A convex hexagon $A_1B_1A_2B_2A_3B_3$ it is inscribed in a circumference $\Omega$ with radius $R$. The diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concurrent in $X$. For each $i=1,2,3$ let $\omega_i$ tangent to the segments $XA_i$ and $XB_i$ and tangent to the arc $A_iB_i$ of $\Omega$ that does not contain the other vertices of the hexagon; let $r_i$ the radius of $\omega_i$. $(a)$ Prove that $R\geq r_1+r_2+r_3$ $(b)$ If $R= r_1+r_2+r_3$, prove that the six points of tangency of the circumferences $\omega_i$ with the diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concyclic

Prove that the number $1\underbrace{\hbox{0...0}}_{\hbox{49}}5\underbrace{\hbox{0...0}}_{\hbox{99}}1$ is not the cube of any integer.

Given regular hexagon $ABCDEF$, compute the probability that a randomly chosen point inside the hexagon is inside triangle $PQR$, where $P$ is the midpoint of $AB$, $Q$ is the midpoint of $CD$, and $R$ is the midpoint of $EF$.

Point $P$ lies inside of parallelogram $ABCD$. Perpendicular lines to $PA,PB,PC$ and $PD$ through $A,B,C$ and $D$ construct convex quadrilateral $XYZT$. Prove that $S_{XYZT}\geq 2S_{ABCD}$. [i]Proposed by Siamak Ahmadpour[/i]

A cube is decomposed in a finite number of rectangular parallelepipeds such that the volume of the cube's circum sphere volume equals the sum of the volumes of all parallelepipeds' circum spheres. Prove that all these parallelepipeds are cubes.

Two non-intersecting circles, $\omega$ and $\Omega$, have centers $C_\omega$ and $C_\Omega$ respectively. It is given that the radius of $\Omega$ is strictly larger than the radius of $\omega$. The two common external tangents of $\Omega$ and $\omega$ intersect at a point $P$, and an internal tangent of the two circles intersects the common external tangents at $X$ and $Y$. Suppose that the radius of $\omega$ is $4$, the circumradius of $\triangle PXY$ is $9$, and $XY$ bisects $\overline{PC_\Omega}$. Compute $XY$.

Let $ABC $be a right triangle with $\angle C = 90^o$ and let $D$ be the foot of the altitude from $C$. Let $E$ be the centroid of triangle $ACD$ and let $F$ be the centroid of triangle $BCD$. The point $P$ satisfies $\angle CEP = 90^o$ and $|CP| = |AP|$, while point $Q$ satisfies $\angle CFQ = 90^o$ and $|CQ| = |BQ|$. Prove that $PQ$ passes through the centroid of triangle $ABC$.

Let $P$ be a $2019-$gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of $P$ a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent knots in a cycle must be on the same diagonal and on every diagonal there are at most two knots from a cycle.)

Circle $\omega_1$ has diameter $AB$, and circle $\omega_2$ has center $A$ and intersects $\omega_1$ at points $C$ and $D$. Let $E$ be the intersection of $AB$ and $CD$. Point $P$ is chosen on $\omega_2$ such that $P C = 8$, $P D = 14$, and $P E = 7$. Find the length of $P B$.

$M$ is any point on the side $AB$ of the triangle $ABC$. $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$. $q$ is the radius of the circle on the opposite side of $AB$ to $C$, touching the three sides of $AB$ and the extensions of $CA$ and $CB$. Similarly, $q_1$ and $q_2$. Prove that $r_1r_2q=rq_1q_2$.

Let $I$ be the center of the circle inscribed in triangle $ABC$. The inscribed circle is tangent to side $BC$ at point $K$. Let $X$ and $Y$ be points on segments $BI$ and $CI$ respectively, such that $KX \perp AB $ and $KY\perp AC$. The circumscribed circle around triangle $XYK$ intersects line $BC$ at point $D$. Prove that $AD \perp BC$. (Matthew Kurskyi)

[color=indigo]Let $ABC$ is an acute angled triangle and let $A_{1},\, B_{1},\, C_{1}$ are points respectively on $BC,\,CA,\,AB$ such that $\triangle ABC$ is similar to $\triangle A_{1}B_{1}C_{1}.$ Prove that orthocenter of $A_{1}B_{1}C_{1}$ coincides with circumcenter of $ABC$.[/color]

A paper equilateral triangle $ABC$ has side length $12$. The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance $9$ from point $B$. The length of the line segment along which the triangle is folded can be written as $\frac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$. [asy] import cse5; size(12cm); pen tpen = defaultpen + 1.337; real a = 39/5.0; real b = 39/7.0; pair B = MP("B", (0,0), dir(200)); pair A = MP("A", (9,0), dir(-80)); pair C = MP("C", (12,0), dir(-20)); pair K = (6,10.392); pair M = (a*B+(12-a)*K) / 12; pair N = (b*C+(12-b)*K) / 12; draw(B--M--N--C--cycle, tpen); draw(M--A--N--cycle); fill(M--A--N--cycle, mediumgrey); pair shift = (-20.13, 0); pair B1 = MP("B", B+shift, dir(200)); pair A1 = MP("A", K+shift, dir(90)); pair C1 = MP("C", C+shift, dir(-20)); draw(A1--B1--C1--cycle, tpen);[/asy]

Consider a square of sidelength $ n$ and $ (n\plus{}1)^2$ interior points. Prove that we can choose $ 3$ of these points so that they determine a triangle (eventually degenerated) of area at most $ \frac12$.

A trapezoid has side lengths as indicated in the figure (the sides with length $11$ and $36$ are parallel). Calculate the area of the trapezoid.[img]https://1.bp.blogspot.com/-5PKrqDG37X4/XzcJtCyUv8I/AAAAAAAAMY0/tB0FObJUJdcTlAJc4n6YNEaVIDfQ91-eQCLcBGAsYHQ/s0/1995%2BMohr%2Bp1.png[/img]

Let $ACD$ and $BCD$ be acute-angled triangles located in different planes. Let $G$ and $H$ be the centroid and the orthocenter respectively of the $BCD$ triangle; Similarly let $G'$ and $H'$ be the centroid and the orthocenter of the $ACD$ triangle. Knowing that $HH'$ is perpendicular to the plane $(ACD)$, show that $GG' $ is perpendicular to the plane $(BCD)$.

Four distinct circles of radius $r$ are on the surface of a unit sphere such that they are pairwise tangent. Find $r$. [i]Proposed by Thomas Lam[/i]

Let $H$ be the orthocenter of an acute-angled triangle $ABC$; $A_1, B_1, C_1$ be the touching points of the incircle with $BC, CA, AB$ respectively; $E_A, E_B, E_C$ be the midpoints of $AH, BH, CH$ respectively. The circle centered at $E_A$ and passing through $A$ meets for the second time the bisector of angle $A$ at $A_2$; points $B_2, C_2$ are defined similarly. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.

A regular octagon is inscribed in a circle of radius 2. Alice and Bob play a game in which they take turns claiming vertices of the octagon, with Alice going first. A player wins as soon as they have selected three points that form a right angle. If all points are selected without either player winning, the game ends in a draw. Given that both players play optimally, find all positive areas of the convex polygon formed by Alice's points at the end of the game.

In acute triangle $ABC$ $(AB < AC)$ let $D$, $E$ and $F$ be foots of perpedicular from $A$, $B$ and $C$ to $BC$, $CA$ and $AB$, respectively. Let $P$ and $Q$ be points on line $EF$ such that $DP \perp EF$ and $BQ=CQ$. Prove that $\angle ADP = \angle PBQ$

Three mutually tangent spheres each with radius $5$ sit on a horizontal plane. A triangular pyramid has a base that is an equilateral triangle with side length $6$, has three congruent isosceles triangles for vertical faces, and has height $12$. The base of the pyramid is parallel to the plane, and the vertex of the pyramid is pointing downward so that it is between the base and the plane. Each of the three vertical faces of the pyramid is tangent to one of the spheres at a point on the triangular face along its altitude from the vertex of the pyramid to the side of length $6$. The distance that these points of tangency are from the base of the pyramid is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(200); defaultpen(linewidth(0.8)); pair X=(-.6,.4),A=(-.4,2),B=(-.7,1.85),C=(-1.1,2.05); picture spherex; filldraw(spherex,unitcircle,white); draw(spherex,(-1,0)..(-.2,-.2)..(1,0)^^(0,1)..(-.2,-.2)..(0,-1)); add(shift(-0.5,0.6)*spherex); filldraw(X--A--C--cycle,gray); draw(A--B--C^^X--B); add(shift(-1.5,0.2)*spherex); add(spherex); [/asy]

Let $H$ and $O$ be the orthocenter and the circumcenter of the triangle $ABC$. Line $OH$ intersects the sides $AB, AC$ at points $X, Y$ correspondingly, so that $H$ belongs to the segment $OX$. It turned out that $XH = HO = OY$. Find $\angle BAC$. [i](Proposed by Oleksii Masalitin)[/i]

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?

Square $ABCD$ is cut by a line segment $EF$ into two rectangles $AEFD$ and $BCFE$. The lengths of the sides of each of these rectangles are positive integers. It is known that the area of the rectangle $AEFD$ is $30$ and it is larger than the area of the rectangle $BCFE$. Find the area of square $ABCD$. [i]Proposed by Bogdan Rublov[/i]

Given that $ABC$ is a triangle where $AB < AC$. On the half-lines $BA$ and $CA$ we take points $F$ and $E$ respectively such that $BF = CE = BC$. Let $M,N$ and $H$ be the mid-points of the segments $BF,CE$ and $BC$ respectively and $K$ and $O$ be the circumcenters of the triangles $ABC$ and $MNH$ respectively. We assume that $OK$ cuts $BE$ and $HN$ at the points $A_1$ and $B_1$ respectively and that $C_1$ is the point of intersection of $HN$ and $FE$. If the parallel line from $A_1$ to $OC_1$ cuts the line $FE$ at $D$ and the perpendicular from $A_1$ to the line $DB_1$ cuts $FE$ at the point $M_1$, prove that $E$ is the orthocenter of the triangle $A_1OM_1$.