Found problems: 2008
2001 AIME Problems, 11
In a rectangular array of points, with 5 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N+1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are selected so that each $P_i$ is in row $i.$ Let $x_i$ be the number associated with $P_i.$ Now renumber the array consecutively from top to bottom, beginning with the first column. Let $y_i$ be the number associated with $P_i$ after the renumbering. It is found that $x_1=y_2,$ $x_2=y_1,$ $x_3=y_4,$ $x_4=y_5,$ and $x_5=y_3.$ Find the smallest possible value of $N.$
2011 Iran Team Selection Test, 8
Let $p$ be a prime and $k$ a positive integer such that $k \le p$. We know that $f(x)$ is a polynomial in $\mathbb Z[x]$ such that for all $x \in \mathbb{Z}$ we have $p^k | f(x)$.
[b](a)[/b] Prove that there exist polynomials $A_0(x),\ldots,A_k(x)$ all in $\mathbb Z[x]$ such that
\[ f(x)=\sum_{i=0}^{k} (x^p-x)^ip^{k-i}A_i(x),\]
[b](b)[/b] Find a counter example for each prime $p$ and each $k > p$.
2005 Germany Team Selection Test, 3
We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.
PEN A Problems, 75
Find all triples $(a,b,c)$ of positive integers such that $2^{c}-1$ divides $2^{a}+2^{b}+1$.
2014 Greece Team Selection Test, 1
Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$
a) Prove that the sequence consists only of natural numbers.
b) Check if there are terms of the sequence divisible by $2011$.
2024 Bosnia and Herzegovina Junior BMO TST, 2.
Determine all $x$, $y$, $k$ and $n$ positive integers such that:
$10^x$ + $10^y$ + $n!$ = $2024^k$
2014 AMC 8, 13
If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?
$\textbf{(A) }n$ and $m$ are even $\qquad\textbf{(B) }n$ and $m$ are odd $\qquad\textbf{(C) }n+m$ is even $\qquad\textbf{(D) }n+m$ is odd $\qquad \textbf{(E) }$ none of these are impossible
2003 IMO Shortlist, 8
Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions:
(i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements;
(ii) for any nonempty subset of $A$, the product of its elements is not a perfect $p$-th power.
What is the largest possible number of elements in $A$ ?
2024 AMC 10, 7
What is the remainder when $7^{2024}+7^{2025}+7^{2026}$ is divided by $19$?
$
\textbf{(A) }0 \qquad
\textbf{(B) }1 \qquad
\textbf{(C) }7 \qquad
\textbf{(D) }11 \qquad
\textbf{(E) }18 \qquad
$
PEN A Problems, 27
Show that the coefficients of a binomial expansion $(a+b)^n$ where $n$ is a positive integer, are all odd, if and only if $n$ is of the form $2^{k}-1$ for some positive integer $k$.
2013 Princeton University Math Competition, 6
Let $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$. Find the remainder when $d$ is divided by $2013$.
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]
2013 ELMO Shortlist, 7
Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define
\[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \]
Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$.
[i]Proposed by Victor Wang[/i]
1999 Baltic Way, 18
Let $m$ be a positive integer such that $m=2\pmod{4}$. Show that there exists at most one factorization $m=ab$ where $a$ and $b$ are positive integers satisfying
\[0<a-b<\sqrt{5+4\sqrt{4m+1}}\]
2005 Iran Team Selection Test, 3
Suppose $S= \{1,2,\dots,n\}$ and $n \geq 3$. There is $f:S^k \longmapsto S$ that if $a,b \in S^k$ and $a$ and $b$ differ in all of elements then $f(a) \neq f(b)$. Prove that $f$ is a function of one of its elements.
2011 Croatia Team Selection Test, 1
We define a sequence $a_n$ so that $a_0=1$ and
\[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + d & \textrm{ otherwise. } \end{cases} \]
for all postive integers $n$.
Find all positive integers $d$ such that there is some positive integer $i$ for which $a_i=1$.
1998 Vietnam Team Selection Test, 2
Let $d$ be a positive divisor of $5 + 1998^{1998}$. Prove that $d = 2 \cdot x^2 + 2 \cdot x \cdot y + 3 \cdot y^2$, where $x, y$ are integers if and only if $d$ is congruent to 3 or 7 $\pmod{20}$.
2014 Middle European Mathematical Olympiad, 8
Determine all quadruples $(x,y,z,t)$ of positive integers such that
\[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\]
2008 Putnam, B6
Let $ n$ and $ k$ be positive integers. Say that a permutation $ \sigma$ of $ \{1,2,\dots n\}$ is $ k$-[i]limited[/i] if $ |\sigma(i)\minus{}i|\le k$ for all $ i.$ Prove that the number of $ k$-limited permutations of $ \{1,2,\dots n\}$ is odd if and only if $ n\equiv 0$ or $ 1\pmod{2k\plus{}1}.$
1990 China Team Selection Test, 3
Prove that for every integer power of 2, there exists a multiple of it with all digits (in decimal expression) not zero.
1997 Turkey MO (2nd round), 1
Find all pairs of integers $(x, y)$ such that $5x^{2}-6xy+7y^{2}=383$.
2006 IberoAmerican Olympiad For University Students, 1
Let $m,n$ be positive integers greater than $1$. We define the sets $P_m=\left\{\frac{1}{m},\frac{2}{m},\cdots,\frac{m-1}{m}\right\}$ and $P_n=\left\{\frac{1}{n},\frac{2}{n},\cdots,\frac{n-1}{n}\right\}$.
Find the distance between $P_m$ and $P_n$, that is defined as
\[\min\{|a-b|:a\in P_m,b\in P_n\}\]
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$.
2009 China Team Selection Test, 3
Prove that for any odd prime number $ p,$ the number of positive integer $ n$ satisfying $ p|n! \plus{} 1$ is less than or equal to $ cp^\frac{2}{3}.$ where $ c$ is a constant independent of $ p.$
1994 AIME Problems, 11
Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contribues $4''$ or $10''$ or $19''$ to the total height of the tower. How many differnt tower heights can be achieved using all 94 of the bricks?