[b]a)[/b] Among the $21$ pairwise distances between the $7$ points of the plane, prove that one and the same number occurs not more than $12$ times. [b]b)[/b] Find a maximum number of times may meet the same number among the $15$ pairwise distances between $6$ points of the plane.

Let $\triangle AB\varGamma$ be an acute-angled triangle with $AB < A\varGamma$, and let $O$ be the center of the circumcircle of the triangle. On the sides $AB$ and $A \varGamma$ we select points $T$ and $P$ respectively such that $OT=OP$. Let $M,K$ and $\varLambda$ be the midpoints of $PT,PB$ and $\varGamma T$ respectively. Prove that $\angle TMK = \angle M\varLambda K$.

$ABC$ is a right-angled triangle, with $\angle ABC = 90^{\circ}$. $B'$ is the reflection of $B$ over $AC$. $M$ is the midpoint of $AC$. We choose $D$ on $\overrightarrow{BM}$, such that $BD = AC$. Prove that $B'C$ is the angle bisector of $\angle MB'D$. NOTE: An important condition not mentioned in the original problem is $AB<BC$. Otherwise, $\angle MB'D$ is not defined or $B'C$ is the external bisector.

Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$. The midpoint of $BC$ is denoted by $M$. $AH$ intersects the circumcircle at $D \ne A$ and $DM$ intersects circumcircle of $\vartriangle ABC$ at $T\ne D$. Now, assume the reflection points of $M$ with respect to $AB,AC,AH$ are $F,E,S$. Show that the midpoints of $BE,CF,AM,TS$ are concyclic. [img]https://3.bp.blogspot.com/-v7D_A66nlD0/XnYNJussW9I/AAAAAAAALeQ/q6DMQ7w6QtI5vLwBcKqp4010c3XTCj3BgCK4BGAYYCw/s1600/imoc2019g2.png[/img]

Let $ABC$ be a triangle with incircle centered at $I$, tangent to sides $AC$ and $AB$ at $E$ and $F$ respectively. Let $N$ be the midpoint of major arc $BAC$. Let $IN$ intersect $EF$ at $K$, and $M$ be the midpoint of $BC$. Prove that $KM\perp EF$.

Let $N$ be the midpoint of arc $ABC$ of the circumcircle of triangle $ABC$, let $M$ be the midpoint of $AC$ and let $I_1, I_2$ be the incentres of triangles $ABM$ and $CBM$. Prove that points $I_1, I_2, B, N$ lie on a circle. [i]M. Kungojin[/i]

Three rays are going out from point $O$ in space, forming pairwise angles $\alpha, \beta$ and $\gamma$ with $0^o<\alpha \le \beta \le \gamma <180^o$. Prove that $\sin \frac{\alpha}{2}+ \sin \frac{\beta}{2} > \sin \frac{\gamma}{2}$.

Let $ABC$ be an acute triangle. The altitudes $BE$ and $CF$ intersect at the orthocenter $H$, and point $O$ denotes the circumcenter. Point $P$ is chosen so that $\angle APH = \angle OPE = 90^{\circ}$, and point $Q$ is chosen so that $\angle AQH = \angle OQF = 90^{\circ}$. Lines $EP$ and $FQ$ meet at point $T$. Prove that points $A$, $T$, $O$ are collinear.

Quadrilateral $ABCD$ is a parallelogram, and $E$ is the midpoint of the side $\overline{AD}$. Let $F$ be the intersection of lines $EB$ and $AC$. What is the ratio of the area of quadrilateral $CDEF$ to the area of triangle $CFB$? $\textbf{(A) } 5 : 4 \qquad \textbf{(B) } 4 : 3 \qquad \textbf{(C) } 3 : 2 \qquad \textbf{(D) } 5 : 3 \qquad \textbf{(E) } 2 : 1$

$PQ$ is a chord of parabola $y^2=2px(p>0)$ and $PQ$ pass its focus $F$. Line $l$ is its directrix. Projection of $PQ$ on $l$ is $MN$. The area of curved surface that $PQ$ rotate around $l$ is $S_1$, the area of spherical surface of the ball with diameter of $MN$ is $S_2$, then $\text{(A)}S_1>S_2\qquad\text{(B)}S_1<S_2\qquad\text{(C)}S_1\geq S_2\qquad\text{(D)}$ Not sure

In a convex quadrilateral $ABCD$, the diagonals are perpendicular to each other and they intersect at $E$. Let $P$ be a point on the side $AD$ which is different from $A$ such that $PE=EC.$ The circumcircle of triangle $BCD$ intersects the side $AD$ at $Q$ where $Q$ is also different from $A$. The circle, passing through $A$ and tangent to line $EP$ at $P$, intersects the line segment $AC$ at $R$. If the points $B, R, Q$ are concurrent then show that $\angle BCD=90^{\circ}$.

Let $\omega$ be a circle and let $A,B,C,D,E$ be five points on $\omega$ in this order. Define $F=BC\cap DE$, such that the points $F$ and $A$ are on opposite sides, with regard to the line $BE$ and the line $AE$ is tangent to the circumcircle of the triangle $BFE$. a) Prove that the lines $AC$ and $DE$ are parallel b) Prove that $AE=CD$

Let $\Gamma_1$ be a circle with centre $A$ and $\Gamma_2$ be a circle with centre $B$, with $A$ lying on $\Gamma_2$. On $\Gamma_2$ there is a (variable) point $P$ not lying on $AB$. A line through $P$ is a tangent of $\Gamma_1$ at $S$, and it intersects $\Gamma_2$ again in $Q$, with $P$ and $Q$ lying on the same side of $AB$. A different line through $Q$ is tangent to $\Gamma_1$ at $T$. Moreover, let $M$ be the foot of the perpendicular to $AB$ through $P$. Let $N$ be the intersection of $AQ$ and $MT$. Show that $N$ lies on a line independent of the position of $P$ on $\Gamma_2$.

Let $ABCD$ be a convex quadrilateral. Construct the points $P,Q,R,$ and $S$ on continue of $AB,BC,CD,$ and $DA$, respectively, such that \[BP=CQ=DR=AS.\] Show that if $PQRS$ is a square, then $ABCD$ is also a square.

The diagram below shows an isosceles triangle with base $21$ and height $28$. Inscribed in the triangle is a square. Find the area of the shaded region inside the triangle and outside of the square. [asy] size(170); defaultpen(linewidth(0.8)); draw((0,0)--(1,1)); pair A=(5,0),B=(-5,0),C=(0,14), invis[]={(1,2),(-1,2)}; pair intsquare[]={extension(origin,invis[0],A,C),extension(origin,invis[1],B,C)}; path triangle=A--B--C--cycle,square=(intsquare[0]--intsquare[1]--(intsquare[1].x,0)--(intsquare[0].x,0)--cycle); fill(triangle,gray); unfill(square); draw(triangle^^square); [/asy]

A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once? $ \textbf{(A) } \frac {1}{2187} \qquad \textbf{(B) } \frac {1}{729} \qquad \textbf{(C) } \frac {2}{243} \qquad \textbf{(D) } \frac {1}{81} \qquad \textbf{(E) } \frac {5}{243}$

As shown in figure , the inscribed circle of $ABC$ is intersects $BC$, $CA$, $AB$ at points $D$, $E$, $F$, repectively, and $P$ is a point inside the inscribed circle. The line segments $PA$, $PB$ and $PC$ intersect respectively the inscribed circle at points $X$, $Y$ and $Z$. Prove that the three lines $XD$, $YE$ and $ZF$ have a common point. [img]https://cdn.artofproblemsolving.com/attachments/e/9/bbfb0394b9db7aa5fb1e9a869134f0bca372c1.png[/img]

Given a triangle $ABC$ with $M$ the midpoint of minor arc $BC$. Let $H$ be the feet of altitude from $A$ to $BC$. Let $S$ and $T$ be the reflections of $B$ and $C$ with respect to line $AM$. Suppose the circle $(HST)$ meets $BC$ again at a point $P$. Prove that $\angle AMP = 90^\circ$. [i](Proposed by Tan Rui Xuen)[/i]

Two given circles $k_1$ and $k_2$ intersect at points $P$ and $Q$. Construct a segment $AB$ through $P$ with the endpoints at $k_1$ and $k_2$ for which $AP \cdot PB$ is maximal.

A circle is inscribed in a triangle $ABC$ with sides $a,b,c$. Tangents to the circle parallel to the sides of the triangle are contructe. Each of these tangents cuts off a triagnle from $\triangle ABC$. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of $a,b,c$).

Let $\Gamma$ be the excircle of triangle $ABC$ opposite to the vertex $A$ (namely, the circle tangent to $BC$ and to the prolongations of the sides $AB$ and $AC$ from the part $B$ and $C$). Let $D$ be the center of $\Gamma$ and $E$, $F$, respectively, the points in which $\Gamma$ touches the prolongations of $AB$ and $AC$. Let $J$ be the intersection between the segments $BD$ and $EF$. Prove that $\angle CJB$ is a right angle.

An isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ is given. Circles with centers $O_1$ and $O_2$ are inscribed in triangles $ABC$ and $ABD$. Prove that line $O_1O_2$ is perpendicular on $BC$.

In a parallelogram $ABCD$, points $E$ and $F$ are the midpoints of $AB$ and $BC$, respectively, and $P$ is the intersection of $EC$ and $FD$. Prove that the segments $AP,BP,CP$ and $DP$ divide the parallelogram into four triangles whose areas are in the ratio $1 : 2 : 3 : 4$.

All sides of triangle $ABC$ are different. On rays $B A$ and $C A$ the segments $B K$ and $CM$ are laid out, equal to side $BC$. Let us denote by $x$ the length of the segment $KM$. In the same way, by plotting the side $AC$ on the rays $AB$ and $CB$ from $A$ and $C$, we obtain a segment of length $y$, and by plotting the side AB on the rays $AC$ and $BC$, we obtain a segment of length $z$. a) Prove that a triangle can be formed from the segments $x$, $y$ and $z$, and this triangle is similar to triangle $ABC$. b) Find the radius of the circumcircle of a triangle with sides $x$, $y$ and $z$, if the radii of the circumscribed and inscribed circles of triangle $ABC$ are equal to $R$ and $r$ respectively.

A triangle $ ABC$ is inscribed in a circle $ C(O,R)$ and has incenter $ I$. Lines $ AI,BI,CI$ meet the circumcircle $ (O)$ of triangle $ ABC$ at points $ D,E,F$ respectively. The circles with diameter $ ID,IE,IF$ meet the sides $ BC,CA, AB$ at pairs of points $ (A_1,A_2), (B_1, B_2), (C_1, C_2)$ respectively. Prove that the six points $ A_1,A_2, B_1, B_2, C_1, C_2$ are concyclic. Babis