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]

Let $n$ be a positive integer. Prove that there is no positive integer solution to thxe equation $(x + 2)^n - x^n = 1 + 7^n$.

Determine all the solutions of the equation $x^3 + y^3 + z^3 = wx^2y^2z^2$ in natural numbers $x, y, z, w$. Justify your answer

Given are two coprime positive integers $a, b$ with $b$ odd and $a>2$. The sequence $(x_n)$ is defined by $x_0=2, x_1=a$ and $x_{n+2}=ax_{n+1}+bx_n$ for $n \geq 1$. Prove that: $a)$ If $a$ is even then there do not exist positive integers $m, n, p$ such that $\frac{x_m} {x_nx_p}$ is a positive integer. $b)$ If $a$ is odd then there do not exist positive integers $m, n, p$ such that $mnp$ is even and $\frac{x_m} {x_nx_p}$ is a perfect square.

Show that any number of the form $$4444 ...44 88...8$$ where there are twice as many $4$s as $8$s is a square number.

Given six irrational numbers, will it be possible to choose three such that the sum of any two of these three is irrational?

Show that: a) There are infinitely many positive integers $n$ such that there exists a square equal to the sum of the squares of $n$ consecutive positive integers (for instance, $2$ is one such number as $5^2=3^2+4^2$). b) If $n$ is a positive integer which is not a perfect square, and if $x$ is an integer number such that $x^2+(x+1)^2+...+(x+n-1)^2$ is a perfect square, then there are infinitely many positive integers $y$ such that $y^2+(y+1)^2+...+(y+n-1)^2$ is a perfect square.

Find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that the numbers \[n, f(n),f(f(n)),\dots,f^{m-1}(n)\] are distinct modulo $m$ for all integers $n,m$ with $m>1$. (Here $f^k$ is defined by $f^0(n)=n$ and $f^{k+1}(n)=f(f^{k}(n))$ for $k \ge 0$.)

A [i]trio [/i] is a set of three distinct integers such that two of the numbers are divisors or multiples of the third. Which [i]trio [/i] contained in $\{1, 2, ... , 2002\}$ has the largest possible sum? Find all [i]trios [/i] with the maximum sum.

For a prime number $p$ and a positive integer $n$, denote by $f(p, n)$ the largest integer $k$ such that $p^k \mid n!$. Let $p$ be a given prime number and let $m$ and $c$ be given positive integers. Prove that there exist infinitely many positive integers $n$ such that $f(p, n) \equiv c \pmod m$.

Let us call "fine" the $2n$-digit number if it is exact square itself and the two numbers represented by its first $n$ digits (first digit may not be zero) and last $n$ digits (first digit may be zero, but it may not be zero itself) are exact squares also. a) Find all two- and four-digit fine numbers. b) Is there any six-digit fine number? c) Prove that there exists $20$-digit fine number. d) Prove that there exist at least ten $100$-digit fine numbers. e) Prove that there exists $30$-digit fine number.

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

Let $S_0 = 0, S_1 = 1,$ and for $n \ge 2,$ let $S_n = S_{n-1}+5S_{n-2}.$ What is the sum of the five smallest primes $p$ such that $p \mid S_{p-1}$?

Find all pairs $(n, p)$ that satisfy the following condition, where $n$ is a positive integer and $p$ is a prime number. [b]Condition)[/b] $2n-1$ is a divisor of $p-1$ and $p$ is a divisor of $4n^2+7$.

Let $P(x)$ and $Q(x)$ be polynomials of degree $p$ and $q$ respectively such that every coefficient is $1$ or $2023$. If $P(x)$ divides $Q(x)$, prove that $p+1$ divides $q+1$.

The sequence $L_1,\ L_2,\ L_3,\ \dots$ is defined by \[ L_1 = 1,\ \ L_2 = 3,\ \ L_n = L_{n - 1} + L_{n - 2}\textrm{ for }n > 2. \] Prove that $L_p - 1$ is divisible by $p$ if $p$ is prime.

The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.

p1. Given a set of $n$ the first natural number. If one of the numbers is removed, then the average number remaining is $21\frac14$ . Specify the number which is deleted. p2. Ipin and Upin play a game of Tic Tac Toe with a board measuring $3 \times 3$. Ipin gets first turn by playing $X$. Upin plays $O$. They must fill in the $X$ or $O$ mark on the board chess in turn. The winner of this game was the first person to successfully compose a sign horizontally, vertically, or diagonally. Determine as many final positions as possible, if Ipin wins in the $4$th step. For example, one of the positions the end is like the picture on the side. [img]https://cdn.artofproblemsolving.com/attachments/6/a/a8946f24f583ca5e7c3d4ce32c9aa347c7e083.png[/img] p3. Numbers $ 1$ to $10$ are arranged in pentagons so that the sum of three numbers on each side is the same. For example, in the picture next to the number the three numbers are $16$. For all possible arrangements, determine the largest and smallest values ​​of the sum of the three numbers. [img]https://cdn.artofproblemsolving.com/attachments/2/8/3dd629361715b4edebc7803e2734e4f91ca3dc.png[/img] p4. Define $$S(n)=\sum_{k=1}^{n}(-1)^{k+1}\,\, , \,\, k=(-1)^{1+1}1+(-1)^{2+1}2+...+(-1)^{n+1}n$$ Investigate whether there are positive integers $m$ and $n$ that satisfy $S(m) + S(n) + S(m + n) = 2011$ p5. Consider the cube $ABCD.EFGH$ with side length $2$ units. Point $A, B, C$, and $D$ lie in the lower side plane. Point $I$ is intersection point of the diagonal lines on the plane of the upper side. Next, make a pyramid $I.ABCD$. If the pyramid $I.ABCD$ is cut by a diagonal plane connecting the points $A, B, G$, and $H$, determine the volume of the truncated pyramid low part.

Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$

Let $k,m,n$ be positive integers such that $n^m$ is divisible by $m^n$, and $m^k$ is divisible by $k^m$. [list=a] [*] Prove that $n^k$ is divisible by $k^n$. [*] Find an example of $k,m,n$ satisfying the above conditions, where all three numbers are distinct and bigger than 1. [/list]

The set A contains exactly $21$ integers. The sum of any $11$ numbers in $A$ is greater than the sum of the remaining numbers. It is known that the set $A$ contain thes number $101$, and the largest number in $A$ is $2014$. Find out the other $19$ numbers in $A$.

Prove that for all positive integers $n\geq 1$ the number $\prod^n_{k=1} k^{2k-n-1}$ is also an integer number. [i]Laurentiu Panaitopol[/i].

[b]p1. [/b] If $f(x) = 3x - 1$, what is $f^6(2) = (f \circ f \circ f \circ f \circ f \circ f)(2)$? [b]p2.[/b] A frog starts at the origin of the $(x, y)$ plane and wants to go to $(6, 6)$. It can either jump to the right one unit or jump up one unit. How many ways are there for the frog to jump from the origin to $(6, 6)$ without passing through point $(2, 3)$? [b]p3.[/b] Alfred, Bob, and Carl plan to meet at a café between noon and $2$ pm. Alfred and Bob will arrive at a random time between noon and $2$ pm. They will wait for $20$ minutes or until $2$ pm for all $3$ people to show up after which they will leave. Carl will arrive at the café at noon and leave at $1:30$ pm. What is the probability that all three will meet together? [b]p4.[/b] Let triangle $ABC$ be isosceles with $AB = AC$. Let $BD$ be the altitude from $ B$ to $AC$, $E$ be the midpoint of $AB$, and $AF$ be the altitude from $ A$ to $BC$. If $AF = 8$ and the area of triangle $ACE$ is $ 8$, find the length of $CD$. [b]p5.[/b] Find the sum of the unique prime factors of $(2018^2 - 121) \cdot (2018^2 - 9)$. [b]p6.[/b] Compute the remainder when $3^{102} + 3^{101} + ... + 3^0$ is divided by $101$. [b]p7.[/b] Take regular heptagon $DUKMATH$ with side length $ 3$. Find the value of $$\frac{1}{DK}+\frac{1}{DM}.$$ [b]p8.[/b] RJ’s favorite number is a positive integer less than $1000$. It has final digit of $3$ when written in base $5$ and final digit $4$ when written in base $6$. How many guesses do you need to be certain that you can guess RJ’s favorite number? [b]p9.[/b] Let $f(a, b) = \frac{a^2+b^2}{ab-1}$ , where $a$ and $b$ are positive integers, $ab \ne 1$. Let $x$ be the maximum positive integer value of $f$, and let $y$ be the minimum positive integer value of f. What is $x - y$ ? [b]p10.[/b] Haoyang has a circular cylinder container with height $50$ and radius $5$ that contains $5$ tennis balls, each with outer-radius $5$ and thickness $1$. Since Haoyang is very smart, he figures out that he can fit in more balls if he cuts each of the balls in half, then puts them in the container, so he is ”stacking” the halves. How many balls would he have to cut up to fill up the container? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all triples $ (a,b,c)$ of integers with $ abc\equal{}1989$ and $ a\plus{}b\minus{}c\equal{}89$.

Prove that there exists a real constant $c$ such that for any pair $(x,y)$ of real numbers, there exist relatively prime integers $m$ and $n$ satisfying the relation \[ \sqrt{(x-m)^2 + (y-n)^2} < c\log (x^2 + y^2 + 2). \]