Found problems: 2008
2011 ELMO Shortlist, 2
Let $p\ge5$ be a prime. Show that
\[\sum_{k=0}^{(p-1)/2}\binom{p}{k}3^k\equiv 2^p - 1\pmod{p^2}.\]
[i]Victor Wang.[/i]
2009 National Olympiad First Round, 30
How many of
$ 11^2 \plus{} 13^2 \plus{} 17^2$, $ 24^2 \plus{} 25^2 \plus{} 26^2$, $ 12^2 \plus{} 24^2 \plus{} 36^2$, $ 11^2 \plus{} 12^2 \plus{} 132^2$ are perfect square ?
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 d)1 \qquad\textbf{(E)}\ 0$
2013 Romanian Masters In Mathematics, 1
For a positive integer $a$, define a sequence of integers $x_1,x_2,\ldots$ by letting $x_1=a$ and $x_{n+1}=2x_n+1$ for $n\geq 1$. Let $y_n=2^{x_n}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_1,\ldots,y_k$ are all prime.
2014 Iran MO (3rd Round), 2
We say two sequence of natural numbers A=($a_1,...,a_n$) , B=($b_1,...,b_n$)are the exchange and we write $A\sim B$.
if $503\vert a_i - b_i$ for all $1\leq i\leq n$.
also for natural number $r$ : $A^r$ = ($a_1^r,a_2^r,...,a_n^r$).
Prove that there are natural number $k,m$ such that :
$i$)$250 \leq k $
$ii$)There are different permutations $\pi _1,...,\pi_k$ from {$1,2,3,...,502$} such that for $1\leq i \leq k-1$ we have $\pi _i^m\sim \pi _{i+1}$
(15 points)
2012 Iran Team Selection Test, 1
Suppose $p$ is an odd prime number. We call the polynomial $f(x)=\sum_{j=0}^n a_jx^j$ with integer coefficients $i$-remainder if $ \sum_{p-1|j,j>0}a_{j}\equiv i\pmod{p}$. Prove that the set $\{f(0),f(1),...,f(p-1)\}$ is a complete residue system modulo $p$ if and only if polynomials $f(x), (f(x))^2,...,(f(x))^{p-2}$ are $0$-remainder and the polynomial $(f(x))^{p-1}$ is $1$-remainder.
[i]Proposed by Yahya Motevassel[/i]
2001 Irish Math Olympiad, 1
Find all positive integer solutions $ (a,b,c,n)$ of the equation: $ 2^n\equal{}a!\plus{}b!\plus{}c!$.
1978 IMO Shortlist, 3
Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.
1993 India Regional Mathematical Olympiad, 5
Show that $19^{93} - 13^{99}$ is a positive integer divisible by $162$.
1989 Polish MO Finals, 1
An even number of politicians are sitting at a round table. After a break, they come back and sit down again in arbitrary places. Show that there must be two people with the same number of people sitting between them as before the break..
[b]Additional problem:[/b]
Solve the problem when the number of people is in a form $6k+3$.
2012 ELMO Shortlist, 5
Let $n>2$ be a positive integer and let $p$ be a prime. Suppose that the nonzero integers are colored in $n$ colors. Let $a_1,a_2,\ldots,a_{n}$ be integers such that for all $1\le i\le n$, $p^i\nmid a_i$ and $p^{i-1}\mid a_i$. In terms of $n$, $p$, and $\{a_i\}_{i=1}^{n}$, determine if there must exist integers $x_1,x_2,\ldots,x_{n}$ of the same color such that $a_1x_1+a_2x_2+\cdots+a_{n}x_{n}=0$.
[i]Ravi Jagadeesan.[/i]
2010 Contests, 2
Given a fixed integer $k>0,r=k+0.5$,define
$f^1(r)=f(r)=r[r],f^l(r)=f(f^{l-1}(r))(l>1)$
where $[x]$ denotes the smallest integer not less than $x$.
prove that there exists integer $m$ such that $f^m(r)$ is an integer.
2005 Manhattan Mathematical Olympiad, 2
What is the largest number of Sundays can be in one year? Explain your answer.
1999 Putnam, 3
Let $A=\{(x,y): 0\le x,y < 1\}.$ For $(x,y)\in A,$ let
\[S(x,y)=\sum_{\frac12\le\frac mn\le2}x^my^n,\]
where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate
\[\lim_{(x,y)\to(1,1),(x,y)\in A}(1-xy^2)(1-x^2y)S(x,y).\]
2010 USA Team Selection Test, 9
Determine whether or not there exists a positive integer $k$ such that $p = 6k+1$ is a prime and
\[\binom{3k}{k} \equiv 1 \pmod{p}.\]
1986 Spain Mathematical Olympiad, 3
Find all natural numbers $n$ such that $5^n+3$ is a power of $2$
2000 JBMO ShortLists, 10
Prove that there are no integers $x,y,z$ such that
\[x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 \]
2010 National Olympiad First Round, 10
How many integers $n$ with $0\leq n < 840$ are there such that $840$ divides $n^8-n^4+n-1$?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 8
$
PEN A Problems, 24
Let $p>3$ is a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that \[{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}\] is divisible by $p^{2}$.
PEN H Problems, 59
Solve the equation $28^x =19^y +87^z$, where $x, y, z$ are integers.
2010 India IMO Training Camp, 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 $n\ge 2$.
1988 IMO Shortlist, 7
Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$
PEN H Problems, 48
Solve the equation $x^2 +7=2^n$ in integers.
2003 Canada National Olympiad, 2
Find the last three digits of the number $2003^{{2002}^{2001}}$.
2001 Baltic Way, 18
Let $a$ be an odd integer. Prove that $a^{2^m}+2^{2^m}$ and $a^{2^n}+2^{2^n}$ are relatively prime for all positive integers $n$ and $m$ with $n\not= m$.
PEN A Problems, 87
Find all positive integers $n$ such that $3^{n}-1$ is divisible by $2^n$.