Determine all solutions $(n, k)$ of the equation $n!+A
n = n^k$ with $n, k \in\mathbb{N}$ for $A = 7$ and for $A = 2012$.

An ancient Inca legend tells that a monster lives among the mountains that when wakes up, eats everyone who read this issue. After such a task, the monster returns to the mountains and sleeps for a number of years equal to the sum of its digits of the year in which you last woke up. The monster woke up for the first time in the year $234$. a) Would the monster have woken up between the years $2005$ and $2015$? b) Will we be safe in the next $10$ years?

A positive integer $c$ is given. The sequence $\{p_{k}\}$ is constructed by the following rule: $p_{1}$ is arbitrary prime and for $k\geq 1$ the number $p_{k+1}$ is any prime divisor of $p_{k}+c$ not present among the numbers $p_{1}$, $p_{2}$, $\dots$, $p_{k}$. Prove that the sequence $\{p_{k}\}$ cannot be infinite. [i]Proposed by A. Golovanov[/i]

Find the number of ordered pairs $(x,y)$ of integers with $0 \le x < 2023$ and $0 \le y < 2023$ such that $y^3 \equiv x^2 \pmod{2023}.$

Let us consider a simle graph with vertex set $ V$. All ordered pair $ (a,b)$ of integers with $ gcd(a,b) \equal{} 1$, are elements of V. $ (a,b)$ is connected to $ (a,b \plus{} kab)$ by an edge and to $ (a \plus{} kab,b)$ by another edge for all integer k. Prove that for all $ (a,b)\in V$, there exists a path fromm $ (1,1)$ to $ (a,b)$.

A staircase has $100$ steps. Kolya wishes to descend the staircase by alternately jumping down some steps and then up some. The possible jumps he can do are through $6$ (i.e. over $5$ and landing on the $6$th) , $7$ or $8$ steps . He also does not wish to land twice on the same step . Can he descend the staircase in this way? ( S . Fomin, Leningrad)

Find all pairs of integers $(m, n)$ such that $$m+n = 3(mn+10).$$

An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $ n,$ expressed in terms of $ a$ and $ b.$

Prove that among $20$ consecutive positive integers there is an integer $d$ such that for every positive integer $n$ the following inequality holds $$n \sqrt{d} \left\{n \sqrt {d} \right \} > \dfrac{5}{2}$$ where by $\left \{x \right \}$ denotes the fractional part of the real number $x$. The fractional part of the real number $x$ is defined as the difference between the largest integer that is less than or equal to $x$ to the actual number $x$. [i](Serbia)[/i]

Let $n$ be a positive integer with $d^2$ positive divisors. We fill a $d\times d$ board with these divisors. At a move, we can choose a row, and shift the divisor from the $i^{\text{th}}$ column to the $(i+1)^{\text{th}}$ column, for all $i=1,2,\ldots, d$ (indices reduced modulo $d$). A configuration of the $d\times d$ board is called [i]feasible[/i] if there exists a column with elements $a_1,a_2,\ldots,a_d,$ in this order, such that $a_1\mid a_2\mid\ldots\mid a_d$ or $a_d\mid a_{d-1}\mid\ldots\mid a_1.$ Determine all values of $n$ for which ragardless of how we initially fill the board, we can reach a feasible configuration after a finite number of moves.

[b]p1.[/b] A straight line is painted in two colors. Prove that there are three points of the same color such that one of them is located exactly at the midpoint of the interval bounded by the other two. [b]p2.[/b] Find all positive integral solutions $x, y$ of the equation $xy = x + y + 3$. [b]p3.[/b] Can one cut a square into isosceles triangles with angle $80^o$ between equal sides? [b]p4.[/b] $20$ children are grouped into $10$ pairs: one boy and one girl in each pair. In each pair the boy is taller than the girl. Later they are divided into pairs in a different way. May it happen now that (a) in all pairs the girl is taller than the boy; (b) in $9$ pairs out of $10$ the girl is taller than the boy? [b]p5.[/b] Mr Mouse got to the cellar where he noticed three heads of cheese weighing $50$ grams, $80$ grams, and $120$ grams. Mr. Mouse is allowed to cut simultaneously $10$ grams from any two of the heads and eat them. He can repeat this procedure as many times as he wants. Can he make the weights of all three pieces equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all solutions of the following equation in integers $x,y: x^4+ y= x^3+ y^2$

Find all positive integers $n$, such that the number \[ \frac{n^{2021}+101}{n^2+n+1}\] is an integer.

[u]Round 1[/u] [b]p1.[/b] Five girls and three boys are sitting in a room. Suppose that four of the children live in California. Determine the maximum possible number of girls that could live somewhere outside California. [b]p2.[/b] A $4$-meter long stick is rotated $60^o$ about a point on the stick $1$ meter away from one of its ends. Compute the positive difference between the distances traveled by the two endpoints of the stick, in meters. [b]p3.[/b] Let $f(x) = 2x(x - 1)^2 + x^3(x - 2)^2 + 10(x - 1)^3(x - 2)$. Compute $f(0) + f(1) + f(2)$. [u]Round 2[/u] [b]p4.[/b] Twenty boxes with weights $10, 20, 30, ... , 200$ pounds are given. One hand is needed to lift a box for every $10$ pounds it weighs. For example, a $40$ pound box needs four hands to be lifted. Determine the number of people needed to lift all the boxes simultaneously, given that no person can help lift more than one box at a time. [b]p5.[/b] Let $ABC$ be a right triangle with a right angle at $A$, and let $D$ be the foot of the perpendicular from vertex$ A$ to side $BC$. If $AB = 5$ and $BC = 7$, compute the length of segment $AD$. [b]p6.[/b] There are two circular ant holes in the coordinate plane. One has center $(0, 0)$ and radius $3$, and the other has center $(20, 21)$ and radius $5$. Albert wants to cover both of them completely with a circular bowl. Determine the minimum possible radius of the circular bowl. [u]Round 3[/u] [b]p7.[/b] A line of slope $-4$ forms a right triangle with the positive x and y axes. If the area of the triangle is 2013, find the square of the length of the hypotenuse of the triangle. [b]p8.[/b] Let $ABC$ be a right triangle with a right angle at $B$, $AB = 9$, and $BC = 7$. Suppose that point $P$ lies on segment $AB$ with $AP = 3$ and that point $Q$ lies on ray $BC$ with $BQ = 11$. Let segments $AC$ and $P Q$ intersect at point $X$. Compute the positive difference between the areas of triangles $AP X$ and $CQX$. [b]p9.[/b] Fresh Mann and Sophy Moore are racing each other in a river. Fresh Mann swims downstream, while Sophy Moore swims $\frac12$ mile upstream and then travels downstream in a boat. They start at the same time, and they reach the finish line 1 mile downstream of the starting point simultaneously. If Fresh Mann and Sophy Moore both swim at $1$ mile per hour in still water and the boat travels at 10 miles per hour in still water, find the speed of the current. [u]Round 4[/u] [b]p10.[/b] The Fibonacci numbers are defined by $F_0 = 0$, $F_1 = 1$, and for $n \ge 1$, $F_{n+1} = F_n + F_{n-1}$. The first few terms of the Fibonacci sequence are $0$, $1$, $1$, $2$, $3$, $5$, $8$, $13$. Every positive integer can be expressed as the sum of nonconsecutive, distinct, positive Fibonacci numbers, for example, $7 = 5 + 2$. Express $121$ as the sum of nonconsecutive, distinct, positive Fibonacci numbers. (It is not permitted to use both a $2$ and a $1$ in the expression.) [b]p11.[/b] There is a rectangular box of surface area $44$ whose space diagonals have length $10$. Find the sum of the lengths of all the edges of the box. [b]p12.[/b] Let $ABC$ be an acute triangle, and let $D$ and $E$ be the feet of the altitudes to $BC$ and $CA$, respectively. Suppose that segments $AD$ and $BE$ intersect at point $H$ with $AH = 20$ and $HD = 13$. Compute $BD \cdot CD$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c4h2809420p24782524]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The sum of several integers (not necessarily distinct) equals $1492$. Decide whether the sum of their seventh powers can equal (a) $1996$; (b) $1998$.

A prime number $p \ge 5$ is given. Write $\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}$ for natural numbers $a$ and $b$. Show that $p$ divides $a$.

Prove that for all integers $n \geq 3$, there exist odd positive integers $x$, $y$ such that $7x^2 + y^2 = 2^n$.

Find all integers $a$ and $b$ satisfying \begin{align*} a^6 + 1 & \mid b^{11} - 2023b^3 + 40b, \qquad \text{and}\\ a^4 - 1 & \mid b^{10} - 2023b^2 - 41. \end{align*}

The number $50$ is written as a sum of several positive integers (not necessarily distinct) whose product is divisible by $100.$ What is the largest possible value of this product?

Determine all pairs $ (a,b)$ of positive integers, each integer having two decimal digits, such that $ 100a\plus{}b$ and $ 201a\plus{}b$ are both perfect squares.

Let $k\ge0$ be a given integer. Suppose there exists positive integer $n,d$ and an odd integer $m>1$ with $d\mid m^{2^k}-1$ and $m\mid n^d+1$. Find all possible values of $\frac{m^{2^k}-1}d$.

Find all pairs $(a, p)$ of positive integers, where $p$ is a prime, such that for any pair of positive integers $m$ and $n$ the remainder obtained when $a^{2^n}$ is divided by $p^n$ is non-zero and equals the remainder obtained when $a^{2^m}$ is divided by $p^m$.

Find with proof all solutions in nonnegative integers $a,b,c,d$ of the equation $$11^a5^b-3^c2^d=1$$

Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$.

A number of three digits is said to be [i]firm [/i]when it is equal to the product of its unit digit by a number formed by the remaining digits. For example, $153$ is firm because $153 = 3 \times 51$. How many [i]firm [/i] numbers are there?