Two points $K$ and $L$ are chosen inside triangle $ABC$ and a point $D$ is chosen on the side $AB$. Suppose that $B$, $K$, $L$, $C$ are concyclic, $\angle AKD = \angle BCK$ and $\angle ALD = \angle BCL$. Prove that $AK = AL$.

Suppose that $1985$ points are given inside a unit cube. Show that one can always choose $32$ of them in such a way that every (possibly degenerate) closed polygon with these points as vertices has a total length of less than $8 \sqrt 3.$

A point $P$ is chosen in the interior of the side $BC$ of triangle $ABC$. The points $D$ and $C$ are symmetric to $P$ with respect to the vertices $B$ and $C$, respectively. The circumcircles of the triangles $ABE$ and $ACD$ intersect at the points $A$ and $X$. The ray $AB$ intersects the segment $XD$ at the point $C_1$ and the ray $AC$ intersects the segment $XE$ at the point $B_1$. Prove that the lines $BC$ and $B_1C_1$ are parallel. [i](A. Voidelevich)[/i]

Let $ABC$ be a triangle and $AD,BE,CF$ be cevians concurrent at a point $P$. Suppose each of the quadrilaterals $PDCE,PEAF$ and $PFBD$ has both circumcircle and incircle. Prove that $ABC$ is equilateral and $P$ coincides with the center of the triangle.

In the plane, three distinct non-collinear points $A,B,C$ are marked. In each step, Ege can do one of the following: [list] [*] For marked points $X,Y$, mark the reflection of $X$ across $Y$. [*]For distinct marked points $X,Y,Z,T$ which do not form a parallelogram, mark the center of spiral similarity which takes segment $XY$ to $ZT$. [*] For distinct marked points $X,Y,Z,T$, mark the intersection of lines $XY$ and $ZT$. [/list] No matter how the points $A,B,C$ are marked in the beginning, can Ege always mark, after finitely many moves, a) The circumcenter of $\triangle ABC$. b) The incenter of $\triangle ABC$. Proposed by [i]Deniz Can Karaçelebi[/i]

In this figure the center of the circle is $O$. $AB \perp BC$, $ADOE$ is a straight line, $AP = AD$, and $AB$ has a length twice the radius. Then: [asy] size(150); defaultpen(linewidth(0.8)+fontsize(10)); real e=350,c=55; pair O=origin,E=dir(e),C=dir(c),B=dir(180+c),D=dir(180+e), rot=rotate(90,B)*O,A=extension(E,D,B,rot); path tangent=A--B; pair P=waypoint(tangent,abs(A-D)/abs(A-B)); draw(unitcircle^^C--B--A--E); dot(A^^B^^C^^D^^E^^P,linewidth(2)); label("$O$",O,dir(290)); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,NE); label("$D$",D,dir(120)); label("$E$",E,SE); label("$P$",P,SW);[/asy] $ \textbf{(A)} AP^2 = PB \times AB\qquad$ $\textbf{(B)}\ AP \times DO = PB \times AD\qquad$ $\textbf{(C)}\ AB^2 = AD \times DE\qquad$ $\textbf{(D)}\ AB \times AD = OB \times AO\qquad$ $\textbf{(E)}\ \text{none of these} $

Let $\vartriangle ABC$ be an equilateral triangle with side length $1$ and let $\Gamma$ the circle tangent to $AB$ and $AC$ at $B$ and $C$, respectively. Let $P$ be on side $AB$ and $Q$ be on side $AC$ so that $PQ // BC$, and the circle through $A, P$, and $Q$ is tangent to $\Gamma$ . If the area of $\vartriangle APQ$ can be written in the form $\frac{\sqrt{a}}{b}$ for positive integers $a$ and $b$, where $a$ is not divisible by the square of any prime, find $a + b$.

$D$ is inner point of triangle $ABC$. $E$ is on $BD$ and $CE=BD$. $\angle ABD=\angle ECD=10,\angle BAD=40,\angle CED=60$ Prove, that $AB>AC$

A 3-dimensional chess board consists of $ 4 \times 4 \times 4$ unit cubes. A rook can step from any unit cube K to any other unit cube that has a common face with K. A bishop can step from any unit cube K to any other unit cube that has a common edge with K, but does not have a common face. One move of both a rook and a bishop consists of an arbitrary positive number of consecutive steps in the same direction. Find the average number of possible moves for either piece, where the average is taken over all possible starting cubes K.

Let $ABC$ be a triangle, $I$ its incenter and $I_a$ its $A$-excenter. Let $\omega$ be its circuncircle and $D$ be the intersection of $AI$ and $\omega$. Let some line $r$ through $D$ cut $BC$ in $E$ and $\omega$ in $F$. The lines $IE$ and $I_aE$ intersect $I_aF$ and $IF$ in $P$ and $Q$, respectively. Furthermore, the circles $PII_a$ and $QII_a$ intersect $I_aE$ and $IE$ in $R$ and $S$, respectively. Prove that there is a circle passing through $F,E,R$ and $S$.

In a circle with center $O$ is inscribed a polygon, which is triangulated. Show that the sum of the squares of the distances from $O$ to the incenters of the formed triangles is independent of the triangulation.

A circle is inscribed in triangle $ABC$. $M$ and $N$ are the points of its tangency with the sides $BC$ and $CA$, respectively. The segment $AM$ intersects $BN$ at point $P$ and the inscribed circle at point $Q$. It is known that $MP = a$, $PQ = b$. Find $AQ$.

Let $x=\frac pq$ for $p$, $q$ coprime. Find $p+q$. [asy] import olympiad; size(200); pen qq=font("phvb"); defaultpen(linewidth(0.6)+fontsize(10pt)); pair A=(-2.25,7),B=(-5,0),C=(5,0),D=waypoint(A--B,3/7), E=waypoint(A--C,1/2),F=intersectionpoint(C--D, B--E); draw(A--B--C--cycle^^B--E^^C--D); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NW); label("$E$",E,NE); label("$F$",F,N); label(scale(2.5)*"X",centroid(A,D,E),qq); label(scale(2.5)*"3",centroid(B,D,F),0.5*N,qq); label(scale(2.5)*"6",centroid(B,F,C),0.25*dir(180),qq); label(scale(2.5)*"2",centroid(C,E,F),dir(140),qq); [/asy]

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

In a triangle $ABC$ with $B = 90^\circ$, $D$ is a point on the segment $BC$ such that the inradii of triangles $ABD$ and $ADC$ are equal. If $\widehat{ADB} = \varphi$ then prove that $\tan^2 (\varphi/2) = \tan (C/2)$.

Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$

Let $P$ be a point inside a triangle $ABC$. A line through $P$ parallel to $AB$ meets $BC$ and $CA$ at points $L$ and $F$, respectively. A line through $P$ parallel to $BC$ meets $CA$ and $BA$ at points $M$ and $D$ respectively, and a line through $P$ parallel to $CA$ meets $AB$ and $BC$ at points $N$ and $E$ respectively. Prove \begin{align*} [PDBL] \cdot [PECM] \cdot [PFAN]=8\cdot [PFM] \cdot [PEL] \cdot
[PDN] \\ \end{align*} [i]Proposed by Steve Dinh[/i]

Square $ ABCD$ has side length $ 2$. A semicircle with diameter $ \overline{AB}$ is constructed inside the square, and the tangent to the semicricle from $ C$ intersects side $ \overline{AD}$ at $ E$. What is the length of $ \overline{CE}$? [asy] defaultpen(linewidth(0.8)); pair A=origin, B=(1,0), C=(1,1), D=(0,1), X=tangent(C, (0.5,0), 0.5, 1), F=C+2*dir(C--X), E=intersectionpoint(C--F, A--D); draw(C--D--A--B--C--E); draw(Arc((0.5,0), 0.5, 0, 180)); pair point=(0.5,0.5); label("$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));[/asy] $ \textbf{(A)}\ \frac {2 \plus{} \sqrt5}{2} \qquad \textbf{(B)}\ \sqrt 5 \qquad \textbf{(C)}\ \sqrt 6 \qquad \textbf{(D)}\ \frac52 \qquad \textbf{(E)}\ 5 \minus{} \sqrt5$

The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?

Two-mutually perpendicular lines $\ell$ and $m$ intersect each other at a point of the circumference of a circle, dividing it into three arcs. A point $M_i$ ($i = 1$,$2$,$3$) is taken on each arc so that the tangent line to the circumference at the point $M_i$ intersects $\ell$ and $m$ in two points at the same distance from $M_i$ (that is $M_i$ is the midpoint of the segment between them). Prove that the triangle $M_1M_2M_3$ is equilateral. (Przhevalsky)

There are six small circles in the figure with a radius of $1$ and tangent to a large circle and the sides of the $ABC$ of an equilateral triangle, where touch points are $K, L$ and $M$ respectively with the midpoints of sides $AB, BC$ and $AC$. Find the radius of the large circle and the side of the triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/3/0/f858dcc5840759993ea2722fd9b9b15c18f491.png[/img]

Let $A_1, A_2, ..., A_m$ and $B_2 , B_3,..., B_n$ be the points on a circle such that $A_1A_2... A_n$ is a regular $m$-gon and $A_1B_2...B_n$ is a regular $n$-gon whereby $n > m$ and the point $B_2$ lies between $A_1$ and $A_2$. Find $\angle B_2A_1A_2$.

The circles $\mathcal C_1(O_1, r_1)$ and $\mathcal C_2(O_2, r_2)$, $r_2 > r_1$, intersect at $A$ and $B$ such that $\angle O_1AO_2 = 90^\circ$. The line $O_1O_2$ meets $\mathcal C_1$ at $C$ and $D$, and $\mathcal C_2$ at $E$ and $F$ (in the order $C$, $E$, $D$, $F$). The line $BE$ meets $\mathcal C_1$ at $K$ and $AC$ at $M$, and the line $BD$ meets $\mathcal C_2$ at $L$ and $AF$ at $N$. Prove that \[ \frac{ r_2}{r_1} = \frac{KE}{KM} \cdot \frac{LN}{LD} . \] [i]Greece[/i]

Three unit circles have a common point $O.$ The other points of (pairwise) intersection are $A, B$ and $C$. Show that the points $A, B$ and $C$ are located on some unit circle.

Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle's area is numerically equal to $6$ times its perimeter.