Let $\phi (n)$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Evaluate $$\sum \limits _{n=1} ^{64} (-1)^{n} \left \lfloor \frac{64}{n} \right \rfloor \phi (n).$$

[u]Round 5[/u] [b]5.1.[/b] A triangle has lengths such that one side is $12$ less than the sum of the other two sides, the semi-perimeter of the triangle is $21$, and the largest and smallest sides have a difference of $2$. Find the area of this triangle. [b]5.2.[/b] A rhombus has side length $85$ and diagonals of integer lengths. What is the sum of all possible areas of the rhombus? [b]5.3.[/b] A drink from YAKSHAY’S SHAKE SHOP is served in a container that consists of a cup, shaped like an upside-down truncated cone, and a semi-spherical lid. The ratio of the radius of the bottom of the cup to the radius of the lid is $\frac23$ , the volume of the combined cup and lid is $296\pi$, and the height of the cup is half of the height of the entire drink container. What is the volume of the liquid in the cup if it is filled up to half of the height of the entire drink container? [u]Round 6[/u] [i]Each answer in the next set of three problems is required to solve a different problem within the same set. There is one correct solution to all three problems; however, you will receive points for any correct answer regardless whether other answers are correct.[/i] [b]6.1.[/b] Let the answer to problem $2$ be $b$. There are b people in a room, each of which is either a truth-teller or a liar. Person $1$ claims “Person $2$ is a liar,” Person $2$ claims “Person $3$ is a liar,” and so on until Person $b$ claims “Person $1$ is a liar.” How many people are truth-tellers? [b]6.2.[/b] Let the answer to problem $3$ be $c$. What is twice the area of a triangle with coordinates $(0, 0)$, $(c, 3)$ and $(7, c)$ ? [b]6.3.[/b] Let the answer to problem $ 1$ be $a$. Compute the smaller zero to the polynomial $x^2 - ax + 189$ which has $2$ integer roots. [u]Round 7[/u] [b]7.1. [/b]Sir Isaac Neeton is sitting under a kiwi tree when a kiwi falls on his head. He then discovers Neeton’s First Law of Kiwi Motion, which states: [i]Every minute, either $\left\lfloor \frac{1000}{d} \right\rfloor$ or $\left\lceil \frac{1000}{d} \right\rceil$ kiwis fall on Neeton’s head, where d is Neeton’s distance from the tree in centimeters.[/i] Over the next minute, $n$ kiwis fall on Neeton’s head. Let $S$ be the set of all possible values of Neeton’s distance from the tree. Let m and M be numbers such that $m < x < M$ for all elements $x$ in $S$. If the least possible value of $M - m$ is $\frac{2000}{16899}$ centimeters, what is the value of $n$? Note that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, and $\lceil x \rceil$ is the least integer greater than or equal to $x$. [b]7.2.[/b] Nithin is playing chess. If one queen is randomly placed on an $ 8 \times 8$ chessboard, what is the expected number of squares that will be attacked including the square that the queen is placed on? (A square is under attack if the queen can legally move there in one move, and a queen can legally move any number of squares diagonally, horizontally or vertically.) [b]7.3.[/b] Nithin is writing binary strings, where each character is either a $0$ or a $1$. How many binary strings of length $12$ can he write down such that $0000$ and $1111$ do not appear? [u]Round 8[/u] [b]8.[/b] What is the period of the fraction $1/2018$? (The period of a fraction is the length of the repeated portion of its decimal representation.) Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.1 |I|}, 13 - \frac{|I-X|}{0.1 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2765571p24215461]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $m$ and $n$ satisfying $2^n+n=m!$.

Find all positive integer $n$ such that for all $i=1,2,\cdots,n$, $\frac{n!}{i!(n-i+1)!}$ is an integer. [i]Proposed by ckliao914[/i]

Let $n$ be a product of 2024 different prime numbers. Find the number of positive integers $k$, such that $$n+gcd(n, k)=k.$$

Let $a,b$ be two positive integers and $a>b$.We know that $\gcd(a-b,ab+1)=1$ and $\gcd(a+b,ab-1)=1$. Prove that $(a-b)^2+(ab+1)^2$ is not a perfect square.

Let $a_1\in\mathbb{Z}$, $a_2=a_1^2-a_1-1$, $\dots$ ,$a_{n+1}=a_n^2-a_n-1$. Prove that $a_{n+1}$ and $2n+1$ are coprime.

Let $a_0, a_1, \ldots , a_k \ (k \geq 1)$ be positive integers. Find all positive integers $y$ such that \[a_0 | y, (a_0 + a_1) | (y + a1), \ldots , (a_0 + a_n) | (y + a_n).\]

Let $1 + 1/2 + 1/3 +... + 1/n = a_n/b_n$, where $a_n$ and $b_n$ are relatively prime. Show that there exist infinitely many positive integers $n$, such that $b_{n+1} < b_n$. (8)

How many integers $n$ are there such that $0 \le n \le 720$ and $n^2 \equiv 1$ (mod $720$)?

Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]

A $5$-tuple $(1,1,1,1,2)$ has the property that the sum of any three of them is divisible by the sum of the remaining two. Is there a $5$-tuple with this property whose all terms are distinct?

$k>2$ is an integer. $a_0,a_1,...$ is an integer sequence such that $a_0=0$, $a_{n+1}=ka_n-a_{n-1}$. Prove that for any positive integer $m$, $(2m)!|a_1a_2...a_{3m}$.

A positive integer $n$ is called [i]good[/i] if for all positive integers $a$ which can be written as $a=n^2 \sum_{i=1}^n {x_i}^2$ where $x_1, x_2, \ldots ,x_n$ are integers, it is possible to express $a$ as $a=\sum_{i=1}^n {y_i}^2$ where $y_1, y_2, \ldots, y_n$ are integers with none of them is divisible by $n.$ Find all good numbers.

Find all quadruplets of positive integers $(m,n,k,l)$ such that $3^m = 2^k + 7^n$ and $m^k = 1 + k + k^2 + k^3 + \ldots + k^l$.

Determine all $3$-digit numbers which are equal to cube of the sum of all its digits.

Find the number of pairs $(m, n)$ of positive integers, both of which are $\le 1000$, such that $\frac{m}{n+1}< \sqrt2 < \frac{m+1}{n}$ (recalling that $ \sqrt2 = 1.414213..$.). (D. Fomin, Leningrad)

Prove that if a lattice triangle has no lattice points on its boundary in addition to its vertices, and one point in its interior, then this interior point is its center of gravity.

Prove that for each $ a\in\mathbb N$, there are infinitely many natural $ n$, such that \[ n\mid a^{n \minus{} a \plus{} 1} \minus{} 1. \]

Compute the remainder when $\underbrace{\hbox{11...1}}_{\hbox{1862}}$ is divided by $2006$

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

The sequence of natural numbers is defined as follows: for any $ k\geq 1 $,$ a_{k+2}= a_{k+1}\cdot a_k+1 $. Prove that for $ k\geq 9 $ the number $ a_k-22 $ is composite.

[b]6.1[/b] Which number is greater $$\underbrace{1000. . . 001}_{1965\, zeroes} / \underbrace{1000 . . . 001}_{1966\, zeroes} \,\,\, or \,\,\, \underbrace{1000. . . 001}_{1966\, zeroes} / \underbrace{1000 . . . 001}_{1967\, zeroes} \,\,?$$ [b]6.2[/b] $30$ teams participate in the football championship. Prove that at any moment there will be two teams that have played at this point the same number of matches. [b]6.3./ 7.1 [/b] All integers from $1$ to $1966$ are written on the board. Allowed is to erase any two numbers by writing their difference instead. Prove that repeating such an operation many times cannot ensure that There are only zeros left on the board. [b]6.4 / 7.5[/b] Black paint was sprayed onto a white surface. Prove that there are three points of the same color lying on the same line, and so, that one of the points lies in the middle between the other two. [b]6.5[/b] In a chess tournament, there are more than three chess players, and each player plays each other the same number of times. There were $26$ rounds in the tournament. After the $13$th round, one of the participants discovered that he had an odd number points, and each of the other participants has an even number of points. How many chess players participated in the tournament? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here[/url].

Prove that the ratio $$\frac{1^1 + 3^3 + 5^5 + ...+ (2^{2013} - 1)^{(2^{2013} - 1)}}{2^{2013}}$$ is an odd integer.

$p$ is a prime number that is greater than $2$. Let $\{ a_{n}\}$ be a sequence such that $ na_{n+1}= (n+1) a_{n}-\left( \frac{p}{2}\right)^{4}$. Show that if $a_{1}=5$, the $16 \mid a_{81}$.