Found problems: 2008
PEN D Problems, 11
During a break, $n$ children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule. He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on. Determine the values of $n$ for which eventually, perhaps after many rounds, all children will have at least one candy each.
2013 IFYM, Sozopol, 2
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?
[i]Proposed by Mahan Malihi[/i]
PEN D Problems, 16
Determine all positive integers $n \ge 2$ that satisfy the following condition; For all integers $a, b$ relatively prime to $n$, \[a \equiv b \; \pmod{n}\Longleftrightarrow ab \equiv 1 \; \pmod{n}.\]
the 14th XMO, P4
In an $n$ by $n$ grid, each cell is filled with an integer between $1$ and $6$. The outmost cells all contain the number $1$, and any two cells that share a vertex has difference not equal to $3$. For any vertex $P$ inside the grid (not including the boundary), there are $4$ cells that have $P$ has a vertex. If these four cells have exactly three distinct numbers $i$, $j$, $k$ (two cells have the same number), and the two cells with the same number have a common side, we call $P$ an $ijk$-type vertex. Let there be $A_{ijk}$ vertices that are $ijk$-type. Prove that $A_{123}\equiv A_{246} \pmod 2$.
2011 Belarus Team Selection Test, 3
Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$
[list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good.
[*][b](b)[/b] Show that all 2010-good pairs are very good.[/list]
[i]Proposed by Okan Tekman, Turkey[/i]
2009 Indonesia TST, 3
Let $ S\equal{}\{1,2,\ldots,n\}$. Let $ A$ be a subset of $ S$ such that for $ x,y\in A$, we have $ x\plus{}y\in A$ or $ x\plus{}y\minus{}n\in A$. Show that the number of elements of $ A$ divides $ n$.
2008 Czech-Polish-Slovak Match, 3
Find all primes $p$ such that the expression
\[\binom{p}1^2+\binom{p}2^2+\cdots+\binom{p}{p-1}^2\]
is divisible by $p^3$.
2007 ITest, 45
Find the sum of all positive integers $B$ such that $(111)_B=(aabbcc)_6$, where $a,b,c$ represent distinct base $6$ digits, $a\neq 0$.
2014 Online Math Open Problems, 26
Qing initially writes the ordered pair $(1,0)$ on a blackboard. Each minute, if the pair $(a,b)$ is on the board, she erases it and replaces it with one of the pairs $(2a-b,a)$, $(2a+b+2,a)$ or $(a+2b+2,b)$. Eventually, the board reads $(2014,k)$ for some nonnegative integer $k$. How many possible values of $k$ are there?
[i]Proposed by Evan Chen[/i]
1999 IMO Shortlist, 3
A game is played by $n$ girls ($n \geq 2$), everybody having a ball. Each of the $\binom{n}{2}$ pairs of players, is an arbitrary order, exchange the balls they have at the moment. The game is called nice [b]nice[/b] if at the end nobody has her own ball and it is called [b]tiresome[/b] if at the end everybody has her initial ball. Determine the values of $n$ for which there exists a nice game and those for which there exists a tiresome game.
2013 North Korea Team Selection Test, 3
Find all $ a, b, c \in \mathbb{Z} $, $ c \ge 0 $ such that $ a^n + 2^n | b^n + c $ for all positive integers $ n $ where $ 2ab $ is non-square.
2011 Federal Competition For Advanced Students, Part 1, 1
Determine all integer triplets $(x,y,z)$ such that
\[x^4+x^2=7^zy^2\mbox{.}\]
2009 Korea - Final Round, 6
Find all pairs of two positive integers $(m,n)$ satisfying $ 3^m - 7^n = 2 $.
2013 NIMO Problems, 2
How many integers $n$ are there such that $(n+1!)(n+2!)(n+3!)\cdots(n+2013!)$ is divisible by $210$ and $1 \le n \le 210$?
[i]Proposed by Lewis Chen[/i]
2010 Germany Team Selection Test, 3
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$.
[i]Proposed by North Korea[/i]
2023 Dutch BxMO TST, 5
Find all pairs of prime numbers $(p,q)$ for which
\[2^p = 2^{q-2} + q!.\]
2009 AMC 10, 21
What is the remainder when $ 3^0\plus{}3^1\plus{}3^2\plus{}\ldots\plus{}3^{2009}$ is divided by $ 8$?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 6$
2011 USAJMO, 6
Consider the assertion that for each positive integer $n\geq2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of $4$. Either prove the assertion or find (with proof) a counterexample.
2023 Brazil Undergrad MO, 5
A drunken horse moves on an infinite board whose squares are numbered in pairs $(a, b) \in \mathbb{Z}^2$. In each movement, the 8 possibilities $$(a, b) \rightarrow (a \pm 1, b \pm 2),$$ $$(a, b) \rightarrow (a \pm 2, b \pm 1)$$ are equally likely. Knowing that the knight starts at $(0, 0)$, calculate the probability that, after $2023$ moves, it is in a square $(a, b)$ with $a \equiv 4 \pmod 8$ and $b \equiv 5 \pmod 8$.
2012 Indonesia TST, 4
Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$.
Remark: "Natural numbers" is the set of positive integers.
2009 India IMO Training Camp, 8
Let $ n$ be a natural number $ \ge 2$ which divides $ 3^n\plus{}4^n$.Prove That $ 7\mid n$.
1998 USAMTS Problems, 3
The integers from $1$ to $9$ can be arranged into a $3\times3$ array (as shown on the right) so that the sum of the numbers in every row, column, and diagonal is a multiple of $9$.
(a.) Prove that the number in the center of the array must be a multiple of $3$.
(b.) Give an example of such an array with $6$ in the center.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));size(100);
int i,j;
for(i=0; i<4; i=i+1) {
draw((0,2i)--(6,2i));
draw((2i,0)--(2i,6));
}
string[] letters={"G", "H", "I", "D", "E", "F", "A", "B", "C"};
for(i=0; i<3; i=i+1) {
for(j=0; j<3; j=j+1) {
label(letters[3i+j], (2j+1, 2i+1));
}}[/asy]
2013 District Olympiad, 4
Let $n\in {{\mathbb{N}}^{*}}$. Prove that $2\sqrt{{{2}^{n}}}\cos \left( n\arccos \frac{\sqrt{2}}{4} \right)$ is an odd integer.
PEN A Problems, 17
Let $m$ and $n$ be natural numbers such that \[A=\frac{(m+3)^{n}+1}{3m}\] is an integer. Prove that $A$ is odd.
1972 IMO Longlists, 22
Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.