Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

One day while the Kubik family attends one of Michael's baseball games, Tony gets bored and walks to the creek a few yards behind the baseball field. One of Tony's classmates Mitchell sees Tony and goes to join him. While playing around the creek, the two boys find an ordinary six-sided die buried in sediment. Mitchell washes it off in the water and challenges Tony to a contest. Each of the boys rolls the die exactly once. Mitchell's roll is $3$ higher than Tony's. "Let's play once more," says Tony. Let $a/b$ be the probability that the difference between the outcomes of the two dice is again exactly $3$ (regardless of which of the boys rolls higher), where $a$ and $b$ are relatively prime positive integers. Find $a+b$.

Find all pairs of nonnegative integers $(x, p)$, where $p$ is prime, that verify $$x(x+1)(x+2)(x+3)=1679^{p-1}+1680^{p-1}+1681^{p-1}.$$

Let $ m,n$ be integers such that $ 0\le m\le 2n$. Then prove that the number $ 2^{2n \plus{} 2} \plus{} 2^{m \plus{} 2} \plus{} 1$ is perfect square iff $ m \equal{} n$.

Solve in nonnegative integers the equation $5^t + 3^x4^y = z^2$.

Given integers $a$ and $b$ such that $a^2+b^2$ is divisible by $11$. Prove that $a$ and $b$ are both divisible by $11$.

[b]p19.[/b] $A_1A_2...A_{12}$ is a regular dodecagon with side length $1$ and center at point $O$. What is the area of the region covered by circles $(A_1A_2O)$, $(A_3A_4O)$, $(A_5A_6O)$, $(A_7A_8O)$, $(A_9A_{10}O)$, and $(A_{11}A_{12}O)$? $(ABC)$ denotes the circle passing through points $A,B$, and $C$. [b]p20.[/b] Let $N = 2000... 0x0 ... 00023$ be a $2023$-digit number where the $x$ is the $23$rd digit from the right. If$ N$ is divisible by $13$, compute $x$. [b]p21.[/b] Alice and Bob each visit the dining hall to get a grilled cheese at a uniformly random time between $12$ PM and $1$ PM (their arrival times are independent) and, after arrival, will wait there for a uniformly random amount of time between $0$ and $30$ minutes. What is the probability that they will meet? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the problem of expressing $42$ as \(42 = x^3 + y^3 + z^3 - w^2\), where \(x, y, z, w\) are integers. Determine the number of ways to represent $42$ in this form and prove your conclusion.

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

Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.

Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$. Prove that [list] [*]$m = 1$ and $l = 2k$; or [*]$l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$. [/list]

Show that there exists a positive real $C$ such that for any naturals $H,N$ satisfying $H \geq 3, N \geq e^{CH}$, for any subset of $\{1,2,\ldots,N\}$ with size $\lceil \frac{CHN}{\ln N} \rceil$, one can find $H$ naturals in it such that the greatest common divisor of any two elements is the greatest common divisor of all $H$ elements.

Let $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0}$, where $a_{0},\ldots,a_{n}$ are integers, $a_{n}>0$, $n\geq 2$. Prove that there exists a positive integer $m$ such that $P(m!)$ is a composite number.

Find all the pairs of the positive integers such that the product of the numbers of any pair plus the half of one of the numbers plus one third of the other number is three times less than $1004$.

Let $\mathcal{A}$ be the set of all $n\in\mathbb{N}$ for which there exist $k\in\mathbb{N}$ and $a_0,a_1,\dots,a_{k-1}\in \{1,2,\dots,9\}$ such that $a_0 \geq a_1 \geq \cdots \geq a_{k-1}$ and $n = a_0 +a_1 \cdot 10^1 +\cdots +a_{k-1}\cdot 10^{k-1}$. Let $\mathcal{B}$ be the set of all $m \in\mathbb{N}$ for which there exist $l \in\mathbb{N}$ and $b_0,b_1,\dots,b_{l-1} \in \{1,2,\dots,9\}$ such that $b_0 \leq b_1 \leq \cdots\leq b_{l-1}$ and $m = b_0 + b_1 \cdot 10^1 + \cdots+ b_{l-1}\cdot 10^{l-1}$. [list=a] [*] Are there infinitely many $n\in \mathcal{A}$ such that $n^2-3\in\mathcal{A} \ ?$ [*] Are there infinitely many $m\in \mathcal{B}$ such that $m^2-3\in\mathcal{B} \ ?$ [/list] [i]Proposed by Pakawut Jiradilok and Wijit Yangjit[/i]

Determine all triples $(a, b, c)$ of positive integers such that $$a! + b! = 2^{c!}.$$ [i](Walther Janous)[/i]

[b]p1.[/b] If $P$ is a (convex) polygon, a triangulation of $P$ is a set of line segments joining pairs of corners of $P$ in such a way that $P$ is divided into non-overlapping triangles, each of which has its corners at corners of $P$. For example, the following are different triangulations of a square. (a) Prove that if $P$ is an $n$-gon with $n > 3$, then every triangulation of $P$ produces at least two triangles $T_1$, $T_2$ such that two of the sides of $T_i$, $i = 1$ or $2$ are also sides of $P$. (b) Find the number of different possible triangulations of a regular hexagon. [img]https://cdn.artofproblemsolving.com/attachments/9/d/0f760b0869fafc882f293846c05d182109fb78.png[/img] [b]p2.[/b] There are $n$ students, $n \ge 2$, and $n + 1$ cubical cakes of volume $1$. They have the use of a knife. In order to divide the cakes equitably they make cuts with the knife. Each cut divides a cake (or a piece of a cake) into two pieces. (a) Show that it is possible to provide each student with a volume $(n + 1)/n$ of a cake while making no more than $n - 1$ cuts. (b) Show that for each integer $k$ with $2 \le k \le n$ it is possible to make $n - 1$ cuts in such a way that exactly $k$ of the $n$ students receive an entire (uncut) cake in their portion. [b]p3. [/b]The vertical lines at $x = 0$, $x = \frac12$ , $x = 1$, $x = \frac32$ ,$...$ and the horizontal lines at $y = 0$, $y = \frac12$ , $y = 1$, $y = \frac32$ ,$ ...$ subdivide the first quadrant of the plane into $\frac12 \times \frac12$ square regions. Color these regions in a checkerboard fashion starting with a black region near the origin and alternating black and white both horizontally and vertically. (a) Let $T$ be a rectangle in the first quadrant with sides parallel to the axes. If the width of $T$ is an integer, prove that $T$ has equal areas of black and white. Note that a similar argument works to show that if the height of $T$ is an integer, then $T$ has equal areas of black and white. (b) Let $R$ be a rectangle with vertices at $(0, 0)$, $(a, 0)$, $(a, b)$, and $(0, b)$ with $a$ and $b$ positive. If $R$ has equal areas of black and white, prove that either $a$ is an integer or that $b$ is an integer. (c) Suppose a rectangle $R$ is tiled by a finite number of rectangular tiles. That is, the rectangular tiles completely cover $R$ but intersect only along their edges. If each of the tiles has at least one integer side, prove that $R$ has at least one integer side. [b]p4.[/b] Call a number [i]simple [/i] if it can be expressed as a product of single-digit numbers (in base ten). (a) Find two simple numbers whose sum is $2014$ or prove that no such numbers exist. (b) Find a simple number whose last two digits are $37$ or prove that no such number exists. [b]p5.[/b] Consider triangles for which the angles $\alpha$, $\beta$, and $\gamma$ form an arithmetic progression. Let $a, b, c$ denote the lengths of the sides opposite $\alpha$, $\beta$, $\gamma$ , respectively. Show that for all such triangles, $$\frac{a}{c}\sin 2\gamma +\frac{c}{a} \sin 2\alpha$$ has the same value, and determine an algebraic expression for this value. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Call a natural number [i]simple[/i] if it is not divisible by any square of a prime number (in other words it is square-free). Prove that there are infinitely many positive integers $n$ such that both $n$ and $n+1$ are [i]simple[/i].

Prove that: there exists a positive constant $K$, and an integer series $\{a_n\}$, satisfying: $(1)$ $0<a_1<a_2<\cdots <a_n<\cdots $; $(2)$ For any positive integer $n$, $a_n<1.01^n K$; $(3)$ For any finite number of distinct terms in $\{a_n\}$, their sum is not a perfect square.

Let $x$ and $y$ be positive integers such that $xy$ divides $x^{2}+y^{2}+1$. Show that \[\frac{x^{2}+y^{2}+1}{xy}=3.\]

Let $p$ be an odd prime. The numbers $1, 2, \ldots, d$ are written on a blackboard, where $d \geq p-1$ is a positive integer. A valid operation is to delete two numbers $x$ and $y$ and write $x + y - c \cdot xy$ in their place, where $c$ is a positive integer. One moment there is only one number $A$ left on the board. Show that if there is an order of operations such that $p$ divides $A$, then $p | d$ or $p | d + 1$.

If $n$ is is an integer such that $4n+3$ is divisible by $11,$ find the form of $n$ and the remainder of $n^{4}$ upon division by $11$.

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

A pair of positive integers $(a,b)$ is called [i]charrua[/i] if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called [i]non-charrua[/i]. a) Prove that there are infinite [i]non-charrua[/i] pairs. b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is [i]charrua[/i].