Let $ A,B $ be two points in a plane and let two numbers $ a,b\in (0,1) . $ For each point $ M $ that is not on the line $ AB $ consider $ P $ on the segment $ AM $ and $ N $ on $ BM $ (both excluding the extremities) such that $ BN=b\cdot BM $ and $ AP=a\cdot AM. $ Find the locus of the points $ M $ for which $ AN=BP. $

Let $P$ be a point on the circumcircle of triangle $ABC$. Let $A_1$ be the reflection of the orthocenter of triangle $PBC$ about the reflection of the perpendicular bisector of $BC$. Points $B_1$ and $C_1$ are defined similarly. Prove that $A_1,B_1,C_1$ are collinear.

Let $ABC$ be a triangle with $\angle BAC > 90 ^{\circ}$, and let $O$ be its circumcenter and $\omega$ be its circumcircle. The tangent line of $\omega$ at $A$ intersects the tangent line of $\omega$ at $B$ and $C$ respectively at point $P$ and $Q$. Let $D,E$ be the feet of the altitudes from $P,Q$ onto $BC$, respectively. $F,G$ are two points on $\overline{PQ}$ different from $A$, so that $A,F,B,E$ and $A,G,C,D$ are both concyclic. Let M be the midpoint of $\overline{DE}$. Prove that $DF,OM,EG$ are concurrent.

Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Let $D$ be a point on the line $AC$ such that $AB = AD$ and $A$ lies between $C$ and $D$. Suppose that there are two points $E \ne F$ on the circumcircle of the triangle $DBC$ such that $AE = AF = BC$. Prove that the line $EF$ passes through the circumcenter of $ABC$.

ABC is a triangle with A=90 and C=30.Let M be the midpoint of BC. Let W be a circle passing through A tangent in M to BC. Let P be the circumcircle of ABC. W is intersecting AC in N and P in M. prove that MN is perpendicular to BC.

Let $ABC$ be an acute triangle and $AD$ a height. The angle bissector of $\angle DAC$ intersects $DC$ at $E$. Let $F$ be a point on $AE$ such that $BF$ is perpendicular to $AE$. If $\angle BAE=45º$, find $\angle BFC$.

A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. THe remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $ 15$ and $ 25$ meters. What fraction of the yard is occupied by the flower beds? [asy]unitsize(2mm); defaultpen(linewidth(.8pt)); fill((0,0)--(0,5)--(5,5)--cycle,gray); fill((25,0)--(25,5)--(20,5)--cycle,gray); draw((0,0)--(0,5)--(25,5)--(25,0)--cycle); draw((0,0)--(5,5)); draw((20,5)--(25,0));[/asy]$ \textbf{(A)}\ \frac18\qquad \textbf{(B)}\ \frac16\qquad \textbf{(C)}\ \frac15\qquad \textbf{(D)}\ \frac14\qquad \textbf{(E)}\ \frac13$

3-There is given a trapezoid $ ABCD$ in the plane with $ BC\parallel{}AD$.We know that the angle bisectors of the angles of the trapezoid are concurrent at $ O$.Let $ T$ be the intersection of the diagonals $ AC,BD$.Let $ Q$ be on $ CD$ such that $ \angle OQD \equal{} 90^\circ$.Prove that if the circumcircle of the triangle $ OTQ$ intersects $ CD$ again at $ P$ then $ TP\parallel{}AD$.

Is it possible to cut a square into five squares?

In a triangle, one excircle touches side $AB$ at point $C_1$ and the other touches side $BC$ at point $A_1$. Prove that on the straight line $A_1C_1$ the constructed excircles cut out equal segments.

In order to earn her vacation spending money, Alexis helped her mother remove weeds from the garden. When she was done, she came into the house to put away her gardening gloves and change into clean clothes. On her way to her room she notices Joshua with his face to the floor in the family room, looking pretty silly. "Josh, did you know you lose IQ points for sniffing the carpet?" "Shut up. I'm $\textit{not}$ sniffing the carpet. I'm $\textit{doing something}$." "Sure, if $\textit{sniffing the carpet}$ counts as $\textit{doing something}.$" At this point Alexis stands over her twin brother grinning, trying to see how silly she can make him feel. Joshua climbs to his feet and stands on his toes to make himself a half inch taller than his sister, who is ordinarily a half inch taller than Joshua. "I'm measuring something. I'm $\textit{designing}$ something." Alexis stands on her toes too, reminding her brother that she is still taller than he. "When you're done, can you design me a dress?" "Very funny." Joshua walks to the table and points to some drawings. "I'm designing the sand castle I want to build at the beach. Everything needs to be measured out so that I can build something awesome." "And this requires sniffing carpet?" inquires Alexis, who is just a little intrigued by her brother's project. "I was imagining where to put the base of a spiral staircase. Everything needs to be measured out correctly. See, the castle walls will be in the shape of a rectangle, like this room. The center of the staircase will be $9$ inches from one of the corners, $15$ inches from another, $16$ inches from another, and some whole number of inches from the furthest corner." Joshua shoots Alexis a wry smile. The twins liked to challenge each other, and Alexis knew she had to find the distance from the center of the staircase to the fourth corner of the castle on her own, or face Joshua's pestering, which might last for hours or days. Find the distance from the center of the staircase to the furthest corner of the rectangular castle, assuming all four of the distances to the corners are described as distances on the same plane (the ground).

Let $\triangle{ABC}$ be an acute, scalene triangle with orthocenter $H$, and let $AH$ meet the circumcircle of $\triangle{ABC}$ at a point $D \neq A$. Points $E$ and $F$ are chosen on $AC$ and $AB$ such that $DE \perp AC$ and $DF \perp AB$. Show that $BE$, $CF$, and the line through $H$ parallel to $EF$ concur.

In the Cartesian plane $XOY$, there is a rhombus $ABCD$ whose side lengths are all $4$ and $|OB|=|OD|=6$, where $O$ is the origin. [b](1)[/b] Prove that $|OA|\cdot |OB|$ is a constant. [b](2)[/b] Find the locus of $C$ if $A$ is a point on the semicircle \[(x-2)^2+y^2=4 \quad (2\le x\le 4).\]

For triangle $ABC, P$ and $Q$ satisfy $\angle BPA + \angle AQC = 90^o$. It is provided that the vertices of the triangle $BAP$ and $ACQ$ are ordered counterclockwise (or clockwise). Let the intersection of the circumcircles of the two triangles be $N$ ($A \ne N$, however if $A$ is the only intersection $A = N$), and the midpoint of segment $BC$ be $M$. Show that the length of $MN$ does not depend on $P$ and $Q$.

Chords $ AB$ and $ CD$ of a circle intersect at a point $ E$ inside the circle. Let $ M$ be an interior point of the segment $ EB$. The tangent line at $ E$ to the circle through $ D$, $ E$, and $ M$ intersects the lines $ BC$ and $ AC$ at $ F$ and $ G$, respectively. If \[ \frac {AM}{AB} \equal{} t, \] find $\frac {EG}{EF}$ in terms of $ t$.

Quadrilateral $ABCD$ has the property that $AD = BD = CD$ and $\angle ADB < \angle CDB$. Let the circumcircle of $ABD$ be $O$. $O$ intersects $BC$ at $E$ and $CD$ at $F$. Next, extend $AB$ and $CD$ to intersect at a point $G$. Suppose that $BE = 1$, $EF = 3$, and $F D = 4$. Compute the perimeter of $\vartriangle ADG$.

Let $ ABCD$ be an isosceles trapezoid with $ \overline{AD}\parallel{}\overline{BC}$ whose angle at the longer base $ \overline{AD}$ is $ \dfrac{\pi}{3}$. The diagonals have length $ 10\sqrt {21}$, and point $ E$ is at distances $ 10\sqrt {7}$ and $ 30\sqrt {7}$ from vertices $ A$ and $ D$, respectively. Let $ F$ be the foot of the altitude from $ C$ to $ \overline{AD}$. The distance $ EF$ can be expressed in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

Let $ABC$ be a scalene triangle with incenter $I$ and circumcircle $\omega$. The internal and external bisectors of angle $\angle BAC$ intersect $BC$ at $D$ and $E$, respectively. Let $M$ be the point on segment $AC$ such that $MC = MB$. The tangent to $\omega$ at $B$ meets $MD$ at $S$. The circumcircles of triangles $ADE$ and $BIC$ intersect each other at $P$ and $Q$. If $AS$ meets $\omega$ at a point $K$ other than $A$, prove that $K$ lies on $PQ$. [i]Proposed by Alexandru Lopotenco (Moldova)[/i]

Consider an acute-angled triangle $ABC$ with $AB<AC$ and let $\omega$ be its circumscribed circle. Let $t_B$ and $t_C$ be the tangents to the circle $\omega$ at points $B$ and $C$, respectively, and let $L$ be their intersection. The straight line passing through the point $B$ and parallel to $AC$ intersects $t_C$ in point $D$. The straight line passing through the point $C$ and parallel to $AB$ intersects $t_B$ in point $E$. The circumcircle of the triangle $BDC$ intersects $AC$ in $T$, where $T$ is located between $A$ and $C$. The circumcircle of the triangle $BEC$ intersects the line $AB$ (or its extension) in $S$, where $B$ is located between $S$ and $A$. Prove that $ST$, $AL$, and $BC$ are concurrent. $\text{Vangelis Psychas and Silouanos Brazitikos}$

Is there exist four points on plane, such that distance between any two of them is integer odd number? May be it is geometry or number theory or combinatoric, I don't know, so it here :blush:

Circle $k_1$ has radius $10$, externally touching circle $k_2$ with radius $18$. Circle $k_3$ touches both circles, as well as the line $e$ determined by their centres. Let $k_4$ be the circle touching $k_2$ and $k_3$ externally (other than $k_1$) whose center lies on line $e$. What is the radius of $k_4$?

Given a line segment $AB$ in the plane, find all possible points $C$ such that in the triangle $ABC$, the altitude from $A$ and the median from $B$ have the same length.

On sides $CB$ and $CD$ of square $ABCD$ are chosen points $M$ and $K$ so that the perimeter of triangle $CMK$ equals double the side of the square. Find angle $\angle MAK$.

Let $ ABC$ be an isoceles triangle with $ AC\equal{}BC$. A point $ P$ lies inside $ ABC$ such that \[ \angle PAB \equal{} \angle PBC, \angle PAC \equal{} \angle PCB.\] Let $ M$ be the midpoint of $ AB$ and $ K$ be the intersection of $ BP$ and $ AC$. Prove that $ AP$ and $ PK$ trisect $ \angle MPC$.

$(BEL 4)$ Let $O$ be a point on a nondegenerate conic. A right angle with vertex $O$ intersects the conic at points $A$ and $B$. Prove that the line $AB$ passes through a fixed point located on the normal to the conic through the point $O.$