ with $a_{1}>0$, is defined by \[a_{n+1}=\frac{a_{n}}{2}\;\;\; \text{if}\;\; n \equiv 0 \;\; \pmod{4},\] \[a_{n+1}=3 a_{n}+1 \;\;\; \text{if}\;\; n \equiv 1 \; \pmod{4},\] \[a_{n+1}=2 a_{n}-1 \;\;\; \text{if}\;\; n \equiv 2 \; \pmod{4},\] \[a_{n+1}=\frac{a_{n}+1}{4}\;\;\; \text{if}\;\; n \equiv 3 \; \pmod{4}.\] Prove that there is an integer $m$ such that $a_{m}=1$.
2019 IMAR Test, 3
Consider a natural number $ n\equiv 9\pmod {25}. $ Prove that there exist three nonnegative integers $ a,b,c $ having the property that:
$$ n=\frac{a(a+1)}{2} +\frac{b(b+1)}{2} +\frac{c(c+1)}{2} $$
2012 Kurschak Competition, 2
Denote by $E(n)$ the number of $1$'s in the binary representation of a positive integer $n$. Call $n$ [i]interesting[/i] if $E(n)$ divides $n$. Prove that
(a) there cannot be five consecutive interesting numbers, and
(b) there are infinitely many positive integers $n$ such that $n$, $n+1$ and $n+2$ are each interesting.
PEN A Problems, 8
The integers $a$ and $b$ have the property that for every nonnegative integer $n$ the number of $2^n{a}+b$ is the square of an integer. Show that $a=0$.
2003 India National Olympiad, 4
Find all $7$-digit numbers which use only the digits $5$ and $7$ and are divisible by $35$.
2014 Greece National Olympiad, 3
For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once).
Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that we can achieve $\frac{S_1}{S_2}=\frac{39}{64}.$
2010 South East Mathematical Olympiad, 2
For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.
1981 Bundeswettbewerb Mathematik, 4
Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.
2012 Purple Comet Problems, 23
Find the greatest seven-digit integer divisible by $132$ whose digits, in order, are $2, 0, x, y, 1, 2, z$ where $x$, $y$, and $z$ are single digits.
2012 AIME Problems, 12
For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{3, 4, 5, 6, 7, 13, 14, 15, 16, 17,23, \ldots \}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe.·
2010 Macedonia National Olympiad, 1
Solve the equation
\[ x^3+2y^3-4x-5y+z^2=2012, \]
in the set of integers.
PEN P Problems, 2
Show that each integer $n$ can be written as the sum of five perfect cubes (not necessarily positive).
2001 National Olympiad First Round, 23
Which of the followings is false for the sequence $9,99,999,\dots$?
$\textbf{(A)}$ The primes which do not divide any term of the sequence are finite.
$\textbf{(B)}$ Infinitely many primes divide infinitely many terms of the sequence.
$\textbf{(C)}$ For every positive integer $n$, there is a term which is divisible by at least $n$ distinct prime numbers.
$\textbf{(D)}$ There is an inteter $n$ such that every prime number greater than $n$ divides infinitely many terms of the sequence.
$\textbf{(E)}$ None of above
2014 National Olympiad First Round, 18
Which one below cannot be expressed in the form $x^2+y^5$, where $x$ and $y$ are integers?
$
\textbf{(A)}\ 59170
\qquad\textbf{(B)}\ 59149
\qquad\textbf{(C)}\ 59130
\qquad\textbf{(D)}\ 59121
\qquad\textbf{(E)}\ 59012
$
2004 Germany Team Selection Test, 3
Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$.
Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i]
[i]Proposed by Laurentiu Panaitopol, Romania[/i]
2014 AIME Problems, 10
Let $z$ be a complex number with $|z| = 2014$. Let $P$ be the polygon in the complex plane whose vertices are $z$ and every $w$ such that $\tfrac{1}{z+w} = \tfrac{1}{z} + \tfrac{1}{w}$. Then the area enclosed by $P$ can be written in the form $n\sqrt{3},$ where $n$ is an integer. Find the remainder when $n$ is divided by $1000$.