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 $\textbf{(A) }2\sqrt{2}\qquad \textbf{(B) }\dfrac{16}{9}\sqrt{3}\qquad \textbf{(C) }2\sqrt{3}\qquad \textbf{(D) }4\qquad \textbf{(E) }4\sqrt{2}$

In trapezoid $ABCD$, $AB < CD$, $AB\perp BC$, $AB\parallel CD$, and the diagonals $AC$, $BD$ are perpendicular at point $P$. There is a point $Q$ on ray $CA$ past $A$ such that $QD\perp DC$. If \[\frac{QP} {AP}+\frac{AP} {QP} = \left( \frac{51}{14}\right)^4 - 2,\]then $\frac{BP} {AP}-\frac{AP}{BP}$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$. Compute $m+n$. [i]Ray Li.[/i]

A polynomial $p$ can be written as \begin{align*} p(x) = x^6+3x^5-3x^4+ax^3+bx^2+cx+d. \end{align*} Given that all roots of $p(x)$ are equal to either $m$ or $n$ where $m$ and $n$ are integers, compute $p(2)$.

Find all functions $f,g: \mathbb{Q}\to \mathbb{Q}$ such that the following two conditions hold: $$f(g(x)-g(y))=f(g(x))-y \ \ (1)$$ $$g(f(x)-f(y))=g(f(x))-y\ \ (2)$$ for all $x,y \in \mathbb{Q}$.

Determine all pairs of prime numbers $(p; q)$ such that $p^2 + 5pq + 4q^2$ is the square of an integer.

Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$ [i]Proposed by Christopher Bradley, United Kingdom[/i]

Given is a positive integer $k$. There are $n$ points chosen on a line, such the distance between any two adjacent points is the same. The points are colored in $k$ colors. For each pair of monochromatic points such that there are no points of the same color between them, we record the distance between these two points. If all distances are distinct, find the largest possible $n$.

Let $ ABC $ be an isosceles triangle with $ AC=BC $ . Take points $ D $ on side $AC$ and $E$ on side $BC$ and $ F $ the intersection of bisectors of angles $ DEB $ and $ADE$ such that $ F$ lies on side $AB$. Prove that $F$ is the midpoint of $AB$.