Take an arbitrary point $D$ on side $BC$ of triangle $ABC$ and draw a circle through point $D$ and the centers of the circles inscribed in triangles $ABD$ and $ACD$. Prove that all circles obtained for different points $D$ of side $BC$ have a common point.

Suppose $P(x)$ is a cubic polynomial with integer coefficients such $P(\sqrt{5})=5$ and $P(\sqrt[3]{5})=5\sqrt[3]{5}$.

We divide up the plane into disjoint regions using a circle, a rectangle and a triangle. What is the greatest number of regions that we can get?

What conditions should real numbers $ a $, $ b $, $ c $, $ d $, $ e $, $ f $ meet in order for a polynomial of second degree $$ax^2 + 2bxy + cy^2 + 2dx + 2ey + f$$ was the product of two first degree polynomials with real coefficients ?

There are $n$ people arranged in a circle, and $n^{n^n}$ coins are distributed among them, where each person has at least $n^n$ coins. Each person is then assigned a random index number in $\{1,2,...n\}$ such that no two people have the same number. Then every minute, if $i$ is the number of minutes passed, the person with index number congruent to $i$ mod $n$ will give a coin to the person on his left or right. After some time, everyone has the same number of coins. For what $n$ is this always possible, regardless of the original distribution of coins and index numbers? [i](Proposed by Ho Janson)[/i]

Let $ABC$ be a triangle. Four distinct points $D,A,B,E$ lie on the line $AB$ in this order such that $DA=AB=BE.$ Find necessary and sufficient condition for lengths $a=BC,b=AC$ such that the angle $\angle DCE$ is right.

Let $ f \ne 0$ be a polynomial with real coefficients. Define the sequence $ f_{0}, f_{1}, f_{2}, \ldots$ of polynomials by $ f_{0}= f$ and $ f_{n+1}= f_{n}+f_{n}'$ for every $ n \ge 0$. Prove that there exists a number $ N$ such that for every $ n \ge N$, all roots of $ f_{n}$ are real.

Find $x$ such that \[2010^{\log_{10}x}=11^{\log_{10}(1+3+5+\cdots +4019).}\]

Two vector fields $\mathbf{F},\mathbf{G}$ are defined on a three dimensional region $W=\{(x,y,z)\in\mathbb{R}^3 : x^2+y^2\leq 1, |z|\leq 1\}$. $$\mathbf{F}(x,y,z) = (\sin xy, \sin yz, 0),\quad \mathbf{G} (x,y,z) = (e^{x^2+y^2+z^2}, \cos xz, 0)$$ Evaluate the following integral. \[\iiint_{W} (\mathbf{G}\cdot \text{curl}(\mathbf{F}) - \mathbf{F}\cdot \text{curl}(\mathbf{G})) dV\]

An equilateral triangle with side 101 is placed on a plane so that one of its sides is horizontal and the triangle is above it. It is divided into smaller equilateral triangles with side 1 by segments parallel to its sides. All sides of these smaller triangles are colored red (including the entire border of the large triangle). An equilateral triangle on a plane is called a "mirror" triangle if its sides are parallel to the sides of the original triangle, but it lies below its horizontal side. What is the smallest number of contours of mirror triangles needed to cover all the red segments? (Mirror triangles may overlap and extend beyond the original triangle.) [i]Proposed by Dmitry Shiryayev[/i]

Sigfried is singing the ABC’s $100$ times straight, for some reason. It takes him $20$ seconds to sing the ABC’s once, and he takes a $5$ second break in between songs. Normally, he sings the ABC’s without messing up, but he gets fatigued when singing correctly repeatedly. For any song, if he sung the previous three songs without messing up, he has a $\frac12$ chance of messing up and taking $30$ seconds for the song instead. What is the expected number of minutes it takes for Sigfried to sing the ABC’s $100$ times? Round your answer to the nearest minute.

Inside the circle $ S $ there is a circle $ T $ and circles $ K_1, K_2, \ldots, K_n $ tangent externally to $ T $ and internally to $ S $, and the circle $ K_1 $ is tangent to $ K_2 $, $ K_2 $ tangent to $ K_3 $ etc. Prove that the points of tangency of the circles $ K_1 $ with $ K_2 $, $ K_2 $ with $ K_3 $ etc. lie on the circle.

Let $ABC$ be a triangle, $I$ its incenter, and $\omega$ a circle of center $I$. Points $A',B', C'$ are on $\omega$ such that rays $IA', IB', IC',$ starting from $I$ intersect perpendicularly sides $BC, CA, AB$, respectively. Prove that lines $AA', BB', CC'$ are concurrent.

Is it possible for some natural number $k$ to divide all natural numbers from $1$ to $k$ into two groups and write down the numbers in each group in a row in some order so that you get two the same numbers? [hide=original wording beacuse it doesn't make much sense]Можно ли при каком-то натуральном k разбить все натуральные числа от 1 до k на две группы и выписать числа в каждой группе подряд в некотором порядке так, чтобы получились два одинаковых числа?[/hide]

Assume that the system of differential equations $y'=-z^3$, $z'=y^3$ with the initial conditions $y(0)=1$, $z(0)=0$ has a unique solution $y=f(x)$, $z=g(x)$ defined for real $x$. Prove that there exists a positive constant $L$ such that for all real $x$, $$f(x+L)=f(x),\enspace g(x+L)=g(x).$$

What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \frac{1}{2}$? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{5}{2}\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 20$

A cube-shaped container has vertices $A$, $B$, $C$, and $D$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of the faces of the cube. Vertex $A$ of the cube is set on a horizontal plane $\mathcal P$ so that the plane of the rectangle $ABCD$ is perpendicular to $\mathcal P$, vertex $B$ is $2$ meters above $\mathcal P$, vertex $C$ is $8$ meters above $\mathcal P$, and vertex $D$ is $10$ meters above $\mathcal P$. The cube contains water whose surface is $7$ meters above $\mathcal P$. The volume of the water is $\tfrac mn$ cubic meters, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(250); defaultpen(linewidth(0.6)); pair A = origin, B = (6,3), X = rotate(40)*B, Y = rotate(70)*X, C = X+Y, Z = X+B, D = B+C, W = B+Y; pair P1 = 0.8*C+0.2*Y, P2 = 2/3*C+1/3*X, P3 = 0.2*D+0.8*Z, P4 = 0.63*D+0.37*W; pair E = (-20,6), F = (-6,-5), G = (18,-2), H = (9,8); filldraw(E--F--G--H--cycle,rgb(0.98,0.98,0.2)); fill(A--Y--P1--P4--P3--Z--B--cycle,rgb(0.35,0.7,0.9)); draw(A--B--Z--X--A--Y--C--X^^C--D--Z); draw(P1--P2--P3--P4--cycle^^D--P4); dot("$A$",A,S); dot("$B$",B,S); dot("$C$",C,N); dot("$D$",D,N); label("$\mathcal P$",(-13,4.5)); [/asy]

Quadrilateral $ABCD$ is circumscribed around circle $k$. Gind the smallest possible value of $$\frac{AB + BC + CD + DA}{AC + BD}$$, as well as all quadrilaterals with the above property where it is reached.

Four small spheres each with radius $6$ are each internally tangent to a large sphere with radius $17$. The four small spheres form a ring with each of the four spheres externally tangent to its two neighboring small spheres. A sixth intermediately sized sphere is internally tangent to the large sphere and externally tangent to each of the four small spheres. Its radius is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/7/2/25955cd6f22bc85f2f3c5ba8cd1ee0821c9d50.png[/img]

A point $P$ is outside of a circle and the distance to the center is $13$. A secant line from $P$ meets the circle at $Q$ and $R$ so that the exterior segment of the secant, $PQ$, is $9$ and $QR$ is $7$. Find the radius of the circle.

A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only $ 1/3$ of the marbles in the bag are blue. Then yellow marbles are added to the bag until only $ 1/5$ of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is doubled. What fraction of the marbles now in the bag are blue? $ \textbf{(A)}\ \frac {1}{5}\qquad \textbf{(B)}\ \frac {1}{4}\qquad \textbf{(C)}\ \frac {1}{3}\qquad \textbf{(D)}\ \frac {2}{5}\qquad \textbf{(E)}\ \frac {1}{2}$

Let $p$ a prime number and $r$ an integer such that $p|r^7-1$. Prove that if there exist integers $a, b$ such that $p|r+1-a^2$ and $p|r^2+1-b^2$, then there exist an integer $c$ such that $p|r^3+1-c^2$.

Let $\omega$ and $\Gamma$ be circles such that $\omega$ is internally tangent to $\Gamma$ at a point $P$. Let $AB$ be a chord of $\Gamma$ tangent to $\omega$ at a point $Q$. Let $R\neq P$ be the second intersection of line $PQ$ with $\Gamma$. If the radius of $\Gamma$ is $17$, the radius of $\omega$ is $7$, and $\frac{AQ}{BQ}=3$, find the circumradius of triangle $AQR$.

Let $A=(a_{ij})$ be an $m\times n$ real matrix with at least one non-zero element. For each $i\in\{1,\ldots,m\}$, let $R_i=\sum_{j=1}^na_{ij}$ be the sum of the $i$-th row of the matrix $A$, and for each $j\in\{1,\ldots,n\}$, let $C_j =\sum_{i=1}^ma_{ij}$ be the sum of the $j$-th column of the matrix $A$. Prove that there exist indices $k\in\{1,\ldots,m\}$ and $l\in\{1,\ldots,n\}$ such that $$a_{kl}>0,\qquad R_k\ge0,\qquad C_l\ge0,$$or $$a_{kl}<0,\qquad R_k\le0,\qquad C_l\le0.$$

In a country, there are $28$ cities and between some cities there are two-way flights. In every city there is exactly one airport and this airport is either small or medium or big. For every route which contains more than two cities, doesn't contain a city twice and ends where it begins; has all types of airports. What is the maximum number of flights in this country?