A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$ a.) Show that every prime number $n$ has property $P.$ b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$

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.

The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by $ 4$, and the second number is divided by $ 5$. The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square? [asy]unitsize(1.5cm); defaultpen(linewidth(.8pt)+fontsize(15pt)); draw(Circle(origin,1)); for(int i = 0;i < 6; ++i) { draw(origin--dir(60i+30)); } label("$7$",midpoint(origin--(dir(0))),E); label("$1$",midpoint(origin--(dir(60))),NE); label("$6$",midpoint(origin--(dir(120))),NW); label("$3$",midpoint(origin--(dir(180))),W); label("$9$",midpoint(origin--(dir(240))),SW); label("$2$",midpoint(origin--(dir(300))),SE); draw((2,0)--(3.5,0)--(3.5,1)--(2,1)--cycle); draw((2,0)--(3.5,0)--(3.5,-1)--(2,-1)--cycle); pair[] V = {(2.5,0.5),(2,0),(3,0),(2.5,-0.5),(2,-1),(3,-1)}; for(int i = 0; i <= 5; ++i) { pair A = V[i]; path p = A--(A.x,A.y + 0.5)--(A.x + 0.5,A.y + 0.5)--(A.x + 0.5, A.y)--cycle; fill(p,mediumgray); draw(p); } path pointer = (-2.5,-0.125)--(-2.5,0.125)--(-1.2,0.125)--(-1.05,0)--(-1.2,-0.125)--cycle; fill(pointer,mediumgray); draw(pointer); label("Pointer",(-1.85,0),fontsize(10pt)); label("$4$",(2,0.5),2N + 2W); label("$3$",(2,0),2N + 2W); label("$2$",(2,-0.5),2N + 2W); label("$1$",(2,-1),2N + 2W); label("$1$",(2,-1),2S + 2E); label("$2$",(2.5,-1),2S + 2E); label("$3$",(3,-1),2S + 2E);[/asy]$ \textbf{(A)}\ \frac {1}{3}\qquad \textbf{(B)}\ \frac {4}{9}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {5}{9}\qquad \textbf{(E)}\ \frac {2}{3}$

Two coprime positive integers $ a, b $ are given. Integer sequence $ \{ a_n \}, \{b_n \} $ satisties \[ (a+b \sqrt2 )^{2n} = a_n + b_n \sqrt2 \] Find all prime numbers $ p $ such that there exist positive integer $ n \le p $ satisfying $ p | b_n $.

$ p >3 $ is a prime number such that $ p | 2^{p-1} -1 $ and $ p \not | 2^x - 1 $ for $ x = 1, 2, \cdots , p-2 $. Let $ p = 2k+3 $. Now we define sequence $ \{ a_n \} $ as \[ a_i = a_{i+k}= 2^i ( 1 \le i \le k ) , \ a_{j+2k} = a_j a_{j+k} \ ( j \ge 1 ) \] Prove that there exist $2k$ consecutive terms of sequence $ a_{x+1} , a_{x+2} , \cdots , a_{x+2k} $ such that for all $ 1 \le i < j \le 2k $, $ a_{x+i} \not \equiv a_{x+j} \ (mod \ p) $.

Let $a,b$ and $c$ be positive integers such that $ab$ divides $c(c^{2}-c+1)$ and $a+b$ is divisible by $c^{2}+1$. Prove that the sets $\{a,b\}$ and $\{c,c^{2}-c+1\}$ coincide.

The integers $0 \leq a < 2^{2008}$ and $0 \leq b < 8$ satisfy the equivalence $7(a+2^{2008}b) \equiv 1 \pmod{2^{2011}}$. Then $b$ is $\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None}$

Find all pairs of positive integers $\left(n;\;k\right)$ such that $n!=\left( n+1\right)^{k}-1$.

Show that there exists $n>2$ such that $1991 | 1999 \ldots 91$ (with $n$ 9's).

For each $ k\in\mathbb{N} ,k\le 2000, $ Let $ r_k $ be the remainder of the division of $ k $ by $ 4, $ and $ r'_k $ be the remainder of the division of $ k $ by $ 3. $ Prove that there is an unique $ m\in\mathbb{N} ,m\le 1999 $ such that $$ r_1+r_2+\cdots +r_m=r'_{m+1} +r'_{m+2} +\cdots r'_{2000} . $$ [i]Mircea Fianu[/i]

In a competition, there were $2n+1$ teams. Every team plays exatly once against every other team. Every match finishes with the victory of one of the teams. We call cyclical a 3-subset of team ${ A,B,C }$ if $A$ won against $B$, $B$ won against $C$ , $C$ won against $A$. (a) Find the minimum of cyclical 3-subset (depending on $n$); (b) Find the maximum of cyclical 3-subset (depending on $n$).

Determine all pairs $(p, q)$ of primes for which $p^{q+1}+q^{p+1}$ is a perfect square.

Let $u_1$, $u_2$, $u_3$, $\dots$ be a sequence of integers satisfying the recurrence relation $u_{n + 2} = u_{n + 1}^2 - u_n$. Suppose $u_1 = 39$ and $u_2 = 45$. Prove that 1986 divides infinitely many terms of the sequence.

Let $n$ be a natural number, for which we define $S(n)=\{1+g+g^2+...+g^{n-1}|g\in{\mathbb{N}},g\geq2\}$ $a)$ Prove that: $S(3)\cap S(4)=\varnothing$ $b)$ Determine: $S(3)\cap S(5)$

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]

Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers.

Find all triples $(a,b,c)$ such that $a,b,c$ are integers in the set $\{2000,2001,\ldots,3000\}$ satisfying $a^2+b^2=c^2$ and $\text{gcd}(a,b,c)=1$.

Let $p$ be a given odd prime. Find the largest integer $k'$ such that it is possible to partition $\{1,2,\cdots,p-1\}$ into two sets $X,Y$ such that for any $k$ with $0 \le k \le k'$, $$\sum_{a \in X}a^k \equiv \sum_{b \in Y}b^k \pmod p$$ [i]houkai[/i]

Prove that for every given positive integer $n$, there exists a prime $p$ and an integer $m$ such that $(a)$ $p \equiv 5 \pmod 6$ $(b)$ $p \nmid n$ $(c)$ $n \equiv m^3 \pmod p$

Let $n$ be a positive integer, and let $A_{n}$ be the the set of all positive integers $a\le n$ such that $n|a^{n}+1$. a) Find all $n$ such that $A_{n}\neq \emptyset$ b) Find all $n$ such that $|{A_{n}}|$ is even and non-zero. c) Is there $n$ such that $|{A_{n}}| = 130$?

Show that \[(-1)^{\frac{p-1}{2}}{p-1 \choose{\frac{p-1}{2}}}\equiv 4^{p-1}\pmod{p^{3}}\] for all prime numbers $p$ with $p \ge 5$.

Find the number of positive integers x satisfying the following two conditions: 1. $x<10^{2006}$ 2. $x^{2}-x$ is divisible by $10^{2006}$

Jerry's favorite number is $97$. He knows all kinds of interesting facts about $97$: [list][*]$97$ is the largest two-digit prime. [*]Reversing the order of its digits results in another prime. [*]There is only one way in which $97$ can be written as a difference of two perfect squares. [*]There is only one way in which $97$ can be written as a sum of two perfect squares. [*]$\tfrac1{97}$ has exactly $96$ digits in the [smallest] repeating block of its decimal expansion. [*]Jerry blames the sock gnomes for the theft of exactly $97$ of his socks.[/list] A repunit is a natural number whose digits are all $1$. For instance, \begin{align*}&1,\\&11,\\&111,\\&1111,\\&\vdots\end{align*} are the four smallest repunits. How many digits are there in the smallest repunit that is divisible by $97?$

On a blackboard, there are $11$ positive integers. Show that one can choose some (maybe all) of these numbers and place "$+$" and "$-$" in between such that the result is divisible by $2011$.

Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$