line $l$ through the point $A$ from triangle $ABC$ . Point $X$ is on line $l$.${\omega}_b$ and ${\omega}_c$ are circles that through points $X,A$ and respectively tanget to $AB$ adn $AC$. tangets from $B,C$ respectively to ${\omega}_b$ and ${\omega}_c$ meet them in $Y,Z$. Prove that by changing $X$, the circumcircle of the circle $XYZ$ passes through two fixed points. [i]Proposed by Ali Zamani [/i]

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

Let $ABCD$ be a convex quadrilateral with $A,B,C,D$ concyclic. Assume $\angle ADC$ is acute and $\frac{AB}{BC}=\frac{DA}{CD}$. Let $\Gamma$ be a circle through $A$ and $D$, tangent to $AB$, and let $E$ be a point on $\Gamma$ and inside $ABCD$. Prove that $AE\perp EC$ if and only if $\frac{AE}{AB}-\frac{ED}{AD}=1$.

Let $ABC$ be a triangle with orthocenter $H$. Let the circle $\omega$ have $BC$ as the diameter. Draw tangents $AP$, $AQ$ to the circle $\omega $ at the point $P, Q$ respectively. Prove that $ P,H,Q$ lie on the same line .

An equiangular hexagon has side lengths $1, 1, a, 1, 1, a$ in that order. Given that there exists a circle that intersects the hexagon at $12$ distinct points, we have $M < a < N$ for some real numbers $M$ and $N$. Determine the minimum possible value of the ratio $\frac{N}{M}$ .

In the triangle $ABC$, $\angle C$ is obtuse and $D$ is a fixed point on the side $BC$, different from $B$ and $C$. For any point $M$ on the side $BC$, different from $D$, the ray $AM$ intersects the circumcircle $S$ of $ABC$ at $N$. The circle through $M, D$ and $N$ meets $S$ again at $P$, different from $N$. Find the location of the point $M$ which minimises $MP$.

A diagonal is drawn in an isosceles trapezoid. By the contour of each of the resulting two triangles creeps its own beetle. The velocities of the beetles are constant and identical. Beetles don't change directions around their contours, and along the diagonal of the trapezoid they crawl in different directions. Prove that for any starting positions of the beetles they will ever meet.

As shown in the figure, quadrilateral $ ABCD$ is inscribed in a circle with $ AC$ as its diameter, $ BD \perp AC,$ and $ E$ the intersection of $ AC$ and $ BD.$ Extend line segment $ DA$ and $ BA$ through $ A$ to $ F$ and $ G$ respectively, such that $ DG \parallel{} BF.$ Extend $ GF$ to $ H$ such that $ CH \perp GH.$ Prove that points $ B, E, F$ and $ H$ lie on one circle. [asy] defaultpen(linewidth(0.8)+fontsize(10));size(150); real a=4, b=6.5, c=9, d=a*c/b, g=14, f=sqrt(a^2+b^2)*sqrt(a^2+d^2)/g; pair E=origin, A=(0,a), B=(-b,0), C=(0,-c), D=(d,0), G=A+g*dir(B--A), F=A+f*dir(D--A), M=midpoint(G--C); path c1=circumcircle(A,B,C), c2=Circle(M, abs(M-G)); pair Hf=F+10*dir(G--F), H=intersectionpoint(F--Hf, c2); dot(A^^B^^C^^D^^E^^F^^G^^H); draw(c1^^c2^^G--D--C--A--G--F--D--B--A^^F--H--C--B--F); draw(H--B^^F--E^^G--C, linetype("2 2")); pair point= E; 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("$F$", F, dir(point--F)); label("$G$", G, dir(point--G)); label("$H$", H, dir(point--H)); label("$E$", E, NE);[/asy]

A circle with area $\tfrac{36}{\pi}$ has the same perimeter as a square with what side length? $\textbf{(A) }\frac{9}{\pi}\qquad\textbf{(B) }3\qquad\textbf{(C) }\pi\qquad\textbf{(D) }6\qquad\textbf{(E) }\pi^2$

The point $O$ is the centre of the circumscribed circle of the acute-angled triangle $ABC$. The line $AO$ cuts the side $BC$ in point $N$, and the line $BO$ cuts the side $AC$ at point $M$. Prove that if $CM=CN$, then $AC=BC$.

Points $P_1,P_2,P_3,P_4$ are vertexes of a regular triangular pyramid, and $P_5,P_6,P_7,P_8,P_9,P_{10}$ midpoints of edges. The number of groups $(P_1,P_i,P_j,P_k)(1<i<j<k\leq10)$ that $P_1,P_i,P_j,P_k$ are coplane is________.

Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

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 $ABC$.

Triangle $\vartriangle BMT$ has $BM = 4$, $BT = 6$, and $MT = 8$. Point $A$ lies on line $\overleftrightarrow{BM}$ and point $Y$ lies on line $\overleftrightarrow{BT}$ such that $\overline{AY}$ is parallel to $\overline{MT}$ and the center of the circle inscribed in triangle $\vartriangle BAY$ lies on $\overline{MT}$. Compute $AY$ .

Two circles with centers $O_{1}$ and $O_{2}$ meet at two points $A$ and $B$ such that the centers of the circles are on opposite sides of the line $AB$. The lines $BO_{1}$ and $BO_{2}$ meet their respective circles again at $B_{1}$ and $B_{2}$. Let $M$ be the midpoint of $B_{1}B_{2}$. Let $M_{1}$, $M_{2}$ be points on the circles of centers $O_{1}$ and $O_{2}$ respectively, such that $\angle AO_{1}M_{1}= \angle AO_{2}M_{2}$, and $B_{1}$ lies on the minor arc $AM_{1}$ while $B$ lies on the minor arc $AM_{2}$. Show that $\angle MM_{1}B = \angle MM_{2}B$. [i]Ciprus[/i]

$(HUN 3)$ In the plane $4000$ points are given such that each line passes through at most $2$ of these points. Prove that there exist $1000$ disjoint quadrilaterals in the plane with vertices at these points.

Let $ABCD$ be a quadrilateral such that the excircles $\omega_{1}$ and $\omega_{2}$ of triangles $ABC$ and $BCD$ touching their sides $AB$ and $BD$ respectively touch the extension of $BC$ at the same point $P$. The segment $AD$ meets $\omega_{2}$ at point $Q$, and the line $AD$ meets $\omega_{1}$ at $R$ and $S$. Prove that one of angles $RPQ$ and $SPQ$ is right Proposed by: I.Kukharchuk

$ABCD$ is a rectangle $\overline{AB} = 5\sqrt{3}$, $\overline{AD} = 30$. Extend $\overline{BC}$ past $C$ and construct point $P$ on this extension such that $\angle APD = 60^{\circ}$. Point $H$ is on $\overline{AP}$ such that $\overline{DH} \perp \overline{AP}$. Find the length of $\overline{DH}$. [i]Proposed by Kevin Wu[/i]

Let ABCD be a cyclic quadrilateral in which AB = BC and AD =CD. Point M is on the small arc CD of the circle circumscribed to the quadrilateral. The lines BM and CD intersect at point P, and the lines AM and BD intersect at point Q. Prove that PQ is parralel to AC.

Length of edges of regular triangle $ABC$ are $4$, $D\in BC,E\in CA,F\in AB$, satisfying: $|AE|=|BF|=|CD|=1$. $BE\cap CF=R, CF\cap AD=Q, AD\cap BE=S$. $P$ is a point inside $\triangle RQS$ or on its sides. Note that $x=d(P,BC),y=d(P,CA),z=d(P,AB)$. [b](a)[/b] $xyz$ get its minumum value when $P=R$ (or$Q,S$). [b](b)[/b] Calculate the minumum value of $xyz$.

Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$? $\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer

Two circles $\omega_1$ and $\omega_2$ touch externally at point $P$.Let $A$ be a point on $\omega_2$ not lying on the line through the centres of the two circles.Let $AB$ and $AC$ be the tangents to $\omega_1$.Lines $BP$ and $CP$ meet $\omega_2$ for the second time at points $E$ and $F$.Prove that the line $EF$,the tangent to $\omega_2$ at $A$ and the common tangent at $P$ concur.

A tetrahedron is to be cut by three planes which form a parallelepiped whose three faces and all vertices lie on the surface of the tetrahedron. (a) Can this be done so that the volume of the parallelepiped is at least $ \frac{9}{40}$ of the volume of the tetrahedron? (b) Determine the common point of the three planes if the volume of the parallelepiped is $ \frac{11}{50}$ of the volume of the tetrahedron.

There are one-way flights between $100$ cities of a country. It is possible to fly starting from the capital city and visiting all other $99$ cities and returning again to the capital city. Let $ N$ be the smallest number of flights inorder to form such a flight combination. Among all flight combinations (satisfying previous condtions), $ N$ can be at most ? $\textbf{(A)}\ 1850 \qquad\textbf{(B)}\ 2100 \qquad\textbf{(C)}\ 2550 \qquad\textbf{(D)}\ 3060 \qquad\textbf{(E)}\ \text{None}$