The polynomial $A(x) = x^2 + ax + b$ with integer coefficients has the following property: for each prime $p$ there is an integer $k$ such that $A(k)$ and $A(k + 1)$ are both divisible by $p$. Proof that there is an integer $m$ such that $A(m) = A(m + 1) = 0$.

$a, b, c, d$ are positive integers, and $r=1-\frac{a}{b}-\frac{c}{d}$. And, $a+c \le 1982, r \ge 0$. Prove that $r>\frac{1}{1983^3}$.

Five guys each have a positive integer (the integers are not necessarily distinct). The greatest common divisor of any two guys' numbers is always more than $1$, but the greatest common divisor of all the numbers is $1$. What is the minimum possible value of the product of the numbers?

Find all Arithmetic progressions $a_{1},a_{2},...$ of natural numbers for which there exists natural number $N>1$ such that for every $k\in \mathbb{N}$: $a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}$

At the vertices of a cube are written eight pairwise distinct natural numbers, and on each of its edges is written the greatest common divisor of the numbers at the endpoints of the edge. Can the sum of the numbers written at the vertices be the same as the sum of the numbers written at the edges? [i]A. Shapovalov[/i]

Let $ P$ be a polynomial with integer coefficients such that there are two distinct integers at which $ P$ takes coprime values. Show that there exists an infinite set of integers, such that the values $ P$ takes at them are pairwise coprime.

A positive interger number $k$ is called “$t-m$”-property if forall positive interger number $a$, there exists a positive integer number $n$ such that ${{1}^{k}}+{{2}^{k}}+{{3}^{k}}+...+{{n}^{k}} \equiv a (\bmod m).$ a) Find all positive integer numbers $k$ which has $t-20$-property. b) Find smallest positive integer number $k$ which has $t-{{20}^{15}}$-property.

Ben picks a positive number $n$ less than $2017$ uniformly at random. Then Rex, starting with the number $ 1$, repeatedly multiplies his number by $n$ and then finds the remainder when dividing by $2017$. Rex does this until he gets back to the number $ 1$. What is the probability that, during this process, Rex reaches every positive number less than $2017$ before returning back to $ 1$?

Find the number of positive integers (m,n) which follows the following: 1) m<n 2) The sum of even numbers between 2m and 2n is 100 greater than the sum of odd numbers between 2m and 2n.

Tetris is a Soviet block game developed in $1984$, probably to torture misbehaving middle school children. Nowadays, Tetris is a game that people play for fun, and we even have a mini-event featuring it, but it shall be used on this test for its original purpose. The $7$ Tetris pieces, which will be used in various problems in this theme, are as follows: [img]https://cdn.artofproblemsolving.com/attachments/b/c/f4a5a2b90fcf87968b8f2a1a848ad32ef52010.png[/img] [b]p1.[/b] Each piece has area $4$. Find the sum of the perimeters of each of the $7$ Tetris pieces. [b]p2.[/b] In a game of Tetris, Qinghan places $4$ pieces every second during the first $2$ minutes, and $2$ pieces every second for the remainder of the game. By the end of the game, her average speed is $3.6$ pieces per second. Find the duration of the game in seconds. [b]p3.[/b] Jeff takes all $7$ different Tetris pieces and puts them next to each other to make a shape. Each piece has an area of $4$. Find the least possible perimeter of such a shape. [b]p4.[/b] Qepsi is playing Tetris, but little does she know: the latest update has added realistic physics! She places two blocks, which form the shape below. Tetrominoes $ABCD$ and $EFGHI J$ are both formed from $4$ squares of side length $1$. Given that $CE = CF$, the distance from point $I$ to the line $AD$ can be expressed as $\frac{A\sqrt{B}-C}{D}$ . Find $1000000A+10000B +100C +D$. [img]https://cdn.artofproblemsolving.com/attachments/9/a/5e96a855b9ebbfd3ea6ebee2b19d7c0a82c7c3.png[/img] [b]p5.[/b] Using the following tetrominoes: [img]https://cdn.artofproblemsolving.com/attachments/3/3/464773d41265819c4f452116c1508baa660780.png[/img] Find the number of ways to tile the shape below, with rotation allowed, but reflection disallowed: [img]https://cdn.artofproblemsolving.com/attachments/d/6/943a9161ff80ba23bb8ddb5acaf699df187e07.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

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).$

A number $x_n$ of the form 10101...1 has $n$ ones. Find all $n$ such that $x_n$ is prime.

Show that there do not exist five consecutive integers whose sum of squares is itself a perfect square.

Let $M=\{x^2+x \mid x\in \mathbb N^{\star} \}$. Prove that for every integer $k\geq 2$ there exist elements $a_{1}, a_{2}, \ldots, a_{k},b_{k}$ from $M$, such that $a_{1}+a_{2}+\cdots+a_{k}=b_{k}$.

Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.

Let $S$ be a finite set of integers, each greater than $1$. Suppose that for each integer $n$ there is some $s\in S$ such that $\gcd(s,n)=1$ or $\gcd(s,n)=s$. Show that there exist $s,t\in S$ such that $\gcd(s,t)$ is prime.

For each $q\in\mathbb Q$, let $\pi(q)$ denote the period of the repeating base-$16$ expansion of $q$, with the convention of $\pi(q)=0$ if $q$ has a terminating base-$16$ expansion. Find the maximum value among \[\pi\left(\frac11\right),~\pi\left(\frac12\right),~\dots,~\pi\left(\frac1{70}\right).\]

Chelsea goes to La Verde's at MIT and buys $100$ coconuts, each weighing $4$ pounds, and $100$ honeydews, each weighing $5$ pounds. She wants to distribute them among $n$ bags, so that each bag contains at most $13$ pounds of fruit. What is the minimum $n$ for which this is possible?

Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer. [i]Note: A monic polynomial has a leading coefficient equal to 1.[/i] [i](Greece - Panagiotis Lolas and Silouanos Brazitikos)[/i]

Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.

Let $p$ be an odd prime number. Prove that $$1^{p-1} + 2^{p-1} +...+ (p-1)^{p-1} \equiv p + (p-1)! \mod p^2$$

Suppose $a$, $b$ and $c$ are integers such that the greatest common divisor of $x^2+ax+b$ and $x^2+bx+c$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^2+ax+b$ and $x^2+bx+c$ $x^3-4x^2+x+6$. Find $a+b+c$.

On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on. If $\gcd(m,n)=1$, and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?

We call an infinite set $S\subseteq\mathbb{N}$ good if for all parwise different integers $a,b,c\in S$, all positive divisors of $\frac{a^c-b^c}{a-b}$ are in $S$. for all positive integers $n>1$, prove that there exists a good set $S$ such that $n \not \in S$. Proposed by Seyed Reza Hosseini Dolatabadi

A sequence of positive integers $a_1, a_2, \ldots$ satisfies $a_k + a_l = a_m + a_n$ for all positive integers $k,l,m,n$ satisfying $kl = mn$. Prove that if $p$ divides $q$ then $a_p \le a_q$.