Given triangle $ABC$ with the inner - bisector $AD$. The line passes through $D$ and perpendicular to $BC$ intersects the outer - bisector of $\angle BAC$ at $I$. Circle $(I,ID)$ intersects $CA$, $AB$ at $E$, $F$, reps. The symmedian line of $\triangle AEF$ intersects the circle $(AEF)$ at $X$. Prove that the circles $(BXC)$ and $(AEF)$ are tangent. [Hide=Diagram] [asy]import graph; size(7.04cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = 7.02, xmax = 14.06, ymin = -1.54, ymax = 4.08; /* image dimensions */ /* draw figures */ draw((8.62,3.12)--(7.58,-0.38)); draw((7.58,-0.38)--(13.68,-0.38)); draw((13.68,-0.38)--(8.62,3.12)); draw((8.62,3.12)--(9.85183961338573,3.5535510951732316)); draw((9.85183961338573,3.5535510951732316)--(9.851839613385732,-0.38)); draw((9.851839613385732,-0.38)--(8.62,3.12)); draw(circle((10.012708209519483,1.129702986881574), 2.4291805937992947)); draw((8.62,3.12)--(9.470868507287285,-1.238276762688951), red); draw(shift((9.85183961338573,3.553551095173232))*xscale(3.9335510951732324)*yscale(3.9335510951732324)*arc((0,0),1,237.85842690125605,309.7357733435313), linetype("4 4")); draw(shift((10.63,3.8274278922585725))*xscale(5.196628663716066)*yscale(5.196628663716066)*arc((0,0),1,234.06132677886183,305.9386732211382), blue); /* dots and labels */ dot((8.62,3.12),linewidth(3.pt) + dotstyle); label("$A$", (8.48,3.24), NE * labelscalefactor); dot((7.58,-0.38),linewidth(3.pt) + dotstyle); label("$B$", (7.3,-0.58), NE * labelscalefactor); dot((13.68,-0.38),linewidth(3.pt) + dotstyle); label("$C$", (13.76,-0.26), NE * labelscalefactor); dot((9.851839613385732,-0.38),linewidth(3.pt) + dotstyle); label("$D$", (9.94,-0.26), NE * labelscalefactor); dot((9.85183961338573,3.5535510951732316),linewidth(3.pt) + dotstyle); label("$I$", (9.94,3.68), NE * labelscalefactor); dot((7.759138898806625,0.22287129406075654),linewidth(3.pt) + dotstyle); label("$F$", (7.46,0.16), NE * labelscalefactor); dot((12.36635458796946,0.5286480122740898),linewidth(3.pt) + dotstyle); label("$E$", (12.44,0.64), NE * labelscalefactor); dot((9.470868507287285,-1.238276762688951),linewidth(3.pt) + dotstyle); label("$X$", (9.56,-1.12), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy] [/Hide]

Ali Baba and the $40$ thieves want to cross Bosporus strait. They made a line so that any two people standing next to each other are friends. Ali Baba is the first, he is also a friend with the thief next to his neighbour. There is a single boat that can carry $2$ or $3$ people and these people must be friends. Can Ali Baba and the $40$ thieves always cross the strait if a single person cannot sail?

Two squares on an $8\times 8$ chessboard are called adjacent if they have a common edge or common corner. Is it possible for a king to begin in some square and visit all squares exactly once in such a way that all moves except the first are made into squares adjacent to an even number of squares already visited?

Find the greatest real number $C$ such that, for all real numbers $x$ and $y \neq x$ with $xy = 2$ it holds that \[\frac{((x + y)^2 - 6)((x - y)^2 + 8)}{(x-y)^2}\geq C.\] When does equality occur?

Let $k$ be a positive integer. Prove that: (a) If $k=m+2mn+n$ for some positive integers $m,n$, then $2k+1$ is composite. (b) If $2k+1$ is composite, then there exist positive integers $m,n$ such that $k=m+2mn+n$.

Prove that for any integer $n \ge 3$, the least common multiple of the numbers $1,2, ... ,n$ is greater than $2^{n-1}$.

Prove that the polynomial $x^4+\lambda x^3+\mu x^2+\nu x+1$ has no real roots if $\lambda, \mu , \nu $ are real numbers satisfying \[|\lambda |+|\mu |+|\nu |\le \sqrt{2} \]

If $a,b,c$ are positive real numbers such that $a+b+c=abc$ prove that $$\frac{abc}{3\sqrt{2}}\left ( \sum_{cyc}\frac{\sqrt{a^3+b^3}}{ab+1} \right )\geq \sum_{cyc}\frac{a}{a^2+1}$$

Let $a_{n+1} = a_n + \frac{1}{{a_n}^{2005}}$ and $a_1=1$. Show that $\sum^{\infty}_{n=1}{\frac{1}{n a_n}}$ converge.

Suppose that $ 4^{x_1} \equal{} 5, 5^{x_2} \equal{} 6, 6^{x_3} \equal{} 7,...,127^{x_{124}} \equal{} 128$. What is $ x_1x_2 \cdots x_{124}$? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ \frac {5}{2}\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ \frac {7}{2}\qquad \textbf{(E)}\ 4$

A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ so that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$. If $\omega_1$ has diameter $1$ and touches the parabola at $(0,0)$, find the diameter of $\omega_{1998}$.

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

What is the minimum possible perimeter of a right triangle with integer side lengths whose perimeter is equal to its area? [i]Individual #4[/i]

Seven people of seven different ages are attending a meeting. The seven people leave the meeting one at a time in random order. Given that the youngest person leaves the meeting sometime before the oldest person leaves the meeting, the probability that the third, fourth, and fifth people to leave the meeting do so in order of their ages (youngest to oldest) is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$.

Let $\Delta ABC$ be an acute triangle. $D,E,F$ are the touch points of incircle with $BC,CA,AB$ respectively. $AD,BE,CF$ intersect incircle at $K,L,M$ respectively. If,$$\sigma = \frac{AK}{KD} + \frac{BL}{LE} + \frac{CM}{MF}$$ $$\tau = \frac{AK}{KD}.\frac{BL}{LE}.\frac{CM}{MF}$$ Then prove that $\tau = \frac{R}{16r}$. Also prove that there exists integers $u,v,w$ such that, $uvw \neq 0$, $u\sigma + v\tau +w=0$.

Prove that if $m,n,r$ are positive integers, and: \[1+m+n\sqrt{3}=(2+\sqrt{3})^{2r-1} \] then $m$ is a perfect square.

$\dfrac{1}{10} + \dfrac{9}{100} + \dfrac{9}{1000} + \dfrac{7}{10000} = $ $\textbf{(A)}\ 0.0026 \qquad \textbf{(B)}\ 0.0197 \qquad \textbf{(C)}\ 0.1997 \qquad \textbf{(D)}\ 0.26 \qquad \textbf{(E)}\ 1.997$

At a circular track, $10$ cyclists started from some point at the same time in the same direction with different constant speeds. If any two cyclists are at some point at the same time again, we say that they meet. No three or more of them have met at the same time. Prove that by the time every two cyclists have met at least once, each cyclist has had at least $25$ meetings.

You are in a strange land and you don’t know the language. You know that ”!” and ”?” stand for addition and subtraction, but you don’t know which is which. Each of these two symbols can be written between two arguments, but for subtraction you don’t know if the left argument is subtracted from the right or vice versa. So, for instance, a?b could mean any of a − b, b − a, and a + b. You don’t know how to write any numbers, but variables and parenthesis work as usual. Given two arguments a and b, how can you write an expression that equals 20a − 18b? (12 points) Nikolay Belukhov

Angles of a given triangle $ABC$ are all smaller than $120^\circ$. Equilateral triangles $AFB, BDC$ and $CEA$ are constructed in the exterior of $ABC$. (a) Prove that the lines $AD, BE$, and $CF$ pass through one point $S.$ (b) Prove that $SD + SE + SF = 2(SA + SB + SC).$

Let ${R^* = R-\{0\}}$ be the set of non-zero real numbers. Find all functions ${f : R^* \rightarrow R^*}$ satisfying ${f(x) + f(y) = f(xy f(x + y))}$, for ${x, y \in R^*}$ and ${ x + y\ne 0 }$.

The squares $ABMN$, $BCKL$ and $ACPQ$ are constructed outside triangle $ABC$. The difference between the areas of $AB MN$ and $BCKL$ is $d$. Find the difference between the areas of the squares with sides $NQ$ and $PK$ respectively, if $\angle ABC$ is (a) a right angle; (b) not necessarily a right angle. (A Gerko)

(Game) Károly and Dezso wish to count up to $m$ and play the following game in the meantime: they start from $0$ and the two players can add a positive number less than $13$ to the previous number, taking turns. However because of their superstition, if one of them added $x$, then the other one in the next step cannot add $13-x$. Whoever reaches (or surpasses) $m$ first, loses. [i]Defeat the organisers in this game twice in a row! A starting position will be given and then you can decide whether you want to go first or second.[/i]

Let $A_1A_2A_3A_4$ be a tetrahedron. For any permutation $(i, j,k,h)$ of $1,2,3,4$ denote: - $P_i$ – the orthogonal projection of $A_i$ on $A_jA_kA_h$; - $B_{ij}$ – the midpoint of the edge $A_iAj$, - $C_{ij}$ – the midpoint of segment $P_iP_j$ - $\beta_{ij}$– the plane $B_{ij}P_hP_k$ - $\delta_{ij}$ – the plane $B_{ij}P_iP_j$ - $\alpha_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_kA_h$ - $\gamma_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_iA_j$. Prove that if the points $P_i$ are not in a plane, then the following sets of planes are concurrent: (a) $\alpha_{ij}$, (b) $\beta_{ij}$, (c) $\gamma_{ij}$, (d) $\delta_{ij}$.

Let $k$ be a positive integer. Compute $$\sum_{n_1=1}^\infty\sum_{n_2=1}^\infty\cdots\sum_{n_k=1}^\infty\frac1{n_1n_2\cdots n_k(n_1+n_2+\ldots+n_k+1)}.$$