Given a number $ a $ and natural number $ n\ge 3 $ having the property that $ x^n-x $ and $ x^2-x $ are integers, prove that $ x $ is integer.

Determine all pairs of prime numbers $(p, q)$, with $2 \leq p, q < 100$, such that $p+6, p+10, q+4, q+10$ and $p+q+1$ are all prime numbers.

We will say that a positive integer is [i]lucky [/i ]if the sum of its digits is divisible by $31$. What is the maximum possible difference between two consecutive [i]lucky [/i ] numbers?

Let $p$ be a prime. How many different (infinite) sequences $\left(a_k\right)_{k\ge0}$ exist such that for every positive integer $n$ \[\frac{a_0}{a_1}+\frac{a_0}{a_2}+\cdots+\frac{a_0}{a_n}+\frac{p}{a_{n+1}}=1?\]

Supoose $A$ is a set of integers which contains all integers that can be written as $2^a-2^b$, $a,b\in \mathbb{Z}_{\ge 1}$ and also has the property that $a+b\in A$ whenever $a,b\in A$. Prove that if $A$ contains at least an odd number, then $A=\mathbb{Z}$. [i] (Andrei Bâra)[/i]

Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the maximum possible value of \[\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007},\] where, for $1\leq i\leq 2007$, $x_i$ is a nonnegative real number, and \[x_1+x_2+x_3+\cdots+x_{2007}=\pi.\] Find the value of $a+b$.

Let $x$, $y$ be two positive integers such that $\displaystyle 3x^2+x=4y^2+y$. Prove that $x-y$ is a perfect square.

[b]p1.[/b] James has $8$ Instagram accounts, $3$ Facebook accounts, $4$ QQ accounts, and $3$ YouTube accounts. If each Instagram account has $19$ pictures, each Facebook account has $5$ pictures and $9$ videos, each QQ account has a total of $17$ pictures, and each YouTube account has $13$ videos and no pictures, how many pictures in total does James have in all these accounts? [b]p2.[/b] If Poonam can trade $7$ shanks for $4$ shinks, and she can trade $10$ shinks for $17$ shenks. How many shenks can Poonam get if she traded all of her $105$ shanks? [b]p3.[/b] Jerry has a bag with $3$ red marbles, $5$ blue marbles and $2$ white marbles. If Jerry randomly picks two marbles from the bag without replacement, the probability that he gets two different colors can be expressed as a fraction $\frac{m}{n}$ in lowest terms. What is $m + n$? [b]p4.[/b] Bob's favorite number is between $1200$ and $4000$, divisible by $5$, has the same units and hundreds digits, and the same tens and thousands digits. If his favorite number is even and not divisible by $3$, what is his favorite number? [b]p5.[/b] Consider a unit cube $ABCDEFGH$. Let $O$ be the center of the face $EFGH$. The length of $BO$ can be expressed in the form $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are simplified to lowest terms. What is $a + b$? [b]p6.[/b] Mr. Eddie Wang is a crazy rich boss who owns a giant company in Singapore. Even though Mr. Wang appears friendly, he finds great joy in firing his employees. His immediately fires them when they say "hello" and/or "goodbye" to him. It is well known that $1/2$ of the total people say "hello" and/or "goodbye" to him everyday. If Mr. Wang had $2050$ employees at the end of yesterday, and he hires $2$ new employees at the beginning of each day, in how many days will Mr. Wang first only have $6$ employees left? [b]p7.[/b] In $\vartriangle ABC$, $AB = 5$, $AC = 6$. Let $D,E,F$ be the midpoints of $\overline{BC}$, $\overline{AC}$, $\overline{AB}$, respectively. Let $X$ be the foot of the altitude from $D$ to $\overline{EF}$. Let $\overline{AX}$ intersect $\overline{BC}$ at $Y$ . Given $DY = 1$, the length of $BC$ is $\frac{p}{q}$ for relatively prime positive integers $p, q$: Find $p + q$. [b]p8.[/b] Given $\frac{1}{2006} = \frac{1}{a} + \frac{1}{b}$ where $a$ is a $4$ digit positive integer and $b$ is a $6$ digit positive integer, find the smallest possible value of $b$. [b]p9.[/b] Pocky the postman has unlimited stamps worth $5$, $6$ and $7$ cents. However, his post office has two very odd requirements: On each envelope, an odd number of $7$ cent stamps must be used, and the total number of stamps used must also be odd. What is the largest amount of postage money Pocky cannot make with his stamps, in cents? [b]p10.[/b] Let $ABCDEF$ be a regular hexagon with side length $2$. Let $G$ be the midpoint of side $DE$. Now let $O$ be the intersection of $BG$ and $CF$. The radius of the circle inscribed in triangle $BOC$ can be expressed in the form $\frac{a\sqrt{b}-\sqrt{c}}{d} $ where $a$, $b$, $c$, $d$ are simplified to lowest terms. What is $a + b + c + d$? [b]p11.[/b] Estimation (Tiebreaker): What is the total number of characters in all of the participants' email addresses in the Accuracy Round? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all two-digit positive integers $n =\overline{ab}$ (in the decimal system) with the property that for all integers $x$ the difference $x^a - x^b$ is divisible by $n$.

Find all positive integer $n$ satisfying the conditions $a)n^2=(a+1)^3-a^3$ $b)2n+119$ is a perfect square.

Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides?

We say that a string of digits from $0$ to $9$ is valid if the following conditions hold: First, for $2 \le k \le 4$, no consecutive run of $k$ digits sums to a multiple of $10$. Second, between any two $0$s, there are at least $3$ other digits. Find the last four digits of the number of valid strings of length $2020$.

The denominators of two irreducible fractions are 600 and 700. Find the minimum value of the denominator of their sum (written as an irreducible fraction).

Show that \[A_n=\prod_{j=0}^{n-1}\cfrac{(3j+1)!}{(n+j)!}\] is an integer, for any positive integer \(n\).

Suppose that $a, b, c, k$ are natural numbers with $a, b, c \ge 3$ which fulfill the equation $abc = k^2 + 1$. Show that at least one between $a - 1, b - 1, c -1$ is composite number.

Find all positive integers $a, b,c$ greater than $1$, such that $ab + 1$ is divisible by $c, bc + 1$ is divisible by $a$ and $ca + 1$ is divisible by $b$.

Prove that among any $1001$ numbers taken from the numbers $1,2,\ldots,1997$ there exist two with the difference $4$.

Find all square roots of integers, namely $ p, $ such that $ \left(\frac{p}{2}\right)^2 <3<\left(\frac{p+1}{2}\right)^2. $

Let $n$ and $k$ be two positive integers. Prove that there exist infinitely many perfect squares of the form $n \cdot 2^k - 7$.

Let $P(x)=a x^{75}+b$ be a polynomial where \(a\) and \(b\) are coprime integers in the set \(\{1,2,\dots,151\}\), and suppose it satisfies the following condition: there exists at most one prime \(p\) such that for every positive integer \(k\), \(p\mid P(k)\). Prove that for every prime \(q \neq p\) there exists a positive integer \(k\) for which $q^2 \mid P(k).$

Amy and Bob play the game. At the beginning, Amy writes down a positive integer on the board. Then the players take moves in turn, Bob moves first. On any move of his, Bob replaces the number $n$ on the blackboard with a number of the form $n-a^2$, where $a$ is a positive integer. On any move of hers, Amy replaces the number $n$ on the blackboard with a number of the form $n^k$, where $k$ is a positive integer. Bob wins if the number on the board becomes zero. Can Amy prevent Bob’s win? [i]Maxim Didin, Russia[/i]

Suppose $m$ and $n$ are relatively prime positive integers. A regular $m$-gon and a regular $n$-gon are inscribed in a circle. Let $d$ be the minimum distance in degrees (of the arc along the circle) between a vertex of the $m$-gon and a vertex of the $n$-gon. What is the maximum possible value of $d$?

Do they exist natural numbers $m,x,y$ such that $$m^2 +25 \vdots (2011^x-1007^y) ?$$ S. Finskii

Let $ k$ and $ m$ be positive integers such that $ \displaystyle\frac12\left(\sqrt{k\plus{}4\sqrt{m}}\minus{}\sqrt{k}\right)$ is an integer. (a) Prove that $ \sqrt{k}$ is rational. (b) Prove that $ \sqrt{k}$ is a positive integer.

The two pairs of consecutive natural numbers $(8, 9)$ and $(288, 289)$ have the following property: in each pair, each number contains each of its prime factors to a power not less than $2$. Prove that there are infinitely many such pairs. (A Andjans, Riga)