Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.

Nicholas and Peter are dividing $(2n+1)$ nuts. Each wants to get more. Three ways for that were suggested. (Each consist of three stages.) First two stages are common. 1 stage: Peter divides nuts onto $2$ heaps, each contain not less than $2$ nuts. 2 stage: Nicholas divides both heaps onto $2$ heaps, each contain not less than $1$ nut. 3 stage: 1 way: Nicholas takes the biggest and the least heaps. 2 way: Nicholas takes two middle size heaps. 3 way: Nicholas takes either the biggest and the least heaps or two middle size heaps, but gives one nut to the Peter for the right of choice. Find the most and the least profitable method for the Nicholas.

Show that we cannot form more than $4096$ binary sequences of length $24$ so that any two differ in at least $8$ positions.

$2000$ people are standing on a line. Each one of them is either a [i]liar[/i], who will always lie, or a [i]truth-teller[/i], who will always tell the truth. Each one of them says: "there are more liars to my left than truth-tellers to my right". Determine, if possible, how many people from each class are on the line.

Let $x$, $y$, and $z$ be real numbers satisfying the equations \begin{align*} 4x+2yz-6z+9xz^2&=4\\ xyz&=1. \end{align*} Find all possible values of $x+y+z$.

Given that \[a^2+b^2+c^2+d^2+\frac{5}{4}=e+\sqrt{a+b+c+d-e},\] the value of $a+b+c+d+e$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Individual #12[/i]

Let $ABC$ be an acute-angled triangle and $\Gamma$ be a circle with $AB$ as diameter intersecting $BC$ and $CA$ at $F ( \not= B)$ and $E (\not= A)$ respectively. Tangents are drawn at $E$ and $F$ to $\Gamma$ intersect at $P$. Show that the ratio of the circumcentre of triangle $ABC$ to that if $EFP$ is a rational number.

A sequence $ (S_n), n \geq 1$ of sets of natural numbers with $ S_1 = \{1\}, S_2 = \{2\}$ and \[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}. \] Determine $ S_{1024}.$

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

Find a triangle $ABC$ with a point $D$ on side $AB$ such that the measures of $AB, BC, CA$ and $CD$ are all integers and $\frac{AD}{DB}=\frac{9}{7}$, or prove that such a triangle does not exist.

Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation \[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\] holds.

$\omega$ is circumcircle of triangle $ABC$ and $BB_1, CC_1$ are bisectors of ABC. $I$ is center incirle . $B_1 C1$ and $\omega$ intersects at $M$ and $N$ . Find the ratio of circumradius of $ABC$ to circumradius $MIN$.

Fibonacci numbers are defined as follows: $F_0 = F_1 = 1, F_{n+2} = F_{n+1}+F_n, n \geq 0$. Let $a_n$ be the number of words that consist of $n$ letters $0$ or $1$ and contain no two letters $1$ at distance two from each other. Express $a_n$ in terms of Fibonacci numbers.

On the board are written $100$ mutually different positive real numbers, such that for any three different numbers $a, b, c$ is $a^2 + bc$ is an integer. Prove that for any two numbers $x, y$ from the board , number $\frac{x}{y}$ is rational.

Let $f:\mathbb{R}\rightarrow\mathbb{R}, f(x,y)=x^2-2y.$ Define the sequences $(a_n)_{n\geq1}$ and $(b_n)_{n\geq1}$ such that $a_{n+1}=f(a_n,b_n), b_{n+1}=f(b_n,a_n).$ If $4a_1-2b_1=7 :$ a) find the smallest $k\in\mathbb{N}$ for which the number $p=2^k\cdot(2^{512}a_9-b_9)$ is an integer. b) prove that $2^{2^{10}}+2^{2^9}+1$ divides $p.$

For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]