Quadrilateral $ABCD$ inscribed in circle with center $O$. Let lines $AD$ and $BC$ intersects at $M$, lines $AB$ and $CD$- at $N$, lines $AC$ and $BD$ -at $P$, lines $OP$ and $MN$ at $K$. Proved that $ \angle AKP = \angle PKC$. As I know, this problem was very short solution by polars, but in olympiad for this solution gives maximum 4 balls (in marking schemes written, that needs to prove all theorems about properties of polars)

Let $P(z) = z^8 + (4\sqrt{3} + 6) z^4 - (4\sqrt{3}+7)$. What is the minimum perimeter among all the 8-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$? $ \textbf{(A)}\ 4\sqrt{3}+4 \qquad \textbf{(B)}\ 8\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{2}+3\sqrt{6} \qquad \textbf{(D)}\ 4\sqrt{2}+4\sqrt{3} \qquad $ $\textbf{(E)}\ 4\sqrt{3}+6 $

Given $4$ lines in the plane such that there are not $2$ parallel to each other or no $3$ concurrent, we consider the following $ 8$ segments: in each line we have $2$ consecutive segments determined by the intersections with the other three lines. Prove that: a) The lengths of the $ 8$ segments cannot be the numbers $1, 2, 3,4, 5, 6, 7, 8$ in some order. b) The lengths of the $ 8$ segments can be $ 8$ different integers.

Let $ABCD$ be a convex quadrilateral with $AB = AD$ and $CB = CD$. The bisector of $\angle BDC$ intersects $BC$ at $L$, and $AL$ intersects $BD$ at $M$, and it is known that $BL = BM$. Determine the value of $2\angle BAD + 3\angle BCD$.

Using $ AB$ and $ AC$ as diameters, two semi-circles are constructed respectively outside the acute triangle $ ABC$. $ AH \perp BC$ at $ H$, $ D$ is any point on side $ BC$ ($ D$ is not coinside with $ B$ or $ C$ ), through $ D$, construct $ DE \parallel AC$ and $ DF \parallel AB$ with $ E$ and $ F$ on the two semi-circles respectively. Show that $ D$, $ E$, $ F$ and $ H$ are concyclic.

Let $ABC$ be a triangle and $D$ a point inside the triangle, located on the median of $A$. Prove that if $\angle BDC = 180^o - \angle BAC$, then $AB \cdot CD = AC \cdot BD$.

What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?

Let $K$ be a point in the interior of an acute triangle $ABC$ and $ARBPCQ$ be a convex hexagon whose vertices lie on the circumcircle $\Gamma$ of the triangle $ABC.$ Let $A_1$ be the second point where the circle passing through $K$ and tangent to $\Gamma$ at $A$ intersects the line $AP.$ The points $B_1$ and $C_1$ are defined similarly. Prove that \[ \min\left\{\frac{PA_1}{AA_1}, \: \frac{QB_1}{BB_1}, \: \frac{RC_1}{CC_1}\right\} \leq 1.\]

a) Given a circle with two inscribed triangles $T_1$ and $T_2$. The vertices of $T_1$ are the midpoints of the arcs with the ends in the vertices of $T_2$. Consider a hexagon -- the intersection of $T_1$ and $T_2$. Prove that its main diagonals are parallel to $T_1$ sides and are intersecting in one point. b) The segment, that connects the midpoints of the arcs $AB$ and $AC$ of the circle circumscribed around the $ABC$ triangle, intersects $[AB]$ and $[AC]$ sides in $D$ and $K$ points. Prove that the points $A,D,K$ and $O$ -- the centre of the circle -- are the vertices of a diamond.

Given a cyclic quadrilateral $ABCD$, define $E$ as $AD \cap BC$ and $F$ as $AB \cap CD$. Let $\Omega_A$ be the circle passing through $A, D$ and tangent to $AB$, and let its center be $O_A$. Let $\Gamma_B$ be the circle passing through $B, C$ and tangent to $AB$, and let its center be $O_B$. Let $\Gamma_C$ be the circle passing through $B, C$ and tangent to $CD$, and let its center be $O_C$. Let $\Omega_D$ be the circle passing through $A, D$ and tangent to $CD$, and let its center be $O_D$. Prove that $O_AO_BO_CO_D$ is cyclic, and prove that it's center lies on $EF$.

We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots,5$ is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.

A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width? [asy]unitsize(15mm); defaultpen(linewidth(.8pt)); path P=(-sqrt(2)/2,1)--(0,1-sqrt(2)/2)--(sqrt(2)/2,1); path Pc=(-sqrt(2)/2,1)--(0,1-sqrt(2)/2)--(sqrt(2)/2,1)--cycle; path S=(-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle; fill(S,gray); for(int i=0;i<4;++i) { fill(rotate(90*i)*Pc,white); draw(rotate(90*i)*P); } draw(S);[/asy]$ \textbf{(A)}\ 2\sqrt {2} \plus{} 1 \qquad \textbf{(B)}\ 3\sqrt {2}\qquad \textbf{(C)}\ 2\sqrt {2} \plus{} 2 \qquad \textbf{(D)}\ 3\sqrt {2} \plus{} 1 \qquad \textbf{(E)}\ 3\sqrt {2} \plus{} 2$

[b]p1.[/b] In triangle $ABC$, the median $BM$ is drawn. The length $|BM| = |AB|/2$. The angle $\angle ABM = 50^o$. Find the angle $\angle ABC$. [b]p2.[/b] Is there a positive integer $n$ which is divisible by each of $1, 2,3,..., 2018$ except for two numbers whose difference is$ 7$? [b]p3.[/b] Twenty numbers are placed around the circle in such a way that any number is the average of its two neighbors. Prove that all of the numbers are equal. [b]p4.[/b] A finite number of frogs occupy distinct integer points on the real line. At each turn, a single frog jumps by $1$ to the right so that all frogs again occupy distinct points. For some initial configuration, the frogs can make $n$ moves in $m$ ways. Prove that if they jump by $1$ to the left (instead of right) then the number of ways to make $n$ moves is also $m$. [b]p5.[/b] A square box of chocolates is divided into $49$ equal square cells, each containing either dark or white chocolate. At each move Alex eats two chocolates of the same kind if they are in adjacent cells (sharing a side or a vertex). What is the maximal number of chocolates Alex can eat regardless of distribution of chocolates in the box? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given triangle $ ABC$. The incircle of triangle $ ABC$ is tangent to $ BC,CA,AB$ at $ D,E,F$ respectively. Construct point $ G$ on $ EF$ such that $ DG$ is perpendicular to $ EF$. Prove that $ \frac{FG}{EG} \equal{} \frac{BF}{CE}$.

Let $ABC$ be a triangle such that $\angle BAC = 90^{\circ}$ and $AB = AC$. Let $M$ be the midpoint of $BC$. A point $D \neq A$ is chosen on the semicircle with diameter $BC$ that contains $A$. The circumcircle of triangle $DAM$ cuts lines $DB$ and $DC$ at $E$ and $F$ respectively. Show that $BE = CF$.

Let $AB$ be a diameter of a circle; let $t_1$ and $t_2$ be the tangents at $A$ and $B$, respectively; let $C$ be any point other than $A$ on $t_1$; and let $D_1D_2. E_1E_2$ be arcs on the circle determined by two lines through $C$. Prove that the lines $AD_1$ and $AD_2$ determine a segment on $t_2$ equal in length to that of the segment on $t_2$ determined by $AE_1$ and $AE_2.$

A cube with an edge of $ n $ cm is divided in $ n^3 $ mini-cubes with edges of legth $ 1 $ cm. Only the exterior of the cube is colored. [b]a)[/b] How many of the mini-cubes haven't any colored face? [b]b)[/b] How many of the mini-cubes have only one colored face? [b]c)[/b] How many of the mini-cubes have, at least, two colored faces? [b]d)[/b] If we draw with blue all the diagonals of all the faces of the cube, upon how many mini-cubes do we find blue segments?

The excircle of right-angled triangle $ABC$ ($\angle B =90^o$) touches side $BC$ at point $A_1$ and touches line $AC$ in point $A_2$. Line $A_1A_2$ meets the incircle of $ABC$ for the first time at point $A'$, point $C'$ is defined similarly. Prove that $AC||A'C'$.

Let $ABCD$ be a rectangle. Let $K,L$ be the midpoints of $BC, AD$ respectively. From point $B$ the perpendicular line on $AK$, intersects $AK$ at point $E$ and $CL$ at point $Z$. a) Prove that the quadrilateral $AKZL$ is an isosceles trapezoid b) Prove that $2S_{ABKZ}=S_{ABCD}$ c) If quadrilateral $ABCD$ is a square of side $a$, calculate the area of the isosceles trapezoid $AKZL$ in terms of side $BC=a$

Convex quadrilateral $ABCD$ has sides $AB=BC=7$, $CD=5$, and $AD=3$. Given additionally that $m\angle ABC=60^\circ$, find $BD$.

Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$

In the land of knights (who always tell the truth) and liars (who always lie), 10 people sit at a round table, each at a vertex of an inscribed regular 10-gon, at least one of them is a liar. A traveler can stand at any point outside the table and ask the people: ”What is the distance from me to the nearest liar at the table?” After that each person at the table gives him an answer. What is the minimal number of questions the traveler has to ask to determine which people at the table are liars? (Both the people at the table and the traveler should be considered as points, and everyone can compute the distance between any two points) (10 points) Maxim Didin

Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$. Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

Let $G, H$ be the centroid and orthocentre of $\vartriangle ABC$ which has an obtuse angle at $\angle B$. Let $\omega$ be the circle with diameter $AG$. $\omega$ intersects $\odot(ABC)$ again at $L \ne A$. The tangent to $\omega$ at $L$ intersects $\odot(ABC)$ at $K \ne L$. Given that $AG = GH$, prove $\angle HKG = 90^o$ . [i]Sam Bealing, United Kingdom[/i]