Found problems: 2008
PEN A Problems, 14
Let $n$ be an integer with $n \ge 2$. Show that $n$ does not divide $2^{n}-1$.
2012 Brazil Team Selection Test, 3
Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.
1996 Turkey MO (2nd round), 2
Prove that $\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})}$ is divisible by $n!$ for all positive integers $n$.
1977 Germany Team Selection Test, 3
Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$
1982 IMO Longlists, 52
We are given $2n$ natural numbers
\[1, 1, 2, 2, 3, 3, \ldots, n - 1, n - 1, n, n.\]
Find all $n$ for which these numbers can be arranged in a row such that for each $k \leq n$, there are exactly $k$ numbers between the two numbers $k$.
2023 Israel TST, P2
Let $n>3$ be an integer. Integers $a_1, \dots, a_n$ are given so that $a_k\in \{k, -k\}$ for all $1\leq k\leq n$. Prove that there is a sequence of indices $1\leq k_1, k_2, \dots, k_n\leq n$, not necessarily distinct, for which the sums
\[a_{k_1}\]
\[a_{k_1}+a_{k_2}\]
\[a_{k_1}+a_{k_2}+a_{k_3}\]
\[\vdots\]
\[a_{k_1}+a_{k_2}+\cdots+a_{k_n}\]
have distinct residues modulo $2n+1$, and so that the last one is divisible by $2n+1$.
2014 Spain Mathematical Olympiad, 2
Let $M$ be the set of all integers in the form of $a^2+13b^2$, where $a$ and $b$ are distinct itnegers.
i) Prove that the product of any two elements of $M$ is also an element of $M$.
ii) Determine, reasonably, if there exist infinite pairs of integers $(x,y)$ so that $x+y\not\in M$ but $x^{13}+y^{13}\in M$.
2010 Turkey Team Selection Test, 1
Let $0 \leq k < n$ be integers and $A=\{a \: : \: a \equiv k \pmod n \}.$ Find the smallest value of $n$ for which the expression
\[ \frac{a^m+3^m}{a^2-3a+1} \]
does not take any integer values for $(a,m) \in A \times \mathbb{Z^+}.$
2010 Belarus Team Selection Test, 6.1
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$.
[i]Proposed by Juhan Aru, Estonia[/i]
2013 USA TSTST, 8
Define a function $f: \mathbb N \to \mathbb N$ by $f(1) = 1$, $f(n+1) = f(n) + 2^{f(n)}$ for every positive integer $n$. Prove that $f(1), f(2), \dots, f(3^{2013})$ leave distinct remainders when divided by $3^{2013}$.
2002 AIME Problems, 3
Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive integers $(d,n)$ are possible?
PEN A Problems, 20
Determine all positive integers $n$ for which there exists an integer $m$ such that $2^{n}-1$ divides $m^{2}+9$.
Oliforum Contest IV 2013, 1
Given a prime $p$, consider integers $0<a<b<c<d<p$ such that $a^4\equiv b^4\equiv c^4\equiv d^4\pmod{p}$. Show that \[a+b+c+d\mid a^{2013}+b^{2013}+c^{2013}+d^{2013}\]
2007 IMC, 5
Let $ n$ be a positive integer and $ a_{1}, \ldots, a_{n}$ be arbitrary integers. Suppose that a function $ f: \mathbb{Z}\to \mathbb{R}$ satisfies $ \sum_{i=1}^{n}f(k+a_{i}l) = 0$ whenever $ k$ and $ l$ are integers and $ l \ne 0$. Prove that $ f = 0$.
PEN E Problems, 4
Prove that $1280000401$ is composite.
1997 AIME Problems, 3
Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?
2007 QEDMO 5th, 1
Let $ a$, $ b$ and $ k$ be three positive integers.
We define two sequences $ \left( a_{n}\right)$ and $ \left( b_{n}\right)$ by the starting values $ a_{1}\equal{}a$ and $ b_{1}\equal{}b$ and the recurrent equations $ a_{n\plus{}1}\equal{}ka_{n}\plus{}b_{n}$ and $ b_{n\plus{}1}\equal{}kb_{n}\plus{}a_{n}$ for each positive integer $ n$.
Prove that if $ a_{1}\perp b_{1}$, $ a_{2}\perp b_{2}$ and $ a_{3}\perp b_{3}$ hold, then $ a_{n}\perp b_{n}$ holds for every positive integer $ n$.
Here, the abbreviation $ x\perp y$ stands for "the numbers $ x$ and $ y$ are coprime".
1994 Baltic Way, 18
There are $n>2$ lines given in the plane. No two of the lines are parallel and no three of them intersect at one point. Every point of intersection of these lines is labelled with a natural number between $1$ and $n-1$. Prove that, if and only if $n$ is even, it is possible to assign the labels in such a way that every line has all the numbers from $1$ to $n-1$ at its points of intersection with the other $n-1$ lines.
2012 AMC 10, 12
A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$) or is divisible by $4$ but not by $100$ (such as $2012$). The $200\text{th}$ anniversary of the birth of novelist Charles Dickens was celebrated on February $7$, $2012$, a Tuesday. On what day of the week was Dickens born?
$ \textbf{(A)}\ \text{Friday}
\qquad\textbf{(B)}\ \text{Saturday}
\qquad\textbf{(C)}\ \text{Sunday}
\qquad\textbf{(D)}\ \text{Monday}
\qquad\textbf{(E)}\ \text{Tuesday}
$
2002 Poland - Second Round, 1
Find all numbers $p\le q\le r$ such that all the numbers
\[pq+r,pq+r^2,qr+p,qr+p^2,rp+q,rp+q^2 \]
are prime.
2000 Kurschak Competition, 3
Let $k\ge 0$ be an integer and suppose the integers $a_1,a_2,\dots,a_n$ give at least $2k$ different residues upon division by $(n+k)$. Show that there are some $a_i$ whose sum is divisible by $n+k$.
2015 JBMO Shortlist, NT5
Check if there exists positive integers $ a, b$ and prime number $p$ such that $a^3-b^3=4p^2$
2009 China Western Mathematical Olympiad, 1
Define a sequence $(x_{n})_{n\geq 1}$ by taking $x_{1}\in\left\{5,7\right\}$; when $k\ge 1$, $x_{k+1}\in\left\{5^{x_{k}},7^{x_{k}}\right\}$. Determine all possible last two digits of $x_{2009}$.
2007 ITest, 20
Find the largest integer $n$ such that $2007^{1024}-1$ is divisible by $2^n$.
$\textbf{(A) }1\hspace{14em}\textbf{(B) }2\hspace{14em}\textbf{(C) }3$
$\textbf{(D) }4\hspace{14em}\textbf{(E) }5\hspace{14em}\textbf{(F) }6$
$\textbf{(G) }7\hspace{14em}\textbf{(H) }8\hspace{14em}\textbf{(I) }9$
$\textbf{(J) }10\hspace{13.7em}\textbf{(K) }11\hspace{13.5em}\textbf{(L) }12$
$\textbf{(M) }13\hspace{13.3em}\textbf{(N) }14\hspace{13.4em}\textbf{(O) }15$
$\textbf{(P) }16\hspace{13.6em}\textbf{(Q) }55\hspace{13.4em}\textbf{(R) }63$
$\textbf{(S) }64\hspace{13.7em}\textbf{(T) }2007$
2013 USAMTS Problems, 4
An infinite sequence $(a_0,a_1,a_2,\dots)$ of positive integers is called a $\emph{ribbon}$ if the sum of any eight consecutive terms is at most $16$; that is, for all $i\ge0$,
\[a_i+a_{i+1}+\dots+a_{i+7}\le16.\]A positive integer $m$ is called a $\emph{cut size}$ if every ribbon contains a set of consecutive elements that sum to $m$; that is, given any ribbon $(a_0,a_1,a_2,\dots)$, there exist nonnegative integers $k\le l$ such that
\[a_k+a_{k+1}+\dots+a_l=m.\]Find, with proof, all cut sizes, or prove that none exist.