Let $AD$ be the bisector of angle $A$ of the triangle $ABC$. One considers the points M, N on the half-lines $(AB$ and $(AC$, respectively, such that $\angle MDA = \angle B$ and $\angle NDA = \angle C$. Let $AD \cap MN=\{P\}$. Prove that: $$AD^3 = AB \cdot AC\cdot AP$$

Let be two natural numbers $ m,n, $ and $ m $ pairwise disjoint sets of natural numbers $ A_0,A_1,\ldots ,A_{m-1}, $ each having $ n $ elements, such that no element of $ A_{i\pmod m} $ is divisible by an element of $ A_{i+1\pmod m} , $ for any natural number $ i. $ Determine the number of ordered pairs $$ (a,b)\in\bigcup_{0\le j < m} A_j\times\bigcup_{0\le j < m} A_j $$ such that $ a|b $ and such that $ \{ a,b \}\not\in A_k, $ for any $ k\in\{ 0,1,\ldots ,m-1 \} . $ [i]Radu Bumbăcea[/i]

Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$) intersect at point $P$. Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1$ on $CD$, the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent. [i]Proposed by Eugene Bilopitov, Ukraine[/i]

Triangle $ABC$ is so that $AB = 15$, $BC = 22$, and $AC = 20$. Let $D, E, F$ lie on $BC$, $AC$, and $AB$, respectively, so $AD$, $BE$, $CF$ all contain a point $K$. Let $L$ be the second intersection of the circumcircles of $BFK$ and $CEK$. Suppose that $\frac{AK}{KD} = \frac{11}{7}$ , and $BD = 6$. If $KL^2 =\frac{a}{b}$, where $a, b$ are relatively prime integers, find $a + b$.

Alex starts at the origin $O$ of a hexagonal lattice. Every second, he moves to one of the six vertices adjacent to the vertex he is currently at. If he ends up at $X$ after $2018$ moves, then let $p$ be the probability that the shortest walk from $O$ to $X$ (where a valid move is from a vertex to an adjacent vertex) has length $2018$. Then $p$ can be expressed as $\tfrac{a^m-b}{c^n}$, where $a$, $b$, and $c$ are positive integers less than $10$; $a$ and $c$ are not perfect squares; and $m$ and $n$ are positive integers less than $10000$. Find $a+b+c+m+n$.

At the base of the pyramid $SABCD$ lies a convex quadrilateral $ABCD$, such that $BC * AD = BD * AC$. Also $ \angle ADS =\angle BDS ,\angle ACS =\angle BCS$. Prove that the plane $SAB$ is perpendicular to the plane of the base.

Let $3(2^n -1)$ points be selected in the interior of a polyhedron $P$ with volume $2^n$, where n is a positive integer. Prove that there exists a convex polyhedron $U$ with volume $1$, contained entirely inside $P$, which contains none of the selected points.

Prove that if the function $ f : \mathbb{R}^2 \rightarrow [0,1]$ is continuous and its average on every circle of radius $ 1$ equals the function value at the center of the circle, then $ f$ is constant. [i]V. Totik[/i]

By a $\emph{tile}$ we mean a polyomino (i.e. a finite edge-connected set of cells in the infinite grid). There are many ways to place a tile in the infinite table (rotation is allowed but we cannot flip the tile). We call a tile $\textbf{T}$ special if we can place a permutation of the positive integers on all cells of the infinite table in such a way that each number would be maximum between all the numbers that tile covers in at most one placement of the tile. 1. Prove that each square is a special tile. 2. Prove that each non-square rectangle is not a special tile. 3. Prove that tile $\textbf{T}$ is special if and only if it looks the same after $90^\circ$ rotation.

Source: 1976 Euclid Part A Problem 2 ----- The sum of the series $2+5+8+11+14+...+50$ equals $\textbf{(A) } 90 \qquad \textbf{(B) } 425 \qquad \textbf{(C) } 416 \qquad \textbf{(D) } 442 \qquad \textbf{(E) } 495$

A truck is initially moving at velocity $v$. The driver presses the brake in order to slow the truck to a stop. The brake applies a constant force $F$ to the truck. The truck rolls a distance $x$ before coming to a stop, and the time it takes to stop is $t$. Which of the following expressions is equal the initial kinetic energy of the truck (i.e. the kinetic energy before the driver starts braking)? $\textbf{(A) } Fx\\ \textbf{(B) } Fvt\\ \textbf{(C) } Fxt\\ \textbf{(D) } Ft\\ \textbf{(E) } \text{Both (a) and (b) are correct}$

Let $H$ be a rectangle with angle between two diagonal $\leq 45^{0}$. Rotation $H$ around the its center with angle $0^{0}\leq x\leq 360^{0}$ we have rectangle $H_{x}$. Find $x$ such that $[H\cap H_{x}]$ minimum, where $[S]$ is area of $S$.

For what values of $n$ is it possible to paint the edges of a prism whose base is an $n$-gon so that there are edges of all three colours at each vertex and all the faces (including the upper and lower bases) have edges of all three colours? (AV Shapovelov)

Let $P$ be a point inside the square $ABCD$ such that $\angle PAC = \angle PCD = 17^o$ (see Figure 1). Calculate $\angle APB$? [img]https://cdn.artofproblemsolving.com/attachments/d/0/0b20ebee1fe28e9c5450d04685ac8537acda07.png[/img]

Solve the following system of equations in real numbers: \[ \begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x \\ \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y \\ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}. \]

As shown in the figure, $AB$ is the diameter of circle $\odot O$, and chords $AC$ and $BD$ intersect at point $E$, $EF\perp AB$ intersects at point $F$, and $FC$ intersects $BD$ at point $G$. Point $M$ lies on $AB$ such that $MD=MG$ . Prove that points $F$, $M$, $D$, $G$ lies on a circle. [img]https://cdn.artofproblemsolving.com/attachments/2/3/614ef5b9e8c8b16a29b8b960290ef9d7297529.jpg[/img]

Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?

Let ABCD be a convex quadrilateral, and let M and N be the midpoints of its sides AD and BC, respectively. Assume that the points A, B, M, N are concyclic, and the circumcircle of triangle BMC touches the line AB. Show that the circumcircle of triangle AND touches the line AB, too. Darij

[b]7.[/b] Prove that any real number x satysfying the inequalities $0<x\leq 1$ can be represented in the form $x= \sum_{k=1}^{\infty}\frac{1}{n_k}$ where $(n_k)_{k=1}^{\infty}$ is a sequence of positive integers such that $\frac{n_{k+1}}{n_k}$ assumes, for each $k$, one of the three values $2,3$ or $4$. [b](N. 14)[/b]

Prove that among arbitrary $13$ points in plane with coordinates as integers, such that no three are collinear, we can pick three points as vertices of triangle such that its centroid has coordinates as integers.

Let $ ABCD$ is a tetrahedron. Denote by $ A'$, $ B'$ the feet of the perpendiculars from $ A$ and $ B$, respectively to the opposite faces. Show that $ AA'$ and $ BB'$ intersect if and only if $ AB$ is perpendicular to $ CD$. Do they intersect if $ AC \equal{} AD \equal{} BC \equal{} BD$?

A positive integer $n$ is called [i]good [/i] if they sum up in one and only one way at least of two positive integers whose product also has the value $n$. Here representations that differ only in the order of the summands are considered the same viewed. Find all good positive integers.

Consider a circle $ \Gamma(O,R)$ and a point $ P$ found in the interior of this circle. Consider a chord $ AB$ of $ \Gamma$ that passes through $ P$. Suppose that the tangents to $ \Gamma$ at the points $ A$ and $ B$ intersect at $ Q$. Let $ M\in QA$ and $ N\in QB$ s.t. $ PM\perp QA$ and $ PN\perp QB$. Prove that the value of $ \frac {1}{PN} \plus{} \frac {1}{PM}$ doesn't depend of choosing the chord $ AB$.

Outside the triangle $ABC$ the isosceles triangles $AFB, BDC$ and $CEA$ with the bases $AB, BC$ and $CA$ respectively, are constructed. Show that the perpendiculars form $A, B$ and $C$ on $(EF), (FD)$ and $(DE)$, respectively, are concurrent.

Triangle $\triangle ABC$ in the figure has area $10$. Points $D$, $E$ and $F$, all distinct from $A$, $B$ and $C$, are on sides $AB$, $BC$ and $CA$ respectively, and $AD = 2$, $DB = 3$. If triangle $\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then that area is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(10,0), C=(8,7), F=7*dir(A--C), E=(10,0)+4*dir(B--C), D=4*dir(A--B); draw(A--B--C--A--E--F--D); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$2$", (2,0), S); label("$3$", (7,0), S);[/asy] $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ \frac{5}{3}\sqrt{10}\qquad\textbf{(E)}\ \text{not uniquely determined}$