$S_{1},S_{2},S_{3}$ are three spheres in $\mathbb R^{3}$ that their centers are not collinear. $k\leq8$ is the number of planes that touch three spheres. $A_{i},B_{i},C_{i}$ is the point that $i$-th plane touch the spheres $S_{1},S_{2},S_{3}$. Let $O_{i}$ be circumcenter of $A_{i}B_{i}C_{i}$. Prove that $O_{i}$ are collinear.

The triangle $ ABC$ is given. On the extension of the side $ AB$ we construct the point $ R$ with $ BR \equal{} BC$, where $ AR > BR$ and on the extension of the side $ AC$ we construct the point $ S$ with $ CS \equal{} CB$, where $ AS > CS$. Let $ A_1$ be the point of intersection of the diagonals of the quadrilateral $ BRSC$. Analogous we construct the point $ T$ on the extension of the side $ BC$, where $ CT \equal{} CA$ and $ BT > CT$ and on the extension of the side $ BA$ we construct the point $ U$ with $ AU \equal{} AC$, where $ BU > AU$. Let $ B_1$ be the point of intersection of the diagonals of the quadrilateral $ CTUA$. Likewise we construct the point $ V$ on the extension of the side $ CA$, where $ AV \equal{} AB$ and $ CV > AV$ and on the extension of the side $ CB$ we construct the point $ W$ with $ BW \equal{} BA$ and $ CW > BW$. Let $ C_1$ be the point of intersection of the diagonals of the quadrilateral $ AVWB$. Show that the area of the hexagon $ AC_1BA_1CB_1$ is equal to the sum of the areas of the triangles $ ABC$ and $ A_1B_1C_1$.

Consider the following statement: for any permutation $\pi_1\not=\mathbb{I}$ of $\{1,2,...,n\}$ there is a permutation $\pi_2$ such that any permutation on these numbers can be obtained by a finite compostion of $\pi_1$ and $\pi_2$. (a) Prove the statement for $n=3$ and $n=5$. (b) Disprove the statement for $n=4$.

Point $P$ lies on the side $CD$ of the cyclic quadrilateral $ABCD$ such that $\angle CBP = 90^{\circ}$. Let $K$ be the intersection of $AC,BP$ such that $AK = AP = AD$. $H$ is the projection of $B$ onto the line $AC$. Prove that $\angle APH = 90^{\circ}$. [i]Proposed by Iman Maghsoudi - Iran[/i]

Let $ ABC$ be a triangle with $ \left|AB\right| \equal{} 14$, $ \left|BC\right| \equal{} 12$, $ \left|AC\right| \equal{} 10$. Let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[BC\right]$ such that $ \left|AD\right| \equal{} 4$ and $ Area\left(ABC\right) \equal{} 2Area\left(CDE\right)$. Find $ Area\left(ABE\right)$. $\textbf{(A)}\ 4\sqrt {6} \qquad\textbf{(B)}\ 6\sqrt {2} \qquad\textbf{(C)}\ 3\sqrt {6} \qquad\textbf{(D)}\ 4\sqrt {2} \qquad\textbf{(E)}\ 4\sqrt {5}$

The figure shows a square and three circles of equal radius tangent to each other and square passes. Find the radius of the circles if the square length is $1$. [img]http://3.bp.blogspot.com/-iIjwupkz7DQ/XnrIRhKIJnI/AAAAAAAALhA/clERrIDqEtcujzvZk_qu975wsTjKaxCLQCK4BGAYYCw/s400/97%2Bestonia%2Bopen%2Bs2.3.png[/img]

Each side of an arbitrary $\triangle ABC$ is divided into equal parts, and lines parallel to $AB,BC,CA$ are drawn through each of these points, thus cutting $\triangle ABC$ into small triangles. Points are assigned a number in the following manner: $(1)$ $A,B,C$ are assigned $1,2,3$ respectively $(2)$ Points on $AB$ are assigned $1$ or $2$ $(3)$ Points on $BC$ are assigned $2$ or $3$ $(4)$ Points on $CA$ are assigned $3$ or $1$ Prove that there must exist a small triangle whose vertices are marked by $1,2,3$.

Let $ABC$ be a scalene triangle with circle $\Gamma$. Let $P,Q,R,S$ distinct points on the $BC$ side, in that order, such that $\angle BAP = \angle CAS$ and $\angle BAQ = \angle CAR$. Let $U, V, W, Z$ be the intersections, distinct from $A$, of the $AP, AQ, AR$ and $AS$ with $\Gamma$, respectively. Let $X = UQ \cap SW$, $Y = PV \cap ZR$, $T = UR \cap VS$ and $K = PW \cap ZQ$. Suppose that the points $M$ and $N$ are well determined, such that $M = KX \cap TY$ and $N = TX \cap KY$. Show that $M, N, A$ are collinear.

Convex pentagon $ABCDE$ satisfies $AB \parallel DE$, $BE \parallel CD$, $BC \parallel AE$, $AB = 30$, $BC = 18$, $CD = 17$, and $DE = 20$. Find its area. [i] Proposed by Michael Tang [/i]

In the plane there are six different points $A, B, C, D, E, F$ such that $ABCD$ and $CDEF$ are parallelograms. What is the maximum number of those points that can be located on one circle?

Let $I$ be the incenter of an acute triangle $\triangle ABC$, and let the incircle be $\Gamma$. Let the circumcircle of $\triangle IBC$ hit $\Gamma$ at $D, E$, where $D$ is closer to $B$ and $E$ is closer to $C$. Let $\Gamma \cap BE = K (\not= E)$, $CD \cap BI = T$, and $CD \cap \Gamma = L (\not= D)$. Let the line passing $T$ and perpendicular to $BI$ meet $\Gamma$ at $P$, where $P$ is inside $\triangle IBC$. Prove that the tangent to $\Gamma$ at $P$, $KL$, $BI$ are concurrent.

Let $P$ be a point inside the triangle $ABC$, such that the angles $\angle CBP$ and $\angle PAC$ are equal. Denote the intersection of the line $AP$ and the segment $BC$ by $D$, and the intersection of the line $BP$ with the segment $AC$ by $E$. The circumcircles of the triangles $ADC$ and $BEC$ meet at $C$ and $F$. Show that the line $CP$ bisects the angle $DFE$. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

Prove that in space there is a sphere containing exactly $1994$ points with integer coordinates.

Consider the equilateral triangle $ABC$ with $|BC| = |CA| = |AB| = 1$. On the extension of side $BC$, we define points $A_1$ (on the same side as B) and $A_2$ (on the same side as C) such that $|A_1B| = |BC| = |CA_2| = 1$. Similarly, we define $B_1$ and $B_2$ on the extension of side $CA$ such that $|B_1C| = |CA| =|AB_2| = 1$, and $C_1$ and $C_2$ on the extension of side $AB$ such that $|C_1A| = |AB| = |BC_2| = 1$. Now the circumcentre of 4ABC is also the centre of the circle that passes through the points $A_1,B_2,C_1,A_2,B_1$ and $C_2$. Calculate the radius of the circle through $A_1,B_2,C_1,A_2,B_1$ and $C_2$. [asy] unitsize(1.5 cm); pair[] A, B, C; A[0] = (0,0); B[0] = (1,0); C[0] = dir(60); A[1] = B[0] + dir(-60); A[2] = C[0] + dir(120); B[1] = C[0] + dir(60); B[2] = A[0] + dir(240); C[1] = A[0] + (-1,0); C[2] = B[0] + (1,0); draw(A[1]--A[2]); draw(B[1]--B[2]); draw(C[1]--C[2]); draw(circumcircle(A[2],B[1],C[2])); dot("$A$", A[0], SE); dot("$A_1$", A[1], SE); dot("$A_2$", A[2], NW); dot("$B$", B[0], SW); dot("$B_1$", B[1], NE); dot("$B_2$", B[2], SW); dot("$C$", C[0], N); dot("$C_1$", C[1], W); dot("$C_2$", C[2], E); [/asy]

A convex $n$-gon is called [i]nice[/i] if its sides are not all the same length, and the sum of the distances of any interior point to the side lines is $1$. Find all integers $n \ge 4$ such that a nice $n$-gon exists .

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

There are $n$ different points $A_1, \ldots , A_n$ in the plain and each point $A_i$ it is assigned a real number $\lambda_i$ distinct from zero in such way that $(\overline{A_i A_j})^2 = \lambda_i + \lambda_j$ for all the $i$,$j$ with $i\neq{}j$} Show that: (1) $n \leq 4$ (2) If $n=4$, then $\frac{1}{\lambda_1} + \frac{1}{\lambda_2} + \frac{1}{\lambda_3}+ \frac{1}{\lambda_4} = 0$

The planet Tetraincognito covered by ocean has the shape of a regular tetrahedron with an edge of $900$ km. What area of the ocean will the tsunami' cover $2$ hours after the earthquake with the epicenter in a) the center of the face, b) the middle of the edge, if the tsunami propagation speed is $300$ km / h?

Let $ABCD$ be a convex quadrilateral. On the sides $AB$ and $CD$ there are interior points $K$ and $L$, respectively, such that $\angle BAL = \angle CDK$. Prove that the following statements are equivalent: $i) \angle BLA= \angle CKD$ $ii) AD \parallel BC $

Let $ABC$ be a triangle. Let $K, L$ and $M$ be points on the sides $BC, AC$ and $AB$, respectively, such that $\frac{|AM|}{|MB|}\cdot \frac{|BK|}{|KC|}\cdot \frac{|CL|}{|LA|} = 1$. Prove that it is possible to choose two triangles out of $ALM, BMK, CKL$ whose inradii sum up to at least the inradius of triangle $ABC$.

Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle B DC = 90^o$. Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$, respectively, with $P$ lying in between $B$ and $Q$. If $AD = 999$ and $PQ = 200$ then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$ ?

A rubber band of negligible thickness encloses three pegs that lie in a perfect line, as shown. Each peg has a diameter of $4$ cm, as shown. What is the length of the rubber band used, in centimeters? All pegs shown are congruent circles. [asy] size(120); draw(circle((0,0),1));draw(circle((0,2),1));draw(circle((0,4),1)); dot((0,0)^^(0,2)^^(0,4)); draw((-1,0)--(-1,4)--arc((0,4),1,180,0)--(1,4)--(1,0)--arc((0,0),1,360,180),linewidth(2)); draw((-1,0)--(1,0),dotted); label("$4$ cm", (-0.38,0)--(1,0), N); [/asy] $\textbf{(A) }8\qquad\textbf{(B) }8+4\pi\qquad\textbf{(C) }16+4\pi\qquad\textbf{(D) }16+8\pi\qquad\textbf{(E) }16\pi$

Starting with the triple $(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14})$, define a sequence of triples $(x_{n},y_{n},z_{n})$ by $x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})}$ $y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})}$ $ z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})}$ for $n\geq 0$.Show that each of the sequences $\langle x_n\rangle _{n\geq 0},\langle y_n\rangle_{n\geq 0},\langle z_n\rangle_{n\geq 0}$ converges to a limit and find these limits.

10) Call a string of letters $S$ an [i]almost-palindrome[/i] if $S$ and the reverse of $S$ differ in exactly $2$ places. Find the number of ways to order the letters in $HMMTTHEMETEAM$ to get an almost-palindrome. 11) Find all integers $n$, not necessarily positive, for which there exist positive integers ${a,b,c}$ satisfying $a^n + b^n = c^n$. 12) Let $a$ and $b$ be positive real numbers. Determine the minimum possible value of $\sqrt{a^2 + b^2} + \sqrt{a^2 + (b-1)^2} + \sqrt{(a-1)^2 + b^2} + \sqrt{(a-1)^2 + (b-1)^2}$. 13) Consider a $4$ x $4$ grid of squares, each originally colored red. Every minute, Piet can jump on any of the squares, changing the color of it and any adjacent squares to blue (two squares are adjacent if they share a side). What is the minimum number of minutes it will take Piet to change the entire grid to blue? 14) Let $ABC$ be an acute triangle with orthocenter $H$. Let ${D,E}$ be the feet of the ${A,B}$-altitudes, respectively. Given that $\overline{AH} = 20$ and $\overline{HD} =16$ and $\overline{BE} = 56$, find the length of $\overline{BH}$. 15) Find the smallest positive integer $b$ such that $1111 _b$ ($1111$ in base $b$) is a perfect square. If no such $b$ exists, write "No Solution" 16) For how many triples $( {x,y,z} )$ of integers between $-10$ and $10$, inclusive, do there exist reals ${a,b,c}$ that satisfy $ab = x$ $ac = y$ $bc = z$? 17) Unit squares $ABCD$ and $EFGH$ have centers $O_1$ and $O_2$, respectively, and are originally oriented so that $B$ and $E$ are at the same position and $C$ and $H$ are at the same position. The squares then rotate clockwise around their centers at a rate of one revolution per hour. After $5$ minutes, what is the area of the intersection of the two squares? 18) A function $f$ satisfies, for all nonnegative integers $x$ and $y$, $f(x,0) = f(0,x) = x$ If $x \ge y \ge 0$, $f(x,y)=f(x-y,y)+1$ If $y \ge x \ge 0$, $f(x,y) = f(x,y-x)+1$ Find the maximum value of $f$ over $0 \le x,y \le 100$.

Check and justify , if in every tetrahedron are concurrent: a) The perpendiculars to the faces at their circumcenters. b) The perpendiculars to the faces at their orthocenters. c) The perpendiculars to the faces at their incenters. If so, characterize with some simple geometric property the point in that attend If not, show an example that clearly shows the not concurrency.