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.

For natural numbers $m,n$, set $a=(n+1)^m-n$ and $b=(n+1)^{m+3}-n$. (a) Prove that $a$ and $b$ are coprime if $m$ is not divisible by $3$. (b) Find all numbers $m,n$ for which $a$ and $b$ are not coprime.

Solve the following equation in $\mathbb{Z}$: \[3^{2a + 1}b^2 + 1 = 2^c\]

Alexander has chosen a natural number $N>1$ and has written down in a line,and in increasing order,all his positive divisors $d_1<d_2<\ldots <d_s$ (where $d_1=1$ and $d_s=N$).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these $s-1$ numbers (the greatest common divisors) is equal to $N-2$.Find all possible values of $N$.

a) Prove that it is possible to choose one number out of any 39 consecutive positive integers, having the sum of its digits divisible by 11; b) Find the first 38 consecutive positive integers none of which have the sum of its digits divisible by 11.

Prove that the product of the first $1000$ positive even integers differs from the product of the first $1000$ positive odd integers by a multiple of $2001$.

Let $p$ be a prime such that $\frac{p-1}{2}$ is also prime. A pair of integers $(x, y)$ with $1\le x, y \le p-1$ is called a [i]commuter[/i] if at least one of $x^y -y^x$ or $x^y +y^x$ is divisible by $p$. Show that the number of commuters is at most $4.2p\sqrt{p}$.

How many positive integers of $n$ digits exist such that each digit is $1, 2$, or $3$? How many of these contain all three of the digits $1, 2$, and $3$ at least once?

[b]6.[/b] Let $a$ be a positive integer and $p$ a prime divisor of $a^3-3a+1$, with $p \neq 3$. Prove that $p$ is of the form $9k+1$ or $9k-1$, where $k$ is integer.

Let $d(n)$ be the number of divisors of a nonnegative integer $n$ (we set $d(0)=0$). Find all positive integers $d$ such that there exists a two-variable polynomial $P(x,y)$ of degree $d$ with integer coefficients such that: [list] [*] for any positive integer $y$, there are infinitely many positive integers $x$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid x$, and [*] for any positive integer $x$, there are infinitely many positive integers $y$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid y$. [/list] [i]Allen Wang[/i]

Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .

Prove that if $n$ and $k$ are positive integers such that $1<k<n-1$,Then the binomial coefficient $\binom nk$ is divisible by at least two different primes.

Determine all pairs of positive integers $(x, y)$ such that: $$x^4 - 6x^2 + 1 = 7\cdot 2^y$$

For every positive integer $n$, let $S (n)$ be the sum of the digits of $n$. Find, if any, a $171$-digit positive integer $n$ such that $7$ divides $S (n)$ and $7$ divides $S (n + 1)$.

Let $k$ be a positive integer. Show that there exist infinitely many positive integers $n$ such that $\frac{n^n-1}{n-1}$ has at least $k$ distinct prime divisors. [i]Proposed by Adnan Sadik[/i]

Determine the value of $(101 \times 99)$ - $(102 \times 98)$ + $(103 \times 97)$ − $(104 \times 96)$ + ... ... + $(149 \times 51)$ − $(150 \times 50)$.

Determine the natural numbers $m,n$ such as $85^m-n^4=4$

Prove that for all positive integers $n$, there exists a positive integer $m$ which is a multiple of $n$ and the sum of the digits of $m$ is equal to $n$.

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.

[b]p1.[/b] A very large lucky number $N$ consists of eighty-eight $8$s in a row. Find the remainder when this number $N$ is divided by $6$. [b]p2.[/b] If $3$ chickens can lay $9$ eggs in $4$ days, how many chickens does it take to lay $180$ eggs in $ 8$ days? [b]p3.[/b] Find the ordered pair $(x, y)$ of real numbers satisfying the conditions $x > y$, $x+y = 10$, and $xy = -119$. [b]p4.[/b] There is pair of similar triangles. One triangle has side lengths $4, 6$, and $9$. The other triangle has side lengths $ 8$, $12$ and $x$. Find the sum of two possible values of $x$. [b]p5.[/b] If $x^2 +\frac{1}{x^2} = 3$, there are two possible values of $x +\frac{1}{x}$. What is the smaller of the two values? [b]p6.[/b] Three flavors (chocolate strawberry, vanilla) of ice cream are sold at Brian’s ice cream shop. Brian’s friend Zerg gets a coupon for $10$ free scoops of ice cream. If the coupon requires Zerg to choose an even number of scoops of each flavor of ice cream, how many ways can he choose his ice cream scoops? (For example, he could have $6$ scoops of vanilla and $4$ scoops of chocolate. The order in which Zerg eats the scoops does not matter.) [b]p7.[/b] David decides he wants to join the West African Drumming Ensemble, and thus he goes to the store and buys three large cylindrical drums. In order to ensure none of the drums drop on the way home, he ties a rope around all of the drums at their mid sections so that each drum is next to the other two. Suppose that each drum has a diameter of $3.5$ feet. David needs $m$ feet of rope. Given that $m = a\pi + b$, where $a$ and $b$ are rational numbers, find sum $a + b$. [b]p8.[/b] Segment $AB$ is the diameter of a semicircle of radius $24$. A beam of light is shot from a point $12\sqrt3$ from the center of the semicircle, and perpendicular to $AB$. How many times does it reflect off the semicircle before hitting $AB$ again? [b]p9.[/b] A cube is inscribed in a sphere of radius $ 8$. A smaller sphere is inscribed in the same sphere such that it is externally tangent to one face of the cube and internally tangent to the larger sphere. The maximum value of the ratio of the volume of the smaller sphere to the volume of the larger sphere can be written in the form $\frac{a-\sqrt{b}}{36}$ , where $a$ and $b$ are positive integers. Find the product $ab$. [b]p10.[/b] How many ordered pairs $(x, y)$ of integers are there such that $2xy + x + y = 52$? [b]p11.[/b] Three musketeers looted a caravan and walked off with a chest full of coins. During the night, the first musketeer divided the coins into three equal piles, with one coin left over. He threw it into the ocean and took one of the piles for himself, then went back to sleep. The second musketeer woke up an hour later. He divided the remaining coins into three equal piles, and threw out the one coin that was left over. He took one of the piles and went back to sleep. The third musketeer woke up and divided the remaining coins into three equal piles, threw out the extra coin, and took one pile for himself. The next morning, the three musketeers gathered around to divide the coins into three equal piles. Strangely enough, they had one coin left over this time as well. What is the minimum number of coins that were originally in the chest? [b]p12.[/b] The diagram shows a rectangle that has been divided into ten squares of different sizes. The smallest square is $2 \times 2$ (marked with *). What is the area of the rectangle (which looks rather like a square itself)? [img]https://cdn.artofproblemsolving.com/attachments/4/a/7b8ebc1a9e3808096539154f0107f3e23d168b.png[/img] [b]p13.[/b] Let $A = (3, 2)$, $B = (0, 1)$, and $P$ be on the line $x + y = 0$. What is the minimum possible value of $AP + BP$? [b]p14.[/b] Mr. Mustafa the number man got a $6 \times x$ rectangular chess board for his birthday. Because he was bored, he wrote the numbers $1$ to $6x$ starting in the upper left corner and moving across row by row (so the number $x + 1$ is in the $2$nd row, $1$st column). Then, he wrote the same numbers starting in the upper left corner and moving down each column (so the number $7$ appears in the $1$st row, $2$nd column). He then added up the two numbers in each of the cells and found that some of the sums were repeated. Given that $x$ is less than or equal to $100$, how many possibilities are there for $x$? [b]p15.[/b] Six congruent equilateral triangles are arranged in the plane so that every triangle shares at least one whole edge with some other triangle. Find the number of distinct arrangements. (Two arrangements are considered the same if one can be rotated and/or reflected onto another.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that $\sum_{n=0}^{2013}\frac{4026!}{(n!(2013-n)!)^2}$ is a perfect square.

Is there a power of $2$ such that it is possible to rearrange the digits, giving another power of $2$?

Find all triples $(x,y,z)$ of positive integers such that $3^x + 4^y = 5^z$.

Find the number of squares in the sequence given by $ a_0\equal{}91$ and $ a_{n\plus{}1}\equal{}10a_n\plus{}(\minus{}1)^n$ for $ n \ge 0.$