Find all positive integers $k$ such that for any positive integer $n$, $2^{(k-1)n+1}$ does not divide $\frac{(kn)!}{n!}$.

Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$. Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set. [i]Dan Schwarz, Romania[/i]

Let $ n>1$ be an odd integer and define: \[ N\equal{}\{\minus{}n,\minus{}(n\minus{}1),\dots,\minus{}1,0,1,\dots,(n\minus{}1),n\}.\] A subset $ P$ of $ N$ is called [i]basis[/i] if we can express every element of $ N$ as the sum of $ n$ different elements of $ P$. Find the smallest positive integer $ k$ such that every $ k\minus{}$elements subset of $ N$ is basis.

Find all quadruples of positive integers $(a,b,c,d)$ such that $a+b=cd$ and $c+d=ab$.

[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].

Find all positive integers $n \geq 2$ such that for all integers $i,j$ that $ 0 \leq i,j\leq n$ , $i+j$ and $ {n\choose i}+ {n \choose j}$ have same parity. [i]Proposed by Mr.Etesami[/i]

Let $n$ be a positive integer. Determine the number of sequences $a_0, a_1, \ldots, a_n$ with terms in the set $\{0,1,2,3\}$ such that $$n=a_0+2a_1+2^2a_2+\ldots+2^na_n.$$

Show that there exist infinitely many pairs $(a, b)$ of positive integers with the property that $a+b$ divides $ab+1$, $a-b$ divides $ab-1$, $b>1$ and $a>b\sqrt{3}-1$

Determine all positive integer numbers $n$ satisfying the following condition: the sum of the squares of any $n$ prime numbers greater than $3$ is divisible by $n$.

Show that there exists a positive integer $N$ such that the decimal representation of $2000^N$ starts with the digits $200120012001.$

Find all positive integer $n$ such that for all positive integers $ x $, $ y $, $ n \mid x^n-y^n \Rightarrow n^2 \mid x^n-y^n $.

Determine all functions $f : \mathbb {N} \rightarrow \mathbb {N}$ such that $f$ is increasing (not necessarily strictly) and the numbers $f(n)+n+1$ and $f(f(n))-f(n)$ are both perfect squares for every positive integer $n$.

A positive integer is written on a blackboard. Carmela can perform the following operation as many times as she wants: replace the current integer $x$ with another positive integer $y$, as long as $|x^2 - y^2|$ is a perfect square. For example, if the number on the blackboard is $17$, Carmela can replace it with $15$, because $|17^2 - 15^2| = 8^2$, then replace it with $9$, because $|15^2 - 9^2| = 12^2$. If the number on the blackboard is initially $3$, determine all integers that Carmela can write on the blackboard after finitely many operations.

Of the vertices of a cube, $7$ of them have assigned the value $0$, and the eighth the value $1$. A [i]move[/i] is selecting an edge and increasing the numbers at its ends by an integer value $k > 0$. Prove that after any finite number of [i]moves[/i], the g.c.d. of the $8$ numbers at vertices is equal to $1$. Russian M.O.

Consider all the $7$-digit numbers formed by the digits $1,2 , 3,...,7$ each digit being used exactly once in all the $7! $ numbers. Prove that no two of them have the property that one divides the other.

Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.

Let $ a,b,c,d$ be the positive integers such that $ a > b > c > d$ and $ (a \plus{} b \minus{} c \plus{} d) | (ac \plus{} bd)$ . Prove that if $ m$ is arbitrary positive integer , $ n$ is arbitrary odd positive integer, then $ a^n b^m \plus{} c^m d^n$ is composite number

Suppose we choose two numbers $x,y\in[0,1]$ uniformly at random. If the probability that the circle with center $(x,y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1]\times [0,1]$ is written as $\tfrac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?

Let $a,b, c$ be $3$ distinct numbers from $\{1, 2,3, 4, 5, 6\}$ Show that $7$ divides $abc + (7 - a)(7 - b)(7 - c)$

Let $m,n$ be distinct positive integers. Prove that $$gcd(m,n) + gcd(m+1,n+1) + gcd(m+2,n+2) \le 2|m-n| + 1. $$ Further, determine when equality holds.

There are $n>3$ different natural numbers, less than $(n-1)!$ For every pair of numbers Ivan divides bigest on lowest and write integer quotient (for example, $100$ divides $7$ $= 14$) and write result on the paper. Prove, that not all numbers on paper are different.

A polynomial $p(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$ with integer coefficients is said to be divisible by an integer $m$ if $p(x)$ is divisible by m for all integers $x$. Prove that if $p(x)$ is divisible by $m$, then $k!a_0$ is also divisible by $m$. Also prove that if $a_0, k,m$ are non-negative integers for which $k!a_0$ is divisible by $m$, there exists a polynomial $p(x) = a_0x^k+\cdots+ a_k$ divisible by $m.$

Call a positive whole number [i]rickety [/i] if it is three times the product of its digits. There are two $2$-digit numbers that are rickety. What is their sum?

$\textbf{N4:} $ Let $a,b$ be two positive integers such that for all positive integer $n>2020^{2020}$, there exists a positive integer $m$ coprime to $n$ with \begin{align*} \text{ $a^n+b^n \mid a^m+b^m$} \end{align*} Show that $a=b$ [i]Proposed by ltf0501[/i]

If the fraction $\frac{an + b}{cn + d}$ may be simplified using $2$ (as a common divisor ), show that the number $ad - bc$ is even. ($a, b, c, d, n$ are natural numbers and the $cn + d$ different from zero).