Two circles $ \Omega_{1}$ and $ \Omega_{2}$ are externally tangent to each other at a point $ I$, and both of these circles are tangent to a third circle $ \Omega$ which encloses the two circles $ \Omega_{1}$ and $ \Omega_{2}$. The common tangent to the two circles $ \Omega_{1}$ and $ \Omega_{2}$ at the point $ I$ meets the circle $ \Omega$ at a point $ A$. One common tangent to the circles $ \Omega_{1}$ and $ \Omega_{2}$ which doesn't pass through $ I$ meets the circle $ \Omega$ at the points $ B$ and $ C$ such that the points $ A$ and $ I$ lie on the same side of the line $ BC$. Prove that the point $ I$ is the incenter of triangle $ ABC$. [i]Alternative formulation.[/i] Two circles touch externally at a point $ I$. The two circles lie inside a large circle and both touch it. The chord $ BC$ of the large circle touches both smaller circles (not at $ I$). The common tangent to the two smaller circles at the point $ I$ meets the large circle at a point $ A$, where the points $ A$ and $ I$ are on the same side of the chord $ BC$. Show that the point $ I$ is the incenter of triangle $ ABC$.

On a $2004 \times 2004$ chess table there are 2004 queens such that no two are attacking each other\footnote[1]{two queens attack each other if they lie on the same row, column or direction parallel with on of the main diagonals of the table}. Prove that there exist two queens such that in the rectangle in which the center of the squares on which the queens lie are two opposite corners, has a semiperimeter of 2004.

In an acute-angled triangle $ABC$, the altitudes $AA_1,BB_1,CC_1$ intersect at point $V$. If the triangles $AC_1V, BA_1V, CB_1V$ have the same area, does it follow that the triangle $ABC$ is equilateral?

Let $\triangle ABC$ have $\angle ABC=67^{\circ}$. Point $X$ is chosen such that $AB = XC$, $\angle{XAC}=32^\circ$, and $\angle{XCA}=35^\circ$. Compute $\angle{BAC}$ in degrees. [i]Proposed by Raina Yang[/i]

Let $a, b, c$ be the roots of the equation $x^3-9x^2+11x-1 = 0$, and define $s =\sqrt{a}+\sqrt{b}+\sqrt{c}$. Compute $s^4 -18s^2 - 8s$ .

Let $AB$ be an arbitrary chord of the circle $(O)$. Two circles $(X)$ and $(Y )$ are on the same side of the chord $AB$ such that they are both internally tangent to $(O)$ and they are tangent to $AB$ at $C,D$ respectively, $C$ is between $A$ and $D$. Let $H$ be the intersection of $XY$ and $AB, M$ the midpoint of arc $AB$ not containing $X$ and $Y$ . Let $HM$ meet $(O)$ again at $I$. Let $IX, IY$ intersect $AB$ again at $K, J$. Prove that the circumcircle of triangle $IKJ$ is tangent to $(O)$. Nguyễn Văn Linh

Find all non-negative integers $x, y$ and $z$ satisfying the equation: \[7^{x}+1=3^{y}+5^z\]

The chords $AB$ and $CD$ of a sphere intersect at $X. A, C$ and $X$ are equidistant from a point $Y$ on the sphere. Show that $BD$ and $XY$ are perpendicular.

Do there exist integers $a$ and $b$ such that : (a) the equation $x^2 + ax + b = 0$ has no real roots, and the equation $\lfloor x^2 \rfloor + ax + b = 0$ has at least one real root? [i](2 points)[/i] (b) the equation $x^2 + 2ax + b$ = 0 has no real roots, and the equation $\lfloor x^2 \rfloor + 2ax + b = 0$ has at least one real root? [i]3 points[/i] (By $\lfloor k \rfloor$ we denote the integer part of $k$, that is, the greatest integer not exceeding $k$.) [i]Alexandr Khrabrov[/i]

If $ \log_{k}{x}\cdot \log_{5}{k} \equal{} 3$, then $ x$ equals: $ \textbf{(A)}\ k^6\qquad \textbf{(B)}\ 5k^3\qquad \textbf{(C)}\ k^3\qquad \textbf{(D)}\ 243\qquad \textbf{(E)}\ 125$

For how many integer values of $x$ is $|2x|\leq 7\pi?$ $\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$

On $\triangle ABC$, let $M$ the midpoint of $AB$ and $N$ the midpoint of $CM$. Let $X$ a point such that $\angle XMC=\angle MBC$ and $\angle XCM=\angle MCB$ with $X,B$ in opposite sides of line $CM$. Let $\Omega$ the circumcircle of triangle $\triangle AMX$ [b]a)[/b] Show that $CM$ is tangent to $\Omega$ [b]b)[/b] Show that the lines $NX$ and $AC$ meet at $\Omega$

$x$ and $y$ are integer $5$-digits numbers, such that in the decimal notation, all ten digits are used exactly once. Also $\tan{x}-\tan{y}=1+\tan{x}\tan{y}$, where $x,y$ are angles in degrees. Find maximum of $x$

An infinite sequence $x_1,x_2,\ldots$ of positive integers satisfies \[ x_{n+2}=\gcd(x_{n+1},x_n)+2006 \] for each positive integer $n$. Does there exist such a sequence which contains exactly $10^{2006}$ distinct numbers?

Solve equation $x^4 = 2x^2 + \lfloor x \rfloor$, where $ \lfloor x \rfloor$ is an integral part of $x$.

Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere? $ \textbf{(A)}\; 3+\frac{\sqrt{30}}2\qquad \textbf{(B)}\; 3+\frac{\sqrt{69}}3\qquad \textbf{(C)}\; 3+\frac{\sqrt{123}}4\qquad \textbf{(D)}\; \frac{52}9\qquad \textbf{(E)}\; 3+2\sqrt{2} $

In the triangle $ABC$, we have that $\angle BAC$ is acute. Let $\Gamma$ be the circle that passes through $A$ and is tangent to the side $BC$ at $C$. Let $M$ be the midpoint of $BC$ and let $D$ be the other point of intersection of $\Gamma$ with $AM$. If $BD$ cuts back to$ \Gamma$ at $E$, show that $AC$ is the bisector of $\angle BAE$.

Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.

Let $N$ be a positive integer. Consider a $N \times N$ array of square unit cells. Two corner cells that lie on the same longest diagonal are colored black, and the rest of the array is white. A [i]move[/i] consists of choosing a row or a column and changing the color of every cell in the chosen row or column. What is the minimal number of additional cells that one has to color black such that, after a finite number of moves, a completely black board can be reached?

Let $n\geqslant 3$ be an integer and $a_1,\ldots,a_n$ be nonzero real numbers, with sum $S{}$. Prove that \[\sum_{i=1}^n\left|\frac{S-a_i}{a_i}\right|\geqslant\frac{n-1}{n-2}.\]

For how many permutations $(a_1, a_2, \cdots, a_{2007})$ of the integers from $1$ to $2007$ is there exactly one $i$ between $1$ and $2006$ such that $a_i > a_{i+1}$? Express your answer as $a \** b^c + d \** e^f$ for integers $a$, $b$, $c$, $d$, $e$, and $f$ with $a \nmid b$ and $d \nmid e$.

Prove that for every positive real $a,b,c$ the inequality holds : $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1 \geq \frac{2\sqrt2}{3} (\sqrt{\frac{a+b}{c}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{c+a}{b}})$ When does the equality hold?

A segment $AB$ is given in (Euclidean) plane. Consider all triangles $XYZ$ such, that $X$ is an inner point of $AB$, triangles $XBY$ and $XZA$ are similar (in this order of vertices), and points $A, B, Y, Z$ lie on a circle in this order. Find the locus of midpoints of all such segments $YZ$. (Day 1, 2nd problem authors: Michal Rolínek, Jaroslav Švrček)

A ball is dropped from a height of $3$ meters. On its first bounce it rises to a height of $2$ meters. It keeps falling and bouncing to $\frac{2}{3}$ of the height it reached in the previous bounce. On which bounce will it not rise to a height of $0.5$ meters? $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$