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.

For a polynomial $ P(x)$ with integer coefficients, $ r(2i \minus{} 1)$ (for $ i \equal{} 1,2,3,\ldots,512$) is the remainder obtained when $ P(2i \minus{} 1)$ is divided by $ 1024$. The sequence \[ (r(1),r(3),\ldots,r(1023)) \] is called the [i]remainder sequence[/i] of $ P(x)$. A remainder sequence is called [i]complete[/i] if it is a permutation of $ (1,3,5,\ldots,1023)$. Prove that there are no more than $ 2^{35}$ different complete remainder sequences.

Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

Let $ p \equiv 3 \pmod{4}$ be a prime. Define $T = \{ (i,j) \mid i, j \in \{ 0, 1, \cdots , p-1 \} \} \smallsetminus \{ (0,0) \} $. For arbitrary subset $ S ( \ne \emptyset ) \subset T $, prove that there exist subset $ A \subset S $ satisfying following conditions: (a) $ (x_i , y_i ) \in A ( 1 \le i \le 3) $ then $ p \not | x_1 + x_2 - y_3 $ or $ p \not | y_1 + y_2 + x_3 $. (b) $ 8 n(A) > n(S) $

Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$, define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv 1 \pmod{p}$ and set $kx_k = 1+ pn_k$. Prove that : \[\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}\]

There is a frog in every vertex of a regular 2n-gon with circumcircle($n \geq 2$). At certain time, all frogs jump to the neighborhood vertices simultaneously (There can be more than one frog in one vertex). We call it as $\textsl{a way of jump}$. It turns out that there is $\textsl{a way of jump}$ with respect to 2n-gon, such that the line connecting any two distinct vertice having frogs on it after the jump, does not pass through the circumcentre of the 2n-gon. Find all possible values of $n$.

On a $2012 \times 2012$ board, some cells on the top-right to bottom-left diagonal are marked. None of the marked cells is in a corner. Integers are written in each cell of this board in the following way. All the numbers in the cells along the upper and the left sides of the board are 1's. All the numbers in the marked cells are 0's. Each of the other cells contains a number that is equal to the sum of its upper neighbour and its left neighbour. Prove that the number in the bottom right corner is not divisible by 2011.

Show that $ n\equiv 0\pmod 9 $ if $ 2^n\equiv -1\pmod n, $ where $ n $ is a natural number greater than $ 3. $

Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.

Let $S = \{2,5,8,11,14,17,20,\dots\}$. Given that one can choose $n$ different numbers from $S$, $\{A_1,A_2,\dots,A_n\}$ s.t. $\sum_{i=1}^n \frac{1}{A_i} = 1$, find the minimum possible value of $n$.

A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$). Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime. [i]Bobby Shen.[/i]

$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$

Define the sequence $ a_1, a_2, a_3, ...$ as follows. $ a_1$ and $ a_2$ are coprime positive integers and $ a_{n \plus{} 2} \equal{} a_{n \plus{} 1}a_n \plus{} 1$. Show that for every $ m > 1$ there is an $ n > m$ such that $ a_m^m$ divides $ a_n^n$. Is it true that $ a_1$ must divide $ a_n^n$ for some $ n > 1$?

Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]

Suppose that the set $\{1,2,\cdots, 1998\}$ has been partitioned into disjoint pairs $\{a_i,b_i\}$ ($1\leq i\leq 999$) so that for all $i$, $|a_i-b_i|$ equals $1$ or $6$. Prove that the sum \[ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| \] ends in the digit $9$.

Suppose that necklace $\, A \,$ has 14 beads and necklace $\, B \,$ has 19. Prove that for any odd integer $n \geq 1$, there is a way to number each of the 33 beads with an integer from the sequence \[ \{ n, n+1, n+2, \dots, n+32 \} \] so that each integer is used once, and adjacent beads correspond to relatively prime integers. (Here a ``necklace'' is viewed as a circle in which each bead is adjacent to two other beads.)

For $a\in \mathbb{Z}$ define \[ n_a=101a-100\cdot 2^a \] Show that, for $0\le a,b,c,d\le 99$ \[ n_a+n_b\equiv n_c+n_d\pmod{10100}\implies \{a,b\}=\{c,d\} \]

Two straight lines perpendicular to each other meet each side of a triangle in points symmetric with respect to the midpoint of that side. Prove that these two lines intersect in a point on the nine-point circle.

An $n$-term sequence $(x_1, x_2, \ldots, x_n)$ in which each term is either 0 or 1 is called a [i]binary sequence of length [/i]$n$. Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.

Let $ p$ be a prime number, $ p\not \equal{} 3$, and integers $ a,b$ such that $p\mid a+b$ and $ p^2\mid a^3 \plus{} b^3$. Prove that $ p^2\mid a \plus{} b$ or $ p^3\mid a^3 \plus{} b^3$.

Let $ N$ be the greatest integer multiple of $ 36$ all of whose digits are even and no two of whose digits are the same. Find the remainder when $ N$ is divided by $ 1000$.

Decimal digits $a,b,c$ satisfy \[ 37\mid (a0a0\ldots a0b0c0c\ldots 0c)_{10} \] where there are $1001$ a's and $1001$ c's. Prove that $b=a+c$.

$a_{1}=5$ and $a_{n+1}=a_{n}^{3}-2a_{n}^{2}+2$ for all $n\geq1$. $p$ is a prime such that $p=3(mod 4)$ and $p|a_{2011}+1$. Show that $p=3$.

Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$ [i]Author: Gerhard Wöginger, Netherlands[/i]

Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n}$ such that (1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$; (2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees; (3) for any integers $ x, |f(x)|$ isn't prime numbers.