Let $n>2$ be a positive integer. Masha writes down $n$ natural numbers along a circle. Next, Taya performs the following operation: Between any two adjacent numbers $a$ and $b$, she writes a divisor of the number $a+b$ greater than $1$, then Taya erases the original numbers and obtains a new set of $n$ numbers along the circle. Can Taya always perform these operations in such a way that after some number of operations, all the numbers are equal? [i]Proposed by T. Korotchenko[/i]

Let $p$ be a prime such that $p^2$ divides $2^{p-1}-1$. Prove that for all positive integers $n$ the number $\left(p-1\right)\left(p!+2^n\right)$ has at least $3$ different prime divisors. [i]Aleksandar Ivanov[/i]

Find all pairs $(a, b)$ of positive integers for which $a + b = \phi (a) + \phi (b) + gcd (a, b)$. Here $ \phi (n)$ is the number of numbers $k$ from $\{1, 2,. . . , n\}$ with $gcd (n, k) = 1$.

Find all positive integers $r$ with the property that there exists positive prime numbers $p$ and $q$ so that $$p^2 + pq + q^2 = r^2 .$$

Find three-digit numbers such that any its positive integer power ends with the same three digits and in the same order.

[b]p1.[/b] How many subsets of $\{D,U,K,E\}$ have an odd number of elements? [b]p2.[/b] Find the coefficient of $x^{12}$ in $(1 + x^2 + x^4 +... + x^{28})(1 + x + x^2 + ...+ x^{14})^2$. [b]p3.[/b] How many $4$-digit numbers have their digits in non-decreasing order from left to right? [b]p4.[/b] A dodecahedron (a polyhedron with $12$ faces, each a regular pentagon) is projected orthogonally onto a plane parallel to one of its faces to form a polygon. Find the measure (in degrees) of the largest interior angle of this polygon. [b]p5.[/b] Justin is back with a $6\times 6$ grid made of $36$ colorless squares. Dr. Kraines wants him to color some squares such that $\bullet$ Each row and column of the grid must have at least one colored square $\bullet$ For each colored square, there must be another colored square on the same row or column What is the minimum number of squares that Justin will have to color? [b]p6.[/b] Inside a circle $C$, we have three equal circles $C_1$, $C_2$, $C_3$, which are pairwise externally tangent to each other and all internally tangent to $C$. What is the ratio of the area of $C_1$ to the area of $C$? [b]p7.[/b] There are $3$ different paths between the Duke Chapel and the Physics building. $6$ students are heading towards the Physics building for a class, so they split into $3$ pairs and each pair takes a separate path from the Chapel. After class, they again split into $3$ pairs and take separate paths back. Find the number of possible scenarios where each student's companion on the way there is different from their companion on the way back. [b]p8.[/b] Let $a_n$ be a sequence that satisfies the recurrence relation $$a_na_{n+2} =\frac{\cos (3a_{n+1})}{\cos (a_{n+1})[2 \cos(2a_{n+1}) - 1]}a_{n+1}$$ with $a_1 = 2$ and $a_2 = 3$. Find the value of $2018a_{2017}$. [b]p9.[/b] Let $f(x)$ be a polynomial with minimum degree, integer coefficients, and leading coefficient of $1$ that satisfies $f(\sqrt7 +\sqrt{13})= 0$. What is the value of $f(10)$? [b]p10.[/b] $1024$ Duke students, indexed $1$ to $1024$, are having a chat. For each $1 \le i \le 1023$, student $i$ claims that student $2^{\lfloor \log_2 i\rfloor +1}$ has a girlfriend. ($\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) Given that exactly $201$ people are lying, find the index of the $61$st liar (ordered by index from smallest to largest). PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $n \ge 2$ coins, each with a different positive integer value. Call an integer $m$ [i]sticky [/i] if some subset of these $n$ coins have total value $m$. We call the entire set of coins a stick if all the sticky numbers form a consecutive range of integers. Compute the minimum total value of a stick across all sticks containing a coin of value $100$.

$A$ is a $2m$-digit positive integer each of whose digits is $1$. $B$ is an $m$-digit positive integer each of whose digits is $4$. Prove that $A+B +1$ is a perfect square.

Let \(p\) a prime number, and \(N\) the number of matrices \(p \times p\) \[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1p}\\ a_{21} & a_{22} & \ldots & a_{2p}\\ \vdots & \vdots & \ddots & \vdots \\ a_{p1} & a_{p2} & \ldots & a_{pp} \end{array}\] such that \(a_{ij} \in \{0,1,2,\ldots,p\} \) and if \(i \leq i^\prime\) and \(j \leq j^\prime\), then \(a_{ij} \leq a_{i^\prime j^\prime}\). Find \(N \pmod{p}\).

Are there positive integers $a; b; c$ and $d$ such that $a^3 + b^3 + c^3 + d^3 =100^{100}$ ? [i](4 points)[/i]

The fraction $\frac{3}{10}$ can be written as a sum of two reciprocals in exactly two ways: \[\frac{3}{10}=\frac{1}{5}+\frac{1}{10}=\frac{1}{4}+\frac{1}{20}\] a) In how many ways can $\frac{3}{2021}$ be written as a sum of two reciprocals? b) Is there a positive integer $n$ not divisible by $3$ with the property that $\frac{3}{n}$ can be written as a sum of two reciprocals in exactly $2021$ ways?

Let $m,n,k$ be positive integers and $1+m+n \sqrt 3=(2+ \sqrt 3)^{2k+1}$. Prove that $m$ is a perfect square.

Rational numbers $a$, $b$, $c$ satisfy the equation \[(a+b+c)(a+b-c)=c^2\,.\] Show that $a+b=c=0$.

If the positive integers $x$ and $y$ are such that $3x + 4y$ and $4x + 3y$ are both perfect squares, prove that both $x$ and $y$ are both divisible with $7$.

Determine all positive integers $k$ and $n$ satisfying the equation $$k^2 - 2016 = 3^n$$ (Stephan Wagner)

Find all pairs $(n,k)$ of positive integers such that $\binom nk=2002$.

Let $p$ be a prime number. Positive integers numbers $a$ and $b$ are such $\frac{p}{a}+\frac{p}{b}=1$ and $a+b$ is divisible by $p$. What values can an expression $\frac{a+b}{p}$ take? [i]Yu.A.Karpenko[/i]

Prove that for coprime each positive integers $a, c$ there is a positive integer $b$ such that $c$ divides $\underbrace{b^{b^{b^{\ldots^b}}}}_\text{b times}-a$

The natural number $n>1$ is called “heavy”, if it is coprime with the sum of its divisors. What’s the maximal number of consecutive “heavy” numbers?

Consider the sequence $u_0, u_1, u_2, ...$ defined by $u_0 = 0, u_1 = 1,$ and $u_n = 6u_{n - 1} + 7u_{n - 2}$ for $n \ge 2$. Show that there are no non-negative integers $a, b, c, n$ such that $$ab(a + b)(a^2 + ab + b^2) = c^{2022} + 42 = u_n.$$

Prove that between every $27$ different positive integers , less than $100$, there exist some two which are[color=red] NOT [/color]relative prime. [u]babis[/u]

Let $a,b,c,d$ be four positive integers such that $a>b>c>d$. Given that $ab+bc+ca+d^2|(a+b)(b+c)(c+a)$. Find the minimal value of $ \Omega (ab+bc+ca+d^2)$. Here $ \Omega(n)$ denotes the number of prime factors $n$ has. e.g. $\Omega(12)=3$

[b]6.1[/b] The capacities of cubic vessels are in the ratio 1:8:27 and the volumes of liquid poured into them are 1: 2: 3. After this, from the first to a certain amount of liquid was poured into the second vessel, and then from the second in the third so that in all three vessels the liquid level became the same. After this, 128 4/7 liters were poured from the first vessel into the second, and from the second in the first back so much that the height of the liquid column in the first vessel became twice as large as in the second. It turned out that in the first vessel there were 100 fewer liters than at first. How much liquid was initially in each vessel? [b]6.2[/b] How many times a day do all three hands on a clock coincide, including the second hand? [b]6.3.[/b] Prove that in Leningrad there are two people who have the same number of familiar Leningraders. [b]6.4 / 7.4[/b] Each of the eight given different natural numbers less than $16$. Prove that among their pairwise differences there is at least at least three are the same. [b]6.5 / 7.6[/b] The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

Let $a, b, c$ be positive integers such that no number divides some other number. If $ab-b+1 \mid abc+1$, prove that $c \geq b$.

$S$ is a subset of $\{1,2, . . . ,100\}$. What is the maximum number of elements in $S$ such that the product of any two of them is not a perfect square?