A circle is inscribed in an equilateral triangle. Three nested sequences of circles are then constructed as follows: each circle touches the previous circle and has two edges of the triangle as tangents. This is represented by the figure below. [asy] import olympiad; pair A, B, C; A = dir(90); B = dir(210); C = dir(330); draw(A--B--C--cycle); draw(incircle(A,B,C)); draw(incircle(A,2/3*A+1/3*B,2/3*A+1/3*C)); draw(incircle(A,8/9*A+1/9*B,8/9*A+1/9*C)); draw(incircle(A,26/27*A+1/27*B,26/27*A+1/27*C)); draw(incircle(A,80/81*A+1/81*B,80/81*A+1/81*C)); draw(incircle(A,242/243*A+1/243*B,242/243*A+1/243*C)); draw(incircle(B,2/3*B+1/3*A,2/3*B+1/3*C)); draw(incircle(B,8/9*B+1/9*A,8/9*B+1/9*C)); draw(incircle(B,26/27*B+1/27*A,26/27*B+1/27*C)); draw(incircle(B,80/81*B+1/81*A,80/81*B+1/81*C)); draw(incircle(B,242/243*B+1/243*A,242/243*B+1/243*C)); draw(incircle(C,2/3*C+1/3*B,2/3*C+1/3*A)); draw(incircle(C,8/9*C+1/9*B,8/9*C+1/9*A)); draw(incircle(C,26/27*C+1/27*B,26/27*C+1/27*A)); draw(incircle(C,80/81*C+1/81*B,80/81*C+1/81*A)); draw(incircle(C,242/243*C+1/243*B,242/243*C+1/243*A)); [/asy] What is the ratio of the area of the largest circle to the combined area of all the other circles? $\text{(A) }\frac{8}{1}\qquad\text{(B) }\frac{8}{3}\qquad\text{(C) }\frac{9}{1}\qquad\text{(D) }\frac{9}{3}\qquad\text{(E) }\frac{10}{3}$

A right hexagonal prism has height $2$. The bases are regular hexagons with side length $1$. Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).

Given is a flat plane $V$ containing a rectangular coordinate system $xOy$. We consider quartets of numbers $(p,q,r,s)$; $p\le 0$, $q \le 0$, $r \le 0$, $s \le 0$. On every quartet we add a point $S$ from $V$ in a way that is in the accompanying figure is displayed. In this figure $OP = p$,$PQ = q$,$QR = r$,$RS = s$, $\angle OPQ = \angle PQR = \angle QRS = 135^o$. (a) What is the set of the points of $V$, which are added to these quartets ? (b) Which of these points has been added to only one quartet? How many quartets have the other points been added? (c) What is the set of points added to the quartets for which $p + q = 1$ and $r = s = 0$? (d) What is the set of points added to the quartets for which $p + 1 = $ and $r + s = 1$? [asy] unitsize(0.6 cm); pair O, P, Q, R, S; O = (0,0); P = (2,0); Q = P + 2*dir(45); R = Q + (0,2.5); S = R + 3*dir(135); draw((-1,0)--(7,0)); draw((0,-1)--(0,8)); draw(P--Q--R--S); label("$O$", O, SW); label("$P$", P, dir(270)); label("$Q$", Q, E); label("$R$", R, E); label("$S$", S, N); label("$X$", (7,0), E); label("$Y$", (0,8), N); [/asy]

A square is divided by horizontal and vertical lines that form $n^2$ squares each with side $1$. What is the greatest possible value of $n$ such that it is possible to select $n$ squares such that any rectangle with area $n$ formed by the horizontal and vertical lines would contain at least one of the selected $n$ squares.

Triangle $ABC$ has sidelengths $AB=14,AC=13,$ and $BC=15.$ Point $D$ is chosen in the interior of $\overline{AB}$ and point $E$ is selected uniformly at random from $\overline{AD}.$ Point $F$ is then defined to be the intersection point of the perpendicular to $\overline{AB}$ at $E$ and the union of segments $\overline{AC}$ and $\overline{BC}.$ Suppose that $D$ is chosen such that the expected value of the length of $\overline{EF}$ is maximized. Find $AD.$

For every $n\geq 1$ define $x_n$ by $$x_n=\int_0^1 \ln(1+x+x^2+\dots +x^n)\cdot\ln\frac{1}{1-x}\mathrm dx.$$ a) Show that $x_n$ is finite for every $n\geq 1$ and $\lim_{n\rightarrow\infty}x_n=2$. b) Calculate $\lim_{n\rightarrow\infty}\frac{n}{\ln n}(2-x_n)$.

Let $a,b,c$ be positive real numbers such that $a+b+c=2005$. Find the minimum value of the expression: $$E=a^{2006}+b^{2006}+c^{2006}+\frac{(ab)^{2004}+(bc)^{2004}+(ca)^{2004}}{(abc)^{2004}}$$

Let $g: R \to R$ be a linear function such that $g (1) = 0$. If $f: R \to R$ is a quadratic function such what $g (x^2) = f (x)$ and $f (x + 1) - f (x - 1) = x$ for all $x \in R$. Determine the value of $f (2019)$.

Give an acute-angled triangle $ABC$ with area $S$, let points $A',B',C'$ be located as follows: $A'$ is the point where altitude from $A$ on $BC$ meets the outwards facing semicirle drawn on $BX$ as diameter.Points $B',C'$ are located similarly. Evaluate the sum $T=($area $\vartriangle BCA')^2+($area $\vartriangle CAB')^2+($area $\vartriangle ABC')^2$.

Consider an acute triangle $ABC$ with $|AB| > |CA| > |BC|$. The vertices $D, E$, and $F$ are the base points of the altitudes from $A, B$, and $C$, respectively. The line through F parallel to $DE$ intersects $BC$ in $M$. The angular bisector of $\angle MF E$ intersects $DE$ in $N$. Prove that $F$ is the circumcentre of $\vartriangle DMN$ if and only if $B$ is the circumcentre of $\vartriangle FMN$.

Find the smallest positive root of the equation $$\{tg x\}=\sin x. $$ ($\{a\}$ is the fractional part of $a$, $\{a\}$ is equal to the difference between $ a$ and the largest integer not exceeding $a$.)

How many non-similar triangle have angles whose degree measures are distinct positive integers in arithmetic progression? $ \textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 59 \qquad \textbf{(D) } 89 \qquad \textbf{(E) } 178$

Let $(a_n)_{n\geq 0}$ be a sequence defined by $a_0\geq 0$ and the recurrence relation $$a_{n+1}=\frac{a_n^2-1}{n+1},$$ for all $n\geq 0$. Prove that here exists a real number $a> 0$ such that: [list] [*] if $a_0\geq a,$ $\lim_{n\rightarrow\infty}a_n = \infty$; [*] if $a_0\in [0,a),$ $\lim_{n\rightarrow\infty}a_n = 0$.

In a tournament, 2011 players meet 2011 times to play a multiplayer game. Every game is played by all 2011 players together and ends with each of the players either winning or losing. The standings are kept in two $2011\times 2011$ matrices, $T=(T_{hk})$ and $W=(W_{hk}).$ Initially, $T=W=0.$ After every game, for every $(h,k)$ (including for $h=k),$ if players $h$ and $k$ tied (that is, both won or both lost), the entry $T_{hk}$ is increased by $1,$ while if player $h$ won and player $k$ lost, the entry $W_{hk}$ is increased by $1$ and $W_{kh}$ is decreased by $1.$ Prove that at the end of the tournament, $\det(T+iW)$ is a non-negative integer divisible by $2^{2010}.$

Given a triangle $ ABC$ with angle $ C \geq 60^{\circ}$. Prove that: $ \left(a \plus{} b\right) \cdot \left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \right) \geq 4 \plus{} \frac {1}{\sin\left(\frac {C}{2}\right)}.$

Suppose that $x^3-x+10^{-6}=0$. Suppose that $x_1<x_2<x_3$ are the solutions for $x$. Find the integers $(a,b,c)$ closest to $10^8x_1$, $10^8x_2$, and $10^8x_3$ respectively.

Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relation $u_{n+1}-u_n=3+4(n-1)$, $n=1,2,3,\cdots$. If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is: $\textbf{(A) }3\qquad \textbf{(B) }4\qquad \textbf{(C) }5\qquad \textbf{(D) }6\qquad \textbf{(E) }11$

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $x,y>0$ and $xy\ge1.$ Prove that $x^3+y^3+4xy\ge x^2+y^2+x+y+2.$ Let $x,y>0$ and $xy\ge1.$ Prove that $2(x^3+y^3+xy+x+y)\ge5(x^2+y^2).$

Sofía has many boxes where she keeps candies. Every morning, she chooses two of these boxes and places one candy in each. However, each night, a thief selects one box and steals all the candies inside it. Sofía dreams of waking up one day and finding a box with 2024 candies. Prove that Sofía can always fulfill her dream if she has enough boxes.

The diagram below is an example of a ${\textit{rectangle tiled by squares}}$: [center][img]http://i.imgur.com/XCPQJgk.png[/img][/center] Each square has been labeled with its side length. The squares fill the rectangle without overlapping. In a similar way, a rectangle can be tiled by nine squares whose side lengths are $2,5,7,9,16,25,28,33$, and $36$. Sketch one such possible arrangement of those squares. They must fill the rectangle without overlapping. Label each square in your sketch by its side length as in the picture above.

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

Let $ f(x)\equal{}x^n\plus{}a_{n\minus{}1} x^{n\minus{}1}\plus{}...\plus{}a_0$ $ (n \ge 1)$ be a polynomial with real coefficients such that $ |f(0)|\equal{}f(1)$ and each root $ \alpha$ of $ f$ is real and lies in the interval $ [0,1]$. Prove that the product of the roots does not exceed $ \frac{1}{2^n}$.

A piece of cheese is located at $ (12,10)$ in a coordinate plane. A mouse is at $ (4, \minus{} 2)$ and is running up the line $ y \equal{} \minus{} 5x \plus{} 18.$ At the point $ (a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $ a \plus{} b?$ $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 22$

Suppose that the sequence $\{a_n\}$ of positive reals satisfies the following conditions: [list] [*]$a_i \leq a_j$ for every positive integers $i <j$. [*]For any positive integer $k \geq 3$, the following inequality holds: $$(a_1+a_2)(a_2+a_3)\cdots(a_{k-1}+a_k)(a_k+a_1)\leq (2^k+2022)a_1a_2\cdots a_k$$ [/list] Prove that $\{a_n\}$ is constant.

Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB=11$ and $CD=19$. Point $P$ is on segment $AB$ with $AP=6$, and $Q$ is on segment $CD$ with $CQ=7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ=27$, find $XY$.