Let $\triangle ABC$ be a triangle with $\angle A$-acute. Let $P$ be a point inside $\triangle ABC$ such that $\angle BAP = \angle ACP$ and $\angle CAP =\angle ABP$. Let $M, N$ be the centers of the incircle of $\triangle ABP$ and $\triangle ACP$, and $R$ the radius of the circumscribed circle of $\triangle AMN$. Prove that $\displaystyle \frac{1}{R}=\frac{1}{AB}+\frac{1}{AC}+\frac{1}{AP}. $

$(NET 1)$ The vertices of an $(n + 1)-$gon are placed on the edges of a regular $n-$gon so that the perimeter of the $n-$gon is divided into equal parts. How does one choose these $n + 1$ points in order to obtain the $(n + 1)-$gon with $(a)$ maximal area; $(b)$ minimal area?

On the board all the 20-digit numbers which have 10 ones and 10 twos in their decimal form are written. It is allowed to choose two different digits in any number and to reverse the order of digits in the interval between them. What is the maximal quantity of equal numbers which is possible to get on the board using such operations?

Given a prime number $p\ge 3,$ and an integer $d \ge 1$. Prove that there exists an integer $n\ge 1,$ such that $\gcd(n,d) = 1,$ and the product \[P=\prod\limits_{1 \le i < j < p} {({i^{n + j}} - {j^{n + i}})} \text{ is not divisible by } p^n.\]

There is $n$ black points in the plane.We do the following algorithm: Start from any point from those $n$ points and colour it red. Then connect this point to the nearest black point available and colour this new point red. Then do the same with this point but at any step , but you are never allowed to draw a line which intersects on of the current drawn segments. If you reach an intersection , the algorithm is over. Is it true that for any $n$ and at any initial position , we can start from a point st in the algorithm , we reach all the points?

The sum of the digits of a natural number $n$ is denoted by $S(n)$. Prove that $S(8n) \ge \frac{1}{8} S(n)$ for each $n$.

An acute triangle $ABC$ is given. Let us denote by $D$ and $E$ the orthogonal projections, respectively of points $ B$ and $C$ on the bisector of the external angle $BAC$. Let $F$ be the point of intersection of the lines $BE$ and $CD$. Show that the lines $AF$ and $DE$ are perpendicular.

Indicate in which one of the following equations $ y$ is neither directly nor inversely proportional to $ x$: $ \textbf{(A)}\ x \plus{} y \equal{} 0 \qquad\textbf{(B)}\ 3xy \equal{} 10 \qquad\textbf{(C)}\ x \equal{} 5y \qquad\textbf{(D)}\ 3x \plus{} y \equal{} 10$ $ \textbf{(E)}\ \frac {x}{y} \equal{} \sqrt {3}$

Let $a,b,c,d x,y,z, w$ be real numbers such that $$\begin{matrix} ax -by-c z-dw =0\\ b x +a y -d z +cw=0\\ c x+ d y +a z -b w=0\\ dx-c y+bz+aw=0 \end{matrix}$$ prove that $$a=b=c=d=0, \ \ or \ \ x=y=z=w=0$$

Given acute triangle $ABC$ with $AB>AC$, let $M$ be the midpoint of $BC$. $P$ is a point in triangle $AMC$ such that $\angle MAB=\angle PAC$. Let $O,O_1,O_2$ be the circumcenters of $\triangle ABC,\triangle ABP,\triangle ACP$ respectively. Prove that line $AO$ passes through the midpoint of $O_1 O_2$.

Let \[S=\sqrt{1+\dfrac1{1^2}+\dfrac1{2^2}}+\sqrt{1+\dfrac1{2^2}+\dfrac1{3^2}}+\cdots+\sqrt{1+\dfrac1{2007^2}+\dfrac1{2008^2}}.\] Compute $\lfloor S^2\rfloor$.

Initially an urn contains 100 black marbles and 100 white marbles. Repeatedly, three marbles are removed from the urn and replaced from a pile outside the urn as follows: \[ \begin{tabular}{ccc} \textbf{\underline{MARBLES REMOVED}} & & \textbf{\underline{REPLACED WITH}} \\ 3 black & & 1 black \\ 2 black, 1 white & &1 black, 1 white\\ 1 black, 2 white & & 2 white \\ 3 white & & 1 black, 1 white \end{tabular} \] Which of the following sets of marbles could be the contents of the urn after repeated applications of this procedure? $ \textbf{(A)}\ \text{2 black marbles} $ $\textbf{(B)}\ \text{2 white marbles} $ $\textbf{(C)}\ \text{1 black marble} $ $\textbf{(D)}\ \text{1 black and 1 white marble} $ $\textbf{(E)}\ \text{1 white marble} $

Let $ f:\mathbb{D}\longrightarrow\mathbb{C} $ be a continuous function, where $ \mathbb{D} $ is the closed unit disk. Suppose that $ f $ is holomorphic on the open unit disk and that $ e^{i\theta } $ are roots, for any $ \theta\in\left[ 0,\pi /4 \right] . $ Show that $ f=0_{\mathbb{D}} . $

Elisenda has a piece of paper in the shape of a triangle with vertices $A, B,$ and $C$ such that $AB = 42.$ She chooses a point $D$ on segment $AC,$ and she folds the paper along line $BD$ so that $A$ lands at a point $E$ on segment $BC.$ Then, she folds the paper along line $DE.$ When she does this, $B$ lands at the midpoint of segment $DC.$ Compute the perimeter of the original unfolded triangle.

Two straight pipes (circular cylinders), with radii $1$ and $\frac{1}{4}$, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both? [asy] size(6cm); draw(circle((0,1),1), linewidth(1.2)); draw((-1,0)--(1.25,0), linewidth(1.2)); draw(circle((1,1/4),1/4), linewidth(1.2)); [/asy] $\textbf{(A)}~\displaystyle\frac{1}{9} \qquad\textbf{(B)}~1 \qquad\textbf{(C)}~\displaystyle\frac{10}{9} \qquad\textbf{(D)}~\displaystyle\frac{11}{9} \qquad\textbf{(E)}~\displaystyle\frac{19}{9}$

In convex hexagon $ABCDEF$, all six sides are congruent, $\angle A$ and $\angle D$ are right angles, and $\angle B$, $\angle C$, $\angle E$, and $\angle F$ are congruent. The area of the hexagonal region is $2116(\sqrt{2}+1)$. Find $AB$.

Point $M$ is a midpoint of side $BC$ of a triangle $ABC$ and $H$ is the orthocenter of $ABC$. $MH$ intersects the $A$-angle bisector at $Q$. Points $X$ and $Y$ are the projections of $Q$ on sides $AB$ and $AC$. Prove that $XY$ passes through $H$.

Find the product of all possible real values for $k$ such that the system of equations $$x^2+y^2= 80$$ $$x^2+y^2= k+2x-8y$$ has exactly one real solution $(x,y)$. [i]Proposed by Nathan Xiong[/i]

A positive integer is [i]happy[/i] if: 1. All its digits are different and not $0$, 2. One of its digits is equal to the sum of the other digits. For example, 253 is a [i]happy[/i] number. How many [i]happy[/i] numbers are there?

When $ x^{13} \plus{} 1$ is divided by $ x \minus{} 1$, the remainder is: $\textbf{(A)}\ 1\qquad \textbf{(B)}\ -1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \text{None of these answers}$

There are 8 coins, 7 of which are real, which weigh the same, and one is fake, which differs in weight from the rest. Cup scales without weights mean that if you put equal weights on their cups, then any of the cups can outweigh, but if the loads are different in mass, then the cup with a heavier load is definitely overpowered. How to definitely identify a counterfeit coin in four weighings and establish is it lighter or heavier than the others?

Suppose that $ ABCD$ is a cyclic quadrilateral. Let $ E \equal{} AC\cap BD$ and $ F \equal{} AB\cap CD$. Denote by $ H_{1}$ and $ H_{2}$ the orthocenters of triangles $ EAD$ and $ EBC$, respectively. Prove that the points $ F$, $ H_{1}$, $ H_{2}$ are collinear. Original formulation: Let $ ABC$ be a triangle. A circle passing through $ B$ and $ C$ intersects the sides $ AB$ and $ AC$ again at $ C'$ and $ B',$ respectively. Prove that $ BB'$, $CC'$ and $ HH'$ are concurrent, where $ H$ and $ H'$ are the orthocentres of triangles $ ABC$ and $ AB'C'$ respectively.

The inner angles of the pentagon inscribed in the circle are equal to each other. Prove that this pentagon is regular.

One day in a room there were several inhabitants of an island where only truth-tellers and liars live. Three of them made the following statements: [list] [*] There are no more than three of us here. We are all liars. [*] There are no more than four of us here. Not all of us are liars. [*] There are five of us here. At least three of us are liars. [/list] How many people are in the room and how many of them are liars?

Let $K$ and $M$ be points on the side $AB$ of a triangle $\triangle{ABC}$, and let $L$ and $N$ be points on the side $AC$. The point $K$ is between $M$ and $B$, and the point $L$ is between $N$ and $C$. If $\frac{BK}{KM}=\frac{CL}{LN}$, then prove that the orthocentres of the triangles $\triangle{ABC}$, $\triangle{AKL}$ and $\triangle{AMN}$ lie on one line.