A traveller visited a village whose inhabitants either always tell the truth or always lie. The villagers stood in a circle facing the centre of the circle, and each villager announced whether the person standing to his right is a truth-teller. On the basis of this information, the traveller was able to determine what fraction of the villagers were liars. What was this fraction? (B, Frenkin)

The [i]Olimpia[/i] country is formed by $n$ islands. The most populated one is called [i]Panacenter[/i], and every island has a different number of inhabitants. We want to build bridges between these islands, which we'll be able to travel in both directions, under the following conditions: a) No pair of islands is joined by more than one bridge. b) Using the bridges we can reach every island from Panacenter. c) If we want to travel from Panacenter to every other island, in such a way that we use each bridge at most once, the number of inhabitants of the islands we visit is strictly decreasing. Determine the number of ways we can build the bridges.

Two persons play the following game with integers. The initial number is $2011^{2011}$. The players move in turns. Each move consists of subtraction of an integer between $1$ and $2010$ inclusive, or division by $2011$, rounding down to the closest integer when necessary. The player who first obtains a non-positive integer wins. Which player has a winning strategy?

Prove that one can partition the set of natural numbers into $100$ nonempty subsets such that among any three natural numbers $a$, $b$, $c$ satisfying $a+99b=c$, there are two that belong to the same subset.

For a sequence $x_1,x_2,\ldots,x_n$ of real numbers, we define its $\textit{price}$ as \[\max_{1\le i\le n}|x_1+\cdots +x_i|.\] Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_1$ such that $|x_1 |$ is as small as possible; among the remaining numbers, he chooses $x_2$ such that $|x_1 + x_2 |$ is as small as possible, and so on. Thus, in the $i$-th step he chooses $x_i$ among the remaining numbers so as to minimise the value of $|x_1 + x_2 + \cdots x_i |$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$. Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G\le cD$. [i]Proposed by Georgia[/i]

In isosceles triangle $ABC(AC=BC)$ the point $M$ is in the segment $AB$ such that $AM=2MB,$ $F$ is the midpoint of $BC$ and $H$ is the orthogonal projection of $M$ in $AF.$ Prove that $\angle BHF=\angle ABC.$

Prove that there are infinitely many pairs $(k,N)$ of positive integers such that $1 + 2 + \cdots + k = (k + 1) + (k + 2)+\cdots + N.$

Exactly $102$ country leaders arrived at the IMO. At the final session, the IMO chairperson wants to introduce some changes to the regulations, which the leaders must approve. To pass the changes, the chairperson must gather at least \(\frac{2}{3}\) of the votes "FOR" out of the total number of leaders. Some leaders do not attend such meetings, and it is known that there will be exactly $81$ leaders present. The chairperson must seat them in a square-shaped conference hall of size \(9 \times 9\), where each leader will be seated in a designated \(1 \times 1\) cell. It is known that exactly $28$ of these $81$ leaders will surely support the chairperson, i.e., they will always vote "FOR." All others will vote as follows: At the last second of voting, they will look at how their neighbors voted up to that moment — neighbors are defined as leaders seated in adjacent cells \(1 \times 1\) (sharing a side). If the majority of neighbors voted "FOR," they will also vote "FOR." If there is no such majority, they will vote "AGAINST." For example, a leader seated in a corner of the hall has exactly $2$ neighbors and will vote "FOR" only if both of their neighbors voted "FOR." (a) Can the IMO chairperson arrange their $28$ supporters so that they vote "FOR" in the first second of voting and thereby secure a "FOR" vote from at least \(\frac{2}{3}\) of all $102$ leaders? (b) What is the maximum number of "FOR" votes the chairperson can obtain by seating their 28 supporters appropriately? [i]Proposed by Bogdan Rublov[/i]

Three circles of radius $ s$ are drawn in the first quadrant of the $ xy$-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the $ x$-axis, and the third is tangent to the first circle and the $ y$-axis. A circle of radius $ r > s$ is tangent to both axes and to the second and third circles. What is $ r/s$? [asy]unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=3; pair O0=(9,9), O1=(1,1), O2=(3,1), O3=(1,3); pair P0=O0+9*dir(-45), P3=O3+dir(70); pair[] ps={O0,O1,O2,O3}; dot(ps); draw(Circle(O0,9)); draw(Circle(O1,1)); draw(Circle(O2,1)); draw(Circle(O3,1)); draw(O0--P0,linetype("3 3")); draw(O3--P3,linetype("2 2")); draw((0,0)--(18,0)); draw((0,0)--(0,18)); label("$r$",midpoint(O0--P0),NE); label("$s$",(-1.5,4)); draw((-1,4)--midpoint(O3--P3));[/asy]$ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

The sequence $a_0,a_1,a_2,\dots$ is recursively defined by \[ a_0 = 1 \quad \text{and} \quad a_n = a_{n-1} \cdot \left(4-\frac{2}{n} \right) \quad \text{for } n \geq 1. \] Prove for each integer $n \geq 1$: (a) The number $a_n$ is a positive integer. (b) Each prime $p$ with $n < p \leq 2n$ is a divisor of $a_n$. (c) If $n$ is a prime, then $a_n-2$ is divisible by $n$.

Let $n \ge 2$ be an integer and let $a_1, a_2, \cdots a_n$ be fixed positive integers (not necessarily all distinct) in such a way that $\gcd(a_1, a_2 \cdots a_n)=1$. In a board the numbers $a_1, a_2 \cdots a_n$ are all written along with a positive integer $x$. A move consists of choosing two numbers $a>b$ from the $n+1$ numbers in the board and replace them with $a-b,2b$. Find all possible values of $x$, with respect of the values of $a_1, a_2 \cdots a_n$, for which it is possible to achieve a finite sequence of moves (possibly none) such that eventually all numbers written in the board are equal.

Let $A$ and $B$ be arbitrary finite sets and let $f: A\longrightarrow B$ and $g: B\longrightarrow A$ be functions such that $g$ is not onto. Prove that there is a subset $S$ of $A$ such that $\frac{A}{S}=g(\frac{B}{f(S)})$.

Find all pairs of coprime positive integers $(p,q)$ such that $p^2+2q^2+334=[p^2,q^2]$ where $[p^2,q^2]$ is the leact common multiple of $p^2,q^2$ .

For all positive integers $n,$ let $r_n$ be defined as \[r_n=\sum_{i=0}^n(-1)^i\binom{n}{i}\frac{1}{(i+1)!}.\]Prove that $\sum_{r=1}^\infty r_i=0.$

How many positive integers $N$ satisfy all of the following three conditions?\\ (i) $N$ is divisible by $2020$.\\ (ii) $N$ has at most $2020$ decimal digits.\\ (iii) The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros.

Let $\mathbb{R}^{\ge 0}$ be the set of all nonnegative real numbers. Find all functions $f:\mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that $$ x+2 \max\{y,f(x),f(z)\} \ge f(f(x))+2 \max\{z,f(y)\}$$ for all nonnegative real numbers $x,y$ and $z$.

An equilateral triangle with side length 2023 has area $A$ and a regular hexagon with side length 289 has area $B$. If $\frac{A}{B}$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime, find $m+n$. [i]Proposed by Andy Xu[/i]

What is the remainder when $201820182018... $ [$2018$ times] is divided by $15$?

For every positive integer $m$ and for all non-negative real numbers $x,y,z$ denote \begin{align*} K_m =x(x-y)^m (x-z)^m + y (y-x)^m (y-z)^m + z(z-x)^m (z-y)^m \end{align*} [list=a] [*] Prove that $K_m \geq 0$ for every odd positive integer $m$ [*] Let $M$ $= \prod_{cyc} (x-y)^2$. Prove, $K_7+M^2 K_1 \geq M K_4$

Let $b\geq 2$ be an integer, and let $s_b(n)$ denote the sum of the digits of $n$ when it is written in base $b$. Show that there are infinitely many positive integers that cannot be represented in the form $n+s_b(n)$, where $n$ is a positive integer.

Find all real numbers $a,b$ satisfying the following system of equations \begin{align*} 2a^2 -2ab+b^2 &=a\\ 4a^2 -5ab +2b^2 & =b. \end{align*}

Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^o,\angle B = 105^o,\angle C = 90^o$ and $AB = 2,BC = CD = DE =\sqrt2$. If the length of $AE$ is $\sqrt{a }- b$ where $a, b$ are integers, what is the value of $a + b$?

Let $a$ and $b$ be prime numbers such that $a+b = 10000$. Find the sum of the smallest possible value of $a$ and the largest possible value of $a$.

Show that if a circumscribable quadrilateral of sides $a,b,c,d$ has area $A= \sqrt{abcd},$ then it is also inscribable.

A triangle with integral sides has perimeter $8$. The area of the triangle is $\textbf{(A) }2\sqrt{2}\qquad \textbf{(B) }\dfrac{16}{9}\sqrt{3}\qquad \textbf{(C) }2\sqrt{3}\qquad \textbf{(D) }4\qquad \textbf{(E) }4\sqrt{2}$