For composite number $n$, let $f(n)$ denote the sum of the least three divisors of $n$, and $g(n)$ the sum of the greatest two divisors of $n$. Find all composite numbers $n$, such that $g(n)=(f(n))^m$ ($m\in N^*$).

Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

Let $p$ be an odd prime. Let $A(n)$ be the number of subsets of $\{1,2,...,n\}$ such that the sum of elements of the subset is a multiple of $p$. Prove that if $2^{p-1}-1$ is not a multiple of $p^2$, there exists infinitely many positive integer $m$ for any integer $k$ that satisfies the following. (The sum of elements of the empty set is 0.) $$\frac{A(m)-k}{p}\in\mathbb{Z}$$

Let a pair of positive integers $(n, m)$ that are relatively prime be called [i]intertwined[/i] if among any two divisors of $n$ greater than $1$, there exists a divisor of $m$ and among any two divisors of $m$ greater than $1$, there exists a divisor of $n$. For example, pair $(63, 64)$ is intertwined. [b]a)[/b] Find the largest integer $k$ for which there exists an intertwined pair $(n, m)$ such that the product $nm$ is equal to the product of the first $k$ prime numbers. [b]b)[/b] Prove that there does [b]not[/b] exist an intertwined pair $(n, m)$ such that the product $nm$ is the product of $2025$ distinct prime numbers. [b]c)[/b] Prove that there exists an intertwined pair $(n, m)$ such that the number of divisors of $n$ is greater than $2025$. [i]Proposed by Stijn Cambie, Belgium[/i]

For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

Given a finite set $X$, two positive integers $n,k$, and a map $f:X\to X$. Define $f^{(1)}(x)=f(x),f^{(i+1)}(x)=f^{(i)}(x)$,$i=1,2,3,\ldots$. It is known that for any $x\in X$,$f^{(n)}(x)=x$. Define $m_j$ the number of $x\in X$ satisfying $f^{(j)}(x)=x$. Prove that: (1)$\frac{1}n \sum_{j=1}^n m_j\sin {\frac{2kj\pi}{n}}=0$ (2)$\frac{1}n \sum_{j=1}^n m_j\cos {\frac{2kj\pi}{n}}$ is a non-negative integer.

Let $d_1,d_2,d_3,d_4$ be the four smallest divisors of a positive integer $n$ (having at least four divisors). Find all $n$ such that $d_1^2+d_2^2+d_3^2+d_4^2 = n$.

[u]Round 9[/u] [b]p25.[/b] Let $a$, $b$, and $c$ be positive numbers with $a +b +c = 4$. If $a,b,c \le 2$ and $$M =\frac{a^3 +5a}{4a^2 +2}+\frac{b^3 +5b}{4b^2 +2}+\frac{c^3 +5c}{4c^2 +2},$$ then find the maximum possible value of $\lfloor 100M \rfloor$. [b]p26.[/b] In $\vartriangle ABC$, $AB = 15$, $AC = 16$, and $BC = 17$. Points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, such that $CE = 1$ and $BF = 3$. A point $D$ is chosen on side $BC$, and let the circumcircles of $\vartriangle BFD$ and $\vartriangle CED$ intersect at point $P \ne D$. Given that $\angle PEF = 30^o$, the length of segment $PF$ can be expressed as $\frac{m}{n}$ . Find $m+n$. [b]p27.[/b] Arnold and Barnold are playing a game with a pile of sticks with Arnold starting first. Each turn, a player can either remove $7$ sticks or $13$ sticks. If there are fewer than $7$ sticks at the start of a player’s turn, then they lose. Both players play optimally. Find the largest number of sticks under $200$ where Barnold has a winning strategy [u]Round 10[/u] [b]p28.[/b] Let $a$, $b$, and $c$ be positive real numbers such that $\log_2(a)-2 = \log_3(b) =\log_5(c)$ and $a +b = c$. What is $a +b +c$? [b]p29.[/b] Two points, $P(x, y)$ and $Q(-x, y)$ are selected on parabola $y = x^2$ such that $x > 0$ and the triangle formed by points $P$, $Q$, and the origin has equal area and perimeter. Find $y$. [b]p30.[/b] $5$ families are attending a wedding. $2$ families consist of $4$ people, $2$ families consist of $3$ people, and $1$ family consists of $2$ people. A very long row of $25$ chairs is set up for the families to sit in. Given that all members of the same family sit next to each other, let the number of ways all the people can sit in the chairs such that no two members of different families sit next to each other be $n$. Find the number of factors of $n$. [u]Round 11[/u] [b]p31.[/b] Let polynomial $P(x) = x^3 +ax^2 +bx +c$ have (not neccessarily real) roots $r_1$, $r_2$, and $r_3$. If $2ab = a^3 -20 = 6c -21$, then the value of $|r^3_1+r^3_2+r^3_3|$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$. [b]p32.[/b] In acute $\vartriangle ABC$, let $H$, $I$ , $O$, and $G$ be the orthocenter, incenter, circumcenter, and centroid of $\vartriangle ABC$, respectively. Suppose that there exists a circle $\omega$ passing through $B$, $I$ , $H$, and $C$, the circumradius of $\vartriangle ABC$ is $312$, and $OG = 80$. Let $H'$, distinct from $H$, be the point on $\omega$ such that $\overline{HH'}$ is a diameter of $\omega$. Given that lines $H'O$ and $BC$ meet at a point $P$, find the length $OP$. [b]p33.[/b] Find the number of ordered quadruples $(x, y, z,w)$ such that $0 \le x, y, z,w \le 1000$ are integers and $$x!+ y! =2^z \cdot w!$$ holds (Note: $0! = 1$). [u]Round 12[/u] [b]p34.[/b] Let $Z$ be the product of all the answers from the teams for this question. Estimate the number of digits of $Z$. If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \lceil 15- |A-E| \rceil \right).$$ Your answer must be a positive integer. [b]p35.[/b] Let $N$ be number of ordered pairs of positive integers $(x, y)$ such that $3x^2 -y^2 = 2$ and $x < 2^{75}$. Estimate $N$. If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \lceil 15- 2|A-E| \rceil \right).$$ [b]p36.[/b] $30$ points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments). If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \left \lceil 15- \ln \frac{A}{E} \right \rceil \right).$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166472p28814057]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166476p28814111]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For an odd prime $ p, $ show that $ \sum_{k=1}^{p-1} \frac{k^p-k}{p}\equiv \frac{1+p}{2}\pmod p . $

Let $u_1=1,u_2=1$ and for all $k \geq 1$'s $$u_{k+2}=u_{k+1}+u_{k}$$ Prove that for all $m \geq 1$'s $5$ divides $u_{5m}$

A [i]perfect number[/i] is an integer that equals half the sum of its positive divisors. For example, because $2 \cdot 28 = 1 + 2 + 4 + 7 + 14 + 28$, $28$ is a perfect number. [list] [*] [b](a)[/b] A [i]square-free[/i] integer is an integer not divisible by a square of any prime number. Find all square-free integers that are perfect numbers. [*] [b](b)[/b] Prove that no perfect square is a perfect number.[/list]

Determine all functions $f: \mathbb{Q} \rightarrow \mathbb{Z} $ satisfying \[ f \left( \frac{f(x)+a} {b}\right) = f \left( \frac{x+a}{b} \right) \] for all $x \in \mathbb{Q}$, $a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.)

[u]Round 1[/u] [b]p1.[/b] This year, the Mathathon consists of $7$ rounds, each with $3$ problems. Another math test, Aspartaime, consists of $3$ rounds, each with $5$ problems. How many more problems are on the Mathathon than on Aspartaime? [b]p2.[/b] Let the solutions to $x^3 + 7x^2 - 242x - 2016 = 0 $be $a, b$, and $c$. Find $a^2 + b^2 + c^2$. (You might find it helpful to know that the roots are all rational.) [b]p3.[/b] For triangle $ABC$, you are given $AB = 8$ and $\angle A = 30^o$ . You are told that $BC$ will be chosen from amongst the integers from $1$ to $10$, inclusive, each with equal probability. What is the probability that once the side length $BC$ is chosen there is exactly one possible triangle $ABC$? [u]Round 2 [/u] [b]p4.[/b] It’s raining! You want to keep your cat warm and dry, so you want to put socks, rain boots, and plastic bags on your cat’s four paws. Note that for each paw, you must put the sock on before the boot, and the boot before the plastic bag. Also, the items on one paw do not affect the items you can put on another paw. How many different orders are there for you to put all twelve items of rain footwear on your cat? [b]p5.[/b] Let $a$ be the square root of the least positive multiple of $2016$ that is a square. Let $b$ be the cube root of the least positive multiple of $2016$ that is a cube. What is $ a - b$? [b]p6.[/b] Hypersomnia Cookies sells cookies in boxes of $6, 9$ or $10$. You can only buy cookies in whole boxes. What is the largest number of cookies you cannot exactly buy? (For example, you couldn’t buy $8$ cookies.) [u]Round 3 [/u] [b]p7.[/b] There is a store that sells each of the $26$ letters. All letters of the same type cost the same amount (i.e. any ‘a’ costs the same as any other ‘a’), but different letters may or may not cost different amounts. For example, the cost of spelling “trade” is the same as the cost of spelling “tread,” even though the cost of using a ‘t’ may be different from the cost of an ‘r.’ If the letters to spell out $1$ cost $\$1001$, the letters to spell out $2$ cost $\$1010$, and the letters to spell out $11$ cost $\$2015$, how much do the letters to spell out $12$ cost? [b]p8.[/b] There is a square $ABCD$ with a point $P$ inside. Given that $PA = 6$, $PB = 9$, $PC = 8$. Calculate $PD$. [b]p9.[/b] How many ordered pairs of positive integers $(x, y)$ are solutions to $x^2 - y^2 = 2016$? [u]Round 4 [/u] [b]p10.[/b] Given a triangle with side lengths $5, 6$ and $7$, calculate the sum of the three heights of the triangle. [b]p11. [/b]There are $6$ people in a room. Each person simultaneously points at a random person in the room that is not him/herself. What is the probability that each person is pointing at someone who is pointing back to them? [b]p12.[/b] Find all $x$ such that $\sum_{i=0}^{\infty} ix^i =\frac34$. PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782837p24446063]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many positive integers less than $1$ million have exactly $5$ positive divisors? $$ \mathrm a. ~ 1\qquad \mathrm b.~5\qquad \mathrm c. ~11 \qquad \mathrm d. ~23 \qquad \mathrm e. ~24 $$

Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is $m/n$, where $m$ and $n$ are relatively prime positive integers. What are the last three digits of $m + n$?

The integers $a, b$ satisfy the following identity $$2a^2 + a = 3b^2 + b.$$ Prove that $a- b$, $2a + 2b + 1$, and $3a + 3b + 1$ are perfect squares.

Let $a > 2$ be a natural number. Show that there are infinitely many natural numbers n such that $a^n \equiv -1$ (mod $n^2$).

Let $ x,y$ be positive integers with $ y>3$ and $ x^2\plus{}y^4\equal{}2((x\minus{}6)^2\plus{}(y\plus{}1)^2).$ Prove that: $ x^2\plus{}y^4\equal{}1994.$

Let $a$ and $b$ be coprime integers, greater than or equal to $1$. Prove that all integers $n$ greater than or equal to $(a - 1)(b - 1)$ can be written in the form: \[n = ua + vb, \qquad \text{with} (u, v) \in \mathbb N \times \mathbb N.\]

Given a square-free integer $n>2$, evaluate the sum $\sum_{k=1}^{(n-2)(n-1)} \lfloor ({kn})^{1/3} \rfloor$.

Determine all positive integers $k$ for which there exist a positive integer $m$ and a set $S$ of positive integers such that any integer $n > m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways.

Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children? $ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$

Find the smallest positive integer $n$ with the following property: there does not exist an arithmetic progression of $1999$ real numbers containing exactly $n$ integers.

Alice has an integer $N > 1$ on the blackboard. Each minute, she deletes the current number $x$ on the blackboard and writes $2x+1$ if $x$ is not the cube of an integer, or the cube root of $x$ otherwise. Prove that at some point of time, she writes a number larger than $10^{100}$. [i]Proposed by Anant Mudgal and Rohan Goyal[/i]

Prove that there exists a natural number $b$ such that for any natural $n>b$ the sum of the digits of $n!$ is not less than $10^{100}$. [i]Proposed by D. Khramtsov[/i]