Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]

[b]p25.[/b] Compute: $$\frac{ \sum^{\infty}_{i=0}\frac{(2\pi)^{4i+1}}{(4i+1)!}}{\sum^{\infty}_{i=0}\frac{(2\pi)^{4i+1}}{(4i+3)!}}$$ [b]p26.[/b] Suppose points $A, B, C, D$ lie on a circle $\omega$ with radius $4$ such that $ABCD$ is a quadrilateral with $AB = 6$, $AC = 8$, $AD = 7$. Let $E$ and $F$ be points on $\omega$ such that $AE$ and $AF$ are respectively the angle bisectors of $\angle BAC$ and $\angle DAC$. Compute the area of quadrilateral $AECF$. [b]p27.[/b] Let $P(x) = x^2 - ax + 8$ with a a positive integer, and suppose that $P$ has two distinct real roots $r$ and $s$. Points $(r, 0)$, $(0, s)$, and $(t, t)$ for some positive integer t are selected on the coordinate plane to form a triangle with an area of $2021$. Determine the minimum possible value of $a + t$. [b]p28.[/b] A quartic $p(x)$ has a double root at $x = -\frac{21}{4}$ , and $p(x) - 1344x$ has two double roots each $\frac14$ less than an integer. What are these two double roots? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We are given the set $$M = \{-2^{2022}, -2^{2021}, . . . , -2^{2}, -2, -1, 1, 2, 2^2, . . . , 2^{2021}, 2^{2022}\}.$$ Let $T$ be a subset of $M$, such that neighbouring numbers have the same difference when the elements are ordered by size. (a) Determine the maximum number of elements that such a set $T$ can contain. (b) Determine all sets $T$ with the maximum number of elements. [i](Walther Janous)[/i]

Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is [i]superabundant[/i] (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$ Prove that there exists an infinity of [i]superabundant[/i] numbers.

Find the addition of all positive integers n that follows the following: $ \frac{\sqrt{n}}{2} + \frac{30}{\sqrt{n}} $ is an integer.

Prove that for all integers $ m$ and $ n$, the inequality \[ \dfrac{\phi(\gcd(2^m \plus{} 1,2^n \plus{} 1))}{\gcd(\phi(2^m \plus{} 1),\phi(2^n \plus{} 1))} \ge \dfrac{2\gcd(m,n)}{2^{\gcd(m,n)}}\] holds. [i]Nanang Susyanto, Jogjakarta [/i]

Find all solutions for (x,y) , both integers such that: $xy=3(\sqrt{x^2+y^2}-1)$

Find the smallest positive integer $n$ such that $2549 | n^{2545} - 2$.

How many numbers greater than $1.000$ but less than $10.000$ have as a product of their digits $256$?

Two farmers agree that pigs are worth $ \$300$ and that goats are worth $ \$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $ \$390$ debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way? $ \textbf{(A) } \$5\qquad \textbf{(B) } \$10\qquad \textbf{(C) } \$30\qquad \textbf{(D) } \$90\qquad \textbf{(E) } \$210$

Prove that for every $n$ there exists a solution of the equation $$a^2 +b^2 +c^2 = 3abc$$ in natural numbers $a,b,c$ greater than $n$.

We define $ f: \mathbb{N} \rightarrow \mathbb{N}$, $ f(n) \equal{} \sum_{k \equal{} 1}^{n}(k,n)$. a) Show that if $ \gcd(m,n)\equal{}1$ then we have $ f(mn)\equal{}f(m)\cdot f(n)$; b) Show that $ \sum_{d|n}f(d) \equal{} nd(n)$.

Suppose $m$ and $n$ are relatively prime positive integers. A regular $m$-gon and a regular $n$-gon are inscribed in a circle. Let $d$ be the minimum distance in degrees (of the arc along the circle) between a vertex of the $m$-gon and a vertex of the $n$-gon. What is the maximum possible value of $d$?

In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.

It is known that the equation $x^3 + px^2 + q = 0$ where $q$ is non-zero, has three different integer roots, the absolute values of two of which are prime numbers. Find the roots of this equation.

Numbers $u_{n,k} \ (1\leq k \leq n)$ are defined as follows \[u_{1,1}=1, \quad u_{n,k}=\binom{n}{k} - \sum_{d \mid n, d \mid k, d>1} u_{n/d, k/d}.\] (the empty sum is defined to be equal to zero). Prove that $n \mid u_{n,k}$ for every natural number $n$ and for every $k \ (1 \leq k \leq n).$

Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$

Find all pairs of integers $(a, b)$ such that $a^2 + ab + b^2 = 1$

An unordered triple of numbers $(a,b,c)$ in one move you can change to either $(a,b,2a+2b-c)$, $(a,2a+2c-b,c)$ or $(2b+2c-a,b,c)$. Can you from the triple $(3,5,14)$ get the triple $(3,13,6)$ in finite amount of moves?

Prove that for any natural number $n{}$ the numbers $1,2,\ldots,n$ can be divided into several groups so that the sum of the numbers in each group is equal to a power of three. [i]Proposed by V. Novikov[/i]

[b] Problem 5. [/b]Denote with $d(a,b)$ the numbers of the divisors of natural $a$, which are greater or equal to $b$. Find all natural $n$, for which $d(3n+1,1)+d(3n+2,2)+\ldots+d(4n,n)=2006.$ [i]Ivan Landgev[/i]

5)Players $A$ and $B$ play the following game: a player writes, in a board, a positive integer $n$, after this they delete a number in the board and write a new number where can be: i)The last number $p$, where the new number will be $p - 2^k$ where $k$ is greatest number such that $p\ge 2^k$ ii) The last number $p$, where the new number will be $\frac{p}{2}$ if $p$ is even. The players play alternately, a player win(s) if the new number is equal to $0$ and player $A$ starts. a)Which player has the winning strategy with $n = 40$?? b)Which player has the winning strategy with $n = 2012$??

[u]Round 5[/u] [b]p13.[/b] Jason flips a coin repeatedly. The probability that he flips $15$ heads before flipping $4$ tails can be expressed as $\frac{a}{2^b}$ where $a$ and $b$ are positive integers and $a$ is odd. Find $a +b$. [b]p14.[/b] Triangle $ABC$ has side lengths $AB = 3$, $BC = 3$, and $AC = 4$. Let D be the intersection of the angle bisector of $\angle B AC$ and segment $BC$. Let the circumcircle of $\vartriangle B AD$ intersect segment $AC$ at a point $E$ distinct from $A$. The length of $AE$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p15.[/b] The sum of the squares of all values of $x$ such that $\{(x -2)(x -3)\} = \{(x -1)(x -6)\}$ and $\lfloor x^2 -6x +6 \rfloor= 9$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a +b$. Note: $\{a\}$ is the fractional part function, and returns $a -\lfloor a \rfloor$ . [u]Round 6[/u] [b]p16.[/b] Maisy the Polar Bear is at the origin of the Polar Plane ($r = 0, \theta = 0$). Maisy’s location can be expressed as $(r,\theta)$, meaning it is a distance of $r$ away from the origin and at a angle of $\theta$ degrees counterclockwise from the $x$-axis. When Maisy is on the point $(m,n)$ then it can jump to either $(m,n +1)$ or $(m+1,n)$. Maisy cannot jump to any point it has been to before. Let $L(r,\theta)$ be the number of paths Maisy can take to reach point $(r,\theta)$. The sum of $L(r,\theta)$ over all points where $r$ is an integer between $1$ and $2020$ and $\theta$ is an integer between $0$ and $359$ can be written as $\frac{n^k-1}{m}$ for some minimum value of $n$, such that $n$, $k$, and $m$ are all positive integers. Find $n +k +m$. [b]p17.[/b] A circle with center $O$ and radius $2$ and a circle with center $P$ and radius $3$ are externally tangent at $A$. Points $B$ and $C$ are on the circle with center $O$ such that $\vartriangle ABC$ is equilateral. Segment $AB$ extends past $B$ to point $D$ and $AC$ extends past $C$ to point $E$ such that $BD = CE = \sqrt3$. A line through $D$ is tangent to circle $P$ at $F$. Find $DF^2$. [img]https://cdn.artofproblemsolving.com/attachments/2/7/0ee8716cebd6701fcae6544d9e39e68fff35f5.png[/img] [b]p18.[/b] Find the number of trailing zeroes at the end of $$\prod^{2021}_{i=1} (2021i -1) = (2020)(4041)...(2021^2 -1).$$ [u]Round 7[/u] [b]p19.[/b] A function $f (n)$ is defined as follows: $$f (n) = \begin{cases} \frac{n}{3} \,\,\, if \,\,\, n \equiv 0 (mod \, 3) \\ n^2 +4n -5 \,\,\,if \,\,\,n \equiv 1 (mod \, 3) \\ n^2 +n -2 \,\,\, if \,\,\,n \equiv 2 (mod \, 3) \end{cases}$$ Find the number of integer values of $n$ between $2$ and $1000$ inclusive such that $f ( f (... f (n))) = 1$ for some number of applications of $f (n)$. [b]p20.[/b] In the diagram below, the larger circle with diameter $AW$ has radius $16$. $ABCD$ and $WXY Z$ are rhombi where $\angle B AD = \angle XWZ = 60^o$ and $AC = CY = YW$. $M$ is the midpoint of minor arc $AW$, as shown. Let $I$ be the center of the circle with diameter $OM$. Circles with center $P$ and $G$ are tangent to lines $AD$ and $WZ$, respectively, and also tangent to the circle with center $I$ . Given that $IP \perp AD$ and $IG \perp WZ$, the area of $\vartriangle PIG$ can be written as $a +b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of a prime. Find $a +b +c$. [b]p21.[/b] In a list of increasing consecutive positive integers, the first item is divisible by $1$, the second item is divisible by $4$, the third item is divisible by $7$, and this pattern increases up to the seventh item being divisible by $19$. Find the remainder when the least possible value of the first item in the list is divided by $100$. [u]Round 8[/u] [b]p22.[/b] Let the answer to Problem $24$ be $C$. Jacob never drinks more than $C$ cups of coffee in a day. He always drinks a positive integer number of cups. The probability that he drinks $C +1-X$ cups is $X$ times the probability he drinks $C$ cups of coffee for any positive number $X$ from $1$ to $C$ inclusive. Find the expected number of cups of coffee he drinks. [b]p23.[/b] Let the answer to Problem $22$ be $A$. Three lines are drawn intersecting the interior of a triangle with side lengths $26$, $28$, and $30$ such that each line is parallel and a distance A away from a respective side. The perimeter of the triangle formed by the three new lines can be expressed as $\frac{a}{b}$ for relatively prime integers $a$ and $b$. Find $a +b$. [b]p24.[/b] Let the answer to Problem $23$ be $B$. Given that $ab-c = bc-a = ca-b$ and $a^2+b^2+c^2 = B +2$, find the sum of all possible values of $|a +b +c|$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166489p28814241]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166500p28814367]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a prime number \( p \geq 3 \) and a positive integer \( m \), find the smallest positive integer \( n \) with the following property: for every positive integer \( a \), which is not divisible by \( p \), the sum of the natural divisors of \( a^n \) greater than 1 is divisible by \( p^m \).

Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that \[\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.\]