The sum of sixth powers of six integers minus $1$ is six times greater than the product of these six integers. Prove that one of them is $1$ or $-1$ and all others are $0$s. (LD Kurliandchik)

Let $m,n$ be positive integers and let $[a]$ denote the residue class$\pmod{mn}$ of the integer $a$ (thus $\{[r]|r\text{ is an integer}\}$ has exactly $mn$ elements). Suppose the set $\{[ar]|r\text{ is an integer}\}$ has exactly $m$ elements. Prove that there is a positive integer $q$ such that $q$ is coprime to $mn$ and $[nq]=[a]$.

Find two positive integers $a$ and $b$, when their sum and their least common multiple is given. Find the numbers when the sum is $3972$ and the least common multiple is $985928$.

The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]

Prove that there are infinitely many positive integers $n$ such that the number of positive divisors in $2^n-1$ is greater than $n$.

Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following: There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements, $$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$ where $S(n)$ denotes sum of digits of decimal representation of $n$.

Find all natural numbers $a$, $b$, $c$ and prime numbers $p$ and $q$, such that: $\blacksquare$ $4\nmid c$ $\blacksquare$ $p\not\equiv 11\pmod{16}$ $\blacksquare$ $p^aq^b-1=(p+4)^c$

Determine whether there exists a positive integer $a$ such that $$2015a,2016a,\dots,2558a$$ are all perfect power.

is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?

We call a positive integer $n{}$ [i]peculiar[/i] if, for any positive divisor $d{}$ of $n{}$ the integer $d(d + 1)$ divides $n(n + 1).$ Prove that for any four different peculiar positive integers $A, B, C$ and $D{}$ the following holds: \[\gcd(A, B, C, D) = 1.\]

Adam has a secret natural number $x$ which Eve is trying to discover. At each stage Eve may only ask questions of the form "is $x+n$ a prime number?" for some natural number $n$ of her choice. Prove that Eve may discover $x$ using finitely many questions.

Find all triples $(x,y,n)$ of positive integers such that \[ (x+y)(1+xy) = 2^{n} \]

Find two positive integers $a,b$ such that $a | b^2, b^2 | a^3, a^3 | b^4, b^4 | a^5$, but $a^5$ does not divide $b^6$

[u]Set 3[/u] [b]p7.[/b] On unit square $ABCD$, a point $P$ is selected on segment $CD$ such that $DP =\frac14$ . The segment $BP$ is drawn and its intersection with diagonal $AC$ is marked as $E$. What is the area of triangle $AEP$? [b]p8.[/b] Five distinct points are arranged on a plane, creating ten pairs of distinct points. Seven pairs of points are distance $1$ apart, two pairs of points are distance $\sqrt3$ apart, and one pair of points is distance $2$ apart. Draw a line segment from one of these points to the midpoint of a pair of these points. What is the longest this line segment can be? [b]p9.[/b] The inhabitants of Mars use a base $8$ system. Mandrew Mellon is competing in the annual Martian College Interesting Competition of Math (MCICM). The first question asks to compute the product of the base $8$ numerals $1245415_8$, $7563265_8$, and $ 6321473_8$. Mandrew correctly computed the product in his scratch work, but when he looked back he realized he smudged the middle digit. He knows that the product is $1014133027\blacksquare 27662041138$. What is the missing digit? PS. You should use hide for answers.

The sequence of positive integers $a_1, a_2, \dots$ has the property that $\gcd(a_m, a_n) > 1$ if and only if $|m - n| = 1$. Find the sum of the four smallest possible values of $a_2$.

Let $a,b,c\in Z$ and $a$ be the even number and $b$ be the odd number. Show that for every integer $n$ there exist one positive integer $x$ such that $2^n\mid ax^2+bx+c$

$$Problem 4:$$The sum of $5$ positive numbers equals $2$. Let $S_k$ be the sum of the $k-th$ powers of these numbers. Determine which of the numbers $2,S_2,S_3,S_4$ can be the greatest among them.

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

Find the smallest positive integer $k$ such that \[\binom{x+kb}{12} \equiv \binom{x}{12} \pmod{b}\] for all positive integers $b$ and $x$. ([i]Note:[/i] For integers $a,b,c$ we say $a \equiv b \pmod c$ if and only if $a-b$ is divisible by $c$.) [i]Alex Zhu.[/i] [hide="Clarifications"][list=1][*]${{y}\choose{12}} = \frac{y(y-1)\cdots(y-11)}{12!}$ for all integers $y$. In particular, ${{y}\choose{12}} = 0$ for $y=1,2,\ldots,11$.[/list][/hide]

Are there exist some positive integers $n$ and $k$, such that the first decimals of $2^n$ (from left to the right) represent the number $5^k$ while the first decimals of $5^n$ represent the number $2^k$ ? (5)

Find all pairs of integers $ a, b $ that satisfy $a ^2-3a = b ^3-2$.

Suppose that $a, b, x$ are positive integers such that \[x^{a+b}=a^bb\] Prove that $a=x$ and $b=x^x$.

Find all positive integers $n$ such that $(n^2 + 11n - 4) \cdot n! + 33 \cdot 13^n + 4$ is a perfect square

[u]Round 5[/u] [b]p13.[/b] Given a (not necessarily simplified) fraction $\frac{m}{n}$ , where $m, n > 6$ are positive integers, when $6$ is subtracted from both the numerator and denominator, the resulting fraction is equal to $\frac45$ of the original fraction. How many possible ordered pairs $(m, n)$ are there? [b]p14.[/b] Jamesu's favorite anime show has $3$ seasons, with $12$ episodes each. For $8$ days, Jamesu does the following: on the $n^{th}$ day, he chooses $n$ consecutive episodes of exactly one season, and watches them in order. How many ways are there for Jamesu to finish all $3$ seasons by the end of these $8$ days? (For example, on the first day, he could watch episode $5$ of the first season; on the second day, he could watch episodes $11$ and $12$ of the third season, etc.) [b]p15.[/b] Let $O$ be the center of regular octagon $ABCDEFGH$ with side length $6$. Let the altitude from $O$ meet side $AB$ at $M$, and let $BH$ meet $OM$ at $K$. Find the value of $BH \cdot BK$. [u]Round 6[/u] [b]p16.[/b] Fhomas writes the ordered pair $(2, 4)$ on a chalkboard. Every minute, he erases the two numbers $(a, b)$, and replaces them with the pair $(a^2 + b^2, 2ab)$. What is the largest number on the board after $10$ minutes have passed? [b]p17.[/b] Triangle $BAC$ has a right angle at $A$. Point $M$ is the midpoint of $BC$, and $P$ is the midpoint of $BM$. Point $D$ is the point where the angle bisector of $\angle BAC$ meets $BC$. If $\angle BPA = 90^o$, what is $\frac{PD}{DM}$? [b]p18.[/b] A square is called legendary if there exist two different positive integers $a, b$ such that the square can be tiled by an equal number of non-overlapping $a$ by $a$ squares and $b$ by $b$ squares. What is the smallest positive integer $n$ such that an $n$ by $n$ square is legendary? [u]Round 7[/u] [b]p19.[/b] Let $S(n)$ be the sum of the digits of a positive integer $n$. Let $a_1 = 2019!$, and $a_n = S(a_{n-1})$. Given that $a_3$ is even, find the smallest integer $n \ge 2$ such that $a_n = an_1$. [b]p20.[/b] The local EMCC bakery sells one cookie for $p$ dollars ($p$ is not necessarily an integer), but has a special offer, where any non-zero purchase of cookies will come with one additional free cookie. With $\$27:50$, Max is able to buy a whole number of cookies (including the free cookie) with a single purchase and no change leftover. If the price of each cookie were $3$ dollars lower, however, he would be able to buy double the number of cookies as before in a single purchase (again counting the free cookie) with no change leftover. What is the value of $p$? [b]p21.[/b] Let circle $\omega$ be inscribed in rhombus $ABCD$, with $\angle ABC < 90^o$. Let the midpoint of side $AB$ be labeled $M$, and let $\omega$ be tangent to side $AB$ at $E$. Let the line tangent to $\omega$ passing through $M$ other than line $AB$ intersect segment $BC$ at $F$. If $AE = 3$ and $BE = 12$, what is the area of $\vartriangle MFB$? [u]Round 8[/u] [b]p22.[/b] Find the remainder when $1010 \cdot 1009! + 1011 \cdot 1008! + ... + 2018 \cdot 1!$ is divided by $2019$. [b]p23.[/b] Two circles $\omega_1$ and $\omega_2$ have radii $1$ and $2$, respectively and are externally tangent to one another. Circle $\omega_3$ is externally tangent to both $\omega_1$ and $\omega_2$. Let $M$ be the common external tangent of $\omega_1$ and $\omega_3$ that doesn't intersect $\omega_2$. Similarly, let $N$ be the common external tangent of $\omega_2$ and $\omega_3$ that doesn't intersect $\omega_1$. Given that $M$ and N are parallel, find the radius of $\omega_3$. [b]p24.[/b] Mana is standing in the plane at $(0, 0)$, and wants to go to the EMCCiffel Tower at $(6, 6)$. At any point in time, Mana can attempt to move $1$ unit to an adjacent lattice point, or to make a knight's move, moving diagonally to a lattice point $\sqrt5$ units away. However, Mana is deathly afraid of negative numbers, so she will make sure never to decrease her $x$ or $y$ values. How many distinct paths can Mana take to her destination? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949411p26408196]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all integers $n$ satisfying $n \geq 2$ and $\dfrac{\sigma(n)}{p(n)-1} = n$, in which $\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$.