In triangle $ABC$, let $O$ and $I_A$ be the centers of the circumcircle and the circle tangent to $AB$ and $AC$ and externally tangent to $BC$, and let $R$ and $R_A$ be their radii. Find $ \frac {I_A A \cdot I_A B \cdot I_A C}{R \cdot R_A^2} $.

A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle (see picture). Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is $r$. Find $100r^2$. [asy] // dragon96, replacing // [img]http://i.imgur.com/08FbQs.png[/img] size(140); defaultpen(linewidth(.7)); real alpha=10, x=-0.12, y=0.025, r=1/sqrt(3); path hex=rotate(alpha)*polygon(6); pair A = shift(x,y)*(r*dir(330+alpha)), B = shift(x,y)*(r*dir(90+alpha)), C = shift(x,y)*(r*dir(210+alpha)); pair X = (-A.x, -A.y), Y = (-B.x, -B.y), Z = (-C.x, -C.y); int i; pair[] H; for(i=0; i<6; i=i+1) { H[i] = dir(alpha+60*i);} fill(X--Y--Z--cycle, rgb(204,255,255)); fill(H[5]--Y--Z--H[0]--cycle^^H[2]--H[3]--X--cycle, rgb(203,153,255)); fill(H[1]--Z--X--H[2]--cycle^^H[4]--H[5]--Y--cycle, rgb(255,203,153)); fill(H[3]--X--Y--H[4]--cycle^^H[0]--H[1]--Z--cycle, rgb(153,203,255)); draw(hex^^X--Y--Z--cycle); draw(H[1]--B--H[2]^^H[3]--C--H[4]^^H[5]--A--H[0]^^A--B--C--cycle, linewidth(0.6)+linetype("5 5")); draw(H[0]--Z--H[1]^^H[2]--X--H[3]^^H[4]--Y--H[5]);[/asy]

Let $AD, BE$ be the altitudes and $CF$ be the angle bissector of acute non-isosceles triangle $ABC$ and $AE+BD=AB.$ Denote by $I_A, I_B, I_C$ the incentres of triangles $AEF,$ $BDF,$ $CDE$ respectively. Prove that points $D, E, F, I_A, I_B$ and $I_C$ lie on the same circle.

In triangle $ABC$ holds $\angle ACB = 90^{\circ}$, $\angle BAC = 30^{\circ}$ and $BC=1$. In triangle $ABC$ is inscribed equilateral triangle (every side of a triangle $ABC$ contains one vertex of inscribed triangle). Find the least possible value of side of inscribed equilateral triangle

In quadrilateral $ABCD$, $CD = 14$, $\angle BAD = 105^o$, $\angle ACD = 35^o$, and $\angle ACB = 40^o$. Let the midpoint of $CD$ be $M$. Points $P$ and $Q$ lie on $\overrightarrow{AM}$ and $\overrightarrow{BM}$, respectively, such that $\angle AP B = 40^o$ and $\angle AQB = 40^o$ . $P B$ intersects $CD$ at point $R$ and $QA$ intersects $CD$ at point $S$. If $CR = 2$, what is the length of $SM$?

A rhombus with its incircle is given. At each vertex of the rhombus a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii $r_1,r_2$, while the incircle has radius $r$. Given that $r_1$ and $r_2$ are natural numbers and that $r_1r_2=r$, find $r_1,r_2,$ and $r$.

Bisector of angle $BAC$ of triangle $ABC$ intersects circumcircle of this triangle in point $D \neq A$. Points $K$ and $L$ are orthogonal projections on line $AD$ of points $B$ and $C$, respectively. Prove that $AD \ge BK + CL$.

Let $ABC$ be a triangle, and let $X$, $Y$, and $Z$ be collinear points such that $AY=AZ$, $BZ=BX$, and $CX=CY$. Points $X'$, $Y'$, and $Z'$ are the reflections of $X$, $Y$, and $Z$ over $BC$, $CA$, and $AB$, respectively. Prove that if $X'Y'Z'$ is a nondegenerate triangle, then its circumcenter lies on the circumcircle of $ABC$. [i]Michael Ren[/i]

Let $ABCD$ be a parallelogram, and let $P$ be a point on the side $AB$. Let the line through $P$ parallel to $BC$ intersect the diagonal $AC$ at point $Q$. Prove that $$|DAQ|^2 = |PAQ| \times |BCD| ,$$ where $|XY Z|$ denotes the area of triangle $XY Z$.

Let $a, b, c$ be side lengths of a triangle, and define $s =\frac{a+b+c}{2}$. Prove that $$\frac{2a(2a-s)}{b + c}+\frac{2b(2b - s)}{c + a}+\frac{2c(2c - s)}{a + b}\ge s.$$

Inside an $11\times 11$ square, Pablo drew a rectangle and extending its sides divided the square into $5$ rectangles, as shown in the figure. [img]https://cdn.artofproblemsolving.com/attachments/5/a/7774da7085f283b3aae74fb5ff472572571827.gif[/img] Sofía did the same, but she also managed to make the lengths of the sides of the $5$ rectangles be whole numbers between $1$ and $10$, all different. Show a figure like the one Sofia made.

Suppose $x, y$, and $ 1$ are side lengths of a triangle$ T$ such that $x < 1$ and $y < 1$. Given $x$ and $y$ are chosen uniformly at random from all possible pairs $(x, y)$, determine the probability that $T$ is obtuse.

Let $n$ be a positive integer. A regular hexagon with side length $n$ is divided into equilateral triangles with side length $1$ by lines parallel to its sides. Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles. [i]UK - Sahl Khan[/i]

Regular hexagon is divided to equal rhombuses, with sides, parallels to hexagon sides. On the three sides of the hexagon, among which there are no neighbors, is set directions in order of traversing the hexagon against hour hand. Then, on each side of the rhombus, an arrow directed just as the side of the hexagon parallel to this side. Prove that there is not a closed path going along the arrows.

The angles of a convex $n$-gon are $a,2a, ... ,na$. Find all possible values of $n$ and the corresponding values of $a$.

Let $\triangle ABC$ be acute. The feet of altitudes from the corners $A, B, C$ are $ D, E, F$, respectively. If $|DF|=3, |FE|=4,$ and $|DE|=5$, then what is the radius of the circle with center $C$ and tangent to $DE$? $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 3$

Suppose circles $\mathit{W}_1$ and $\mathit{W}2$, with centres $\mathit{O}_1$ and $\mathit{O}_2$ respectively, intersect at points $\mathit{M}$ and $\mathit{N}$. Let the tangent on $\mathit{W}_2$ at point $\mathit{N}$ intersect $\mathit{W}_1$ for the second time at $\mathit{B}_1$. Similarly, let the tangent on $\mathit{W}_1$ at point $\mathit{N}$ intersect $\mathit{W}_2$ for the second time at $\mathit{B}_2$. Let $\mathit{A}_1$ be a point on $\mathit{W}_1$ which is on arc $\mathit{B}_1\mathit{N}$ not containing $\mathit{M}$ and suppose line $\mathit{A}_1\mathit{N}$ intersects $\mathit{W}_2$ at point $\mathit{A}_2$. Denote the incentres of triangles $\mathit{B}_1\mathit{A}_1\mathit{N}$ and $\mathit{B}_2\mathit{A}_2\mathit{N}$ by $\mathit{I}_1$ and $\mathit{I}_2$, respectively.* [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(10.1cm); 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 = -0.9748626324969808, xmax = 13.38440254515721, ymin = 0.5680051903627492, ymax = 10.99430986899034; /* image dimensions */ pair O_2 = (7.682929606970993,6.084708172218866), O_1 = (2.180000000000002,6.760000000000007), M = (4.560858774883258,8.585242858926296), B_2 = (10.07334553576748,9.291873850408265), A_2 = (11.49301008867042,4.866805580476367), B_1 = (2.113311869970955,9.759258690628950), A_1 = (0.2203184186713625,4.488514120712773); /* draw figures */ draw(circle(O_2, 4.000000000000000)); draw(circle(O_1, 3.000000000000000)); draw((4.048892687647541,4.413249028538064)--B_2); draw(B_2--A_2); draw(A_2--(4.048892687647541,4.413249028538064)); draw((4.048892687647541,4.413249028538064)--B_1); draw(B_1--A_1); draw(A_1--(4.048892687647541,4.413249028538064)); /* dots and labels */ dot(O_2,dotstyle); label("$O_2$", (7.788512439159622,6.243082420501817), NE * labelscalefactor); dot(O_1,dotstyle); label("$O_1$", (2.298205165350667,6.929370829727937), NE * labelscalefactor); dot(M,dotstyle); label("$M$", (4.383466101076183,8.935444641311980), NE * labelscalefactor); dot((4.048892687647541,4.413249028538064),dotstyle); label("$N$", (3.855551940133015,3.761885864068922), NE * labelscalefactor); dot(B_2,dotstyle); label("$B_2$", (10.19052187145104,9.463358802255147), NE * labelscalefactor); dot(A_2,dotstyle); label("$A_2$", (11.80066006232771,4.659339937672310), NE * labelscalefactor); dot(B_1,dotstyle); label("$B_1$", (1.981456668784765,10.09685579538695), NE * labelscalefactor); dot(A_1,dotstyle); label("$A_1$", (0.08096568938935705,3.973051528446190), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy] Show that \[\angle\mathit{I}_1\mathit{MI}_2=\angle\mathit{O}_1\mathit{MO}_2.\] *[size=80]Given a triangle ABC, the incentre of the triangle is defined to be the intersection of the angle bisectors of A, B, and C. To avoid cluttering, the incentre is omitted in the provided diagram. Note also that the diagram serves only as an aid and is not necessarily drawn to scale.[/size]

In a cyclic quadrilateral $ABCD$ we have $|AD| > |BC|$ and the vertices $C$ and $D$ lie on the shorter arc $AB$ of the circumcircle. Rays $AD$ and $BC$ intersect at point $K$, diagonals $AC$ and $BD$ intersect at point $P$. Line $KP$ intersects the side $AB$ at point $L$. Prove that $\angle ALK$ is acute.

Let $ABC$ is isosceles triangle $AB=AC$. The point $P$ inside $ABC$ triangle such that angle $\widehat{BCP}=30^o$ , $\widehat{APB}=150^o$ and $\widehat{CAP}=39^o$ . Find $\widehat{BAP}$

Angle $ABC$ of $\triangle ABC$ is a right angle. The sides of $\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\overline{AB}$ equals $8\pi$, and the arc of the semicircle on $\overline{AC}$ has length $8.5\pi$. What is the radius of the semicircle on $\overline{BC}$? [asy] import graph; draw((0,8)..(-4,4)..(0,0)--(0,8)); draw((0,0)..(7.5,-7.5)..(15,0)--(0,0)); real theta = aTan(8/15); draw(arc((15/2,4),17/2,-theta,180-theta)); draw((0,8)--(15,0)); label("$A$", (0,8), NW); label("$B$", (0,0), SW); label("$C$", (15,0), SE);[/asy] $\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 7.5 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 8.5 \qquad \textbf{(E)}\ 9$

Let $ABCD$ be a cyclic quadrilateral with $AB<CD$. The diagonals intersect at the point $F$ and lines $AD$ and $BC$ intersect at the point $E$. Let $K$ and $L$ be the orthogonal projections of $F$ onto lines $AD$ and $BC$ respectively, and let $M$, $S$ and $T$ be the midpoints of $EF$, $CF$ and $DF$ respectively. Prove that the second intersection point of the circumcircles of triangles $MKT$ and $MLS$ lies on the segment $CD$. [i](Greece - Silouanos Brazitikos)[/i]

Each of two regular polygons $P$ and $Q$ was divided by a line into two parts. One part of $P$ was attached to one part of $Q$ along the dividing line so that the resulting polygon was regular and not congruent to $P$ or $Q$. How many sides can it have?

An [i]arithmetic sequence[/i] is an infinite sequence of the form $a_n=a_0+n\cdot d$ with $d\neq 0$. A [i]geometric sequence[/i] is an infinite sequence of the form $b_n=b_0 \cdot q^n$ where $q\neq 1,0,-1$. [list=a] [*] Does every arithmetic sequence of [b]integers[/b] have an infinite subsequence which is geometric? [*] Does every arithmetic sequence of [b]real numbers[/b] have an infinite subsequence which is geometric? [/list]

The incircle of the quadrilateral $ABCD$ touches $AB,BC, CD$ and $DA$ at $E, F,G$ and $H$ respectively. Prove that the line joining the incentres of triangles $HAE$ and $FCG$ is perpendicular to the line joining the incentres of triangles $EBF$ and $GDH$. (4)

Each point of the plane is coloured in one of two colours. Given an odd integer number $n \geq 3$, prove that there exist (at least) two similar triangles whose similitude ratio is $n$, each of which has a monochromatic vertex-set. [i]Vasile Pop[/i]