Found problems: 2008
2002 National Olympiad First Round, 34
How many positive integers $n$ are there such that $3n^2 + 3n + 7$ is a perfect cube?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 3
\qquad\textbf{d)}\ 7
\qquad\textbf{e)}\ \text{Infinitely many}
$
2014 Online Math Open Problems, 15
In Prime Land, there are seven major cities, labelled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e. we take the indices modulo $7$. Al initially starts at city $C_0$.
Each minute for ten minutes, Al flips a fair coin. If the coin land heads, and he is at city $C_k$, he moves to city $C_{2k}$; otherwise he moves to city $C_{2k+1}$. If the probability that Al is back at city $C_0$ after $10$ moves is $\tfrac{m}{1024}$, find $m$.
[i]Proposed by Ray Li[/i]
2009 China Team Selection Test, 2
Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$
2006 India IMO Training Camp, 2
Let $u_{jk}$ be a real number for each $j=1,2,3$ and each $k=1,2$ and let $N$ be an integer such that
\[\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N\]
Let $M$ and $l$ be positive integers such that $l^2 <(M+1)^3$. Prove that there exist integers $\xi_1,\xi_2,\xi_3$ not all zero, such that
\[\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}\]
PEN O Problems, 46
Suppose $p$ is a prime with $p \equiv 3 \; \pmod{4}$. Show that for any set of $p-1$ consecutive integers, the set cannot be divided two subsets so that the product of the members of the one set is equal to the product of the members of the other set.
2005 MOP Homework, 4
Prove that there does not exist an integer $n>1$ such that $n$ divides $3^n-2^n$.
2013 ELMO Shortlist, 5
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that
\[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \]
and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$.
[i]Proposed by Victor Wang[/i]
2014 PUMaC Number Theory B, 8
Find the number of positive integers $n \le 2014$ such that there exists integer $x$ that satisfies the condition that $\frac{x+n}{x-n}$ is an odd perfect square.
PEN H Problems, 53
Suppose that $a, b$, and $p$ are integers such that $b \equiv 1 \; \pmod{4}$, $p \equiv 3 \; \pmod{4}$, $p$ is prime, and if $q$ is any prime divisor of $a$ such that $q \equiv 3 \; \pmod{4}$, then $q^{p}\vert a^{2}$ and $p$ does not divide $q-1$ (if $q=p$, then also $q \vert b$). Show that the equation \[x^{2}+4a^{2}= y^{p}-b^{p}\] has no solutions in integers.
PEN A Problems, 19
Let $f(x)=x^3 +17$. Prove that for each natural number $n \ge 2$, there is a natural number $x$ for which $f(x)$ is divisible by $3^n$ but not $3^{n+1}$.
2012 Spain Mathematical Olympiad, 1
Find all positive integers $n$ and $k$ such that $(n+1)^n=2n^k+3n+1$.
2014 Bundeswettbewerb Mathematik, 4
Find all postive integers $n$ for which the number $\frac{4n+1}{n(2n-1)}$ has a terminating decimal expansion.
2015 Postal Coaching, Problem 3
Let $a$ and $n$ denote positive integers such that $n|a^n-1$. Prove that the numbers $a+1,a^2+2, \cdots a^n+n$ all leave different remainders when divided by $n$.
2009 District Round (Round II), 1
given a 4-digit number $(abcd)_{10}$ such that both$(abcd)_{10}$and$(dcba)_{10}$ are multiples of $7$,having the same remainder modulo $37$.find $a,b,c,d$.
1996 IMO Shortlist, 1
We are given a positive integer $ r$ and a rectangular board $ ABCD$ with dimensions $ AB \equal{} 20, BC \equal{} 12$. The rectangle is divided into a grid of $ 20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $ \sqrt {r}$. The task is to find a sequence of moves leading from the square with $ A$ as a vertex to the square with $ B$ as a vertex.
(a) Show that the task cannot be done if $ r$ is divisible by 2 or 3.
(b) Prove that the task is possible when $ r \equal{} 73$.
(c) Can the task be done when $ r \equal{} 97$?
2002 AMC 8, 5
Carlos Montado was born on Saturday, November 9, 2002. On what day of the week will Carlos be 706 days old?
$ \text{(A)}\ \text{Monday}\qquad\text{(B)}\ \text{Wednesday}\qquad\text{(C)}\ \text{Friday}\qquad\text{(D)}\ \text{Saturday}\qquad\text{(E)}\ \text{Sunday} $
2012 Bosnia Herzegovina Team Selection Test, 4
Define a function $f:\mathbb{N}\rightarrow\mathbb{N}$, \[f(1)=p+1,\] \[f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,\] where $p$ is a prime number. Find all $p$ such that there exists a natural number $k$ such that $f(k)$ is a perfect square.
2009 Germany Team Selection Test, 2
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.
[i]Proposed by Mohsen Jamaali, Iran[/i]
MathLinks Contest 7th, 3.2
Prove that for positive integers $ x,y,z$ the number $ x^2 \plus{} y^2 \plus{} z^2$ is not divisible by $ 3(xy \plus{} yz \plus{} zx)$.
2011 Laurențiu Duican, 4
Let be two natural numbers $ m\ge n $ and a nonnegative integer $ r<2^n. $ How many numbers of $ m $ digits, each digit being either the number $ 1 $ or $ 2, $ are there whose residue modulo $ 2^n $ is $ r? $
[i]Dorel Miheț[/i]
2013 USAMO, 5
Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten.
2007 Princeton University Math Competition, 5
For how many integers $x \in [0, 2007]$ is $\frac{6x^3+53x^2+61x+7}{2x^2+17x+15}$ reducible?
2007 AIME Problems, 8
A rectangular piece of of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called [i]basic [/i]if
(i) all four sides of the rectangle are segments of drawn line segments, and
(ii) no segments of drawn lines lie inside the rectangle.
Given that the total length of all lines drawn is exactly 2007 units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by 1000.
1999 USAMTS Problems, 1
The digits of the three-digit integers $a, b,$ and $c$ are the nine nonzero digits $1,2,3,\cdots 9$ each of them appearing exactly once. Given that the ratio $a:b:c$ is $1:3:5$, determine $a, b,$ and $c$.
2007 Tuymaada Olympiad, 1
What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?