Three different collinear points are given. What is the number of isosceles triangles such that these points are their circumcenter, incenter and excenter (in some order)?

Let $ABC$ be a triangle with $AB \ne AC$ and $ I$ its incenter. Let $M$ be the midpoint of the side $BC$ and $D$ the projection of $I$ on $BC.$ The line $AI$ intersects the circle with center $M$ and radius $MD$ at $P$ and $Q.$ Prove that $\angle BAC + \angle PMQ = 180^{\circ}.$

In acute triangle $ABC$, points $D$ and $E$ are the feet of the perpendiculars from $A$ to $BC$ and $B$ to $CA$, respectively. Segment $AD$ is a diameter of circle $\omega$. Circle $\omega$ intersects sides $AC$ and $AB$ at $F$ and $G$ (other than $A$), respectively. Segment $BE$ intersects segments $GD$ and $GF$ at $X$ and $Y$ respectively. Ray $DY$ intersects side $AB$ at $Z$. Prove that lines $XZ$ and $BC$ are perpendicular

Among the $5$-sided polygons, as many vertices as possible collinear , that is, belonging to a single line, is three, as shown below. What is the largest number of collinear vertices a $12$-sided polygon can have? [img]https://cdn.artofproblemsolving.com/attachments/1/1/53d419efa4fc4110730a857ae6988fc923eb13.png[/img] Attention: In addition to drawing a $12$-sided polygon with the maximum number of vertices collinear , remember to show that there is no other $12$-sided polygon with more vertices collinear than this one.

Determine all bounded closed subsets $F$ of the plane with the following property: $F$ consists of at least two points and always contains two points $A$ and $B$ as well as at least one of the two semicircular arcs over the segment $AB$. Definitions: A subset of the $F$ of the plane is said to be closed if: For every point $P$ of the plane that is not an element of $F$ , there is a (non-degenerate) disc with center $P$ that has no elements of $F$.

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

In triangle $ABC$, $D$, $E$ and $F$ are the points of tangency of incircle with the center of $I$ to $BC$, $CA$ and $AB$ respectively. Let $M$ be the foot of the perpendicular from $D$ to $EF$. $P$ is on $DM$ such that $DP = MP$. If $H$ is the orthocenter of $BIC$, prove that $PH$ bisects $ EF$.

Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?

Let $M $be the point of intersection of the diagonals of the inscribed quadrilateral $ABCD$. Prove that if $AB = AM,$ then a line passing through $M$ perpendicular to $AD$ passes through the midpoint of the arc $BC$.

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

An ant is at vertex of a cube. Every $10$ minutes it moves to an adjacent vertex along an edge. If $N$ is the number of one hour journeys that end at the starting vertex, find the sum of the squares of the digits of $N$.

Alice and Bob stand atop two different towers in the Arctic. Both towers are a positive integer number of meters tall and are a positive (not necessarily integer) distance away from each other. One night, the sea between them has frozen completely into reflective ice. Alice shines her flashlight directly at the top of Bob's tower, and Bob shines his flashlight at the top of Alice's tower by first reflecting it off the ice. The light from Alice's tower travels $16$ meters to get to Bob's tower, while the light from Bob's tower travels $26$ meters to get to Alice's tower. Assuming that the lights are both shown from exactly the top of their respective towers, what are the possibilities for the height of Alice's tower?

$ABC$ is a triangle with incenter $I$. The foot of the perpendicular from $I$ to $BC$ is $D$, and the foot of the perpendicular from $I$ to $AD$ is $P$. Prove that $\angle BPD = \angle DPC$. [i]Alex Zhu.[/i]

Triangle $\Delta ABC$ is inscribed in circle $\Omega$. The tangency point of $\Omega$ and the $A$-mixtilinear circle of $\Delta ABC$ is $T$. Points $E$, $F$ were chosen on $AC$, $AB$ respectively so that $EF\parallel BC$ and $(TEF)$ is tangent to $\Omega$. Let $\omega$ denote the $A$-excircle of $\Delta AEF$, which is tangent to sides $EF$, $AE$, $AF$ at $K$, $Y$, $Z$ respectively. Line $AT$ intersects $\omega$ at two points $P$, $Q$ with $P$ between $A$ and $Q$. Let $QK$ and $YZ$ intersect at $V$, and let the tangent to $\omega$ at $P$ and the tangent to $\Omega$ at $T$ intersect at $U$. Prove that $UV\parallel BC$.

Let $ABCD$ be a cyclic quadrilateral and let its diagonals $AC$ and $BD$ cross at $X$. Let $I$ be the incenter of $XBC$, and let $J$ be the center of the circle tangent to the side $BC$ and the extensions of sides $AB$ and $DC$ beyond $B$ and $C$. Prove that the line $IJ$ bisects the arc $BC$ of circle $ABCD$, not containing the vertices $A$ and $D$ of the quadrilateral.

Consider a circle $K(O,r)$ and a point $A$ outside $K.$ A line $\epsilon$ different from $AO$ cuts $K$ at $B$ and $C,$ where $B$ lies between $A$ and $C.$ Now the symmetric line of $\epsilon$ with respect to axis of symmetry the line $AO$ cuts $K$ at $E$ and $D,$ where $E$ lies between $A$ and $D.$ Show that the diagonals of the quadrilateral $BCDE$ intersect in a fixed point.

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Snowhite wrote on a piece of paper a whole number greater than $1$ and multiplied it by itself. She obtained a number, all digits of which are $1$: $n^2 = 111...111$ Does she know how to multiply? [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a bishop on an arbitrary square. Then the second player can put another bishop on a free square that is not controlled by the first bishop. Then the first player can put a new bishop on a free square that is not controlled by the bishops on the board. Then the second player can do the same, etc. A player who cannot put a new bishop on the board loses the game. Who has a winning strategy? [b]p4.[/b] Four girls Marry, Jill, Ann and Susan participated in the concert. They sang songs. Every song was performed by three girls. Mary sang $8$ songs, more then anybody. Susan sang $5$ songs less then all other girls. How many songs were performed at the concert? [b]p5.[/b] Pinocchio has a $10\times 10$ table of numbers. He took the sums of the numbers in each row and each such sum was positive. Then he took the sum of the numbers in each columns and each such sum was negative. Can you trust Pinocchio's calculations? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$AD$ and $BE$ are the altitudes of the triangle $ABC$. It turned out that the point $C'$, symmetric to the vertex $C$ wrt to the midpoint of the segment $DE$, lies on the side $AB$. Prove that $AB$ is tangent to the circle circumscribed around the triangle $DEC'$.

Suppose that two circles $\alpha, \beta$ with centers $P,Q$, respectively , intersect orthogonally at $A$,$B$. Let $CD$ be a diameter of $\beta$ that is exterior to $\alpha$. Let $E,F$ be points on $\alpha$ such that $CE,DF$ are tangent to $\alpha$ , with $C,E$ on one side of $PQ$ and $D,F$ on the other side of $PQ$. Let $S$ be the intersection of $CF,AQ$ and $T$ be the intersection of $DE,QB$. Prove that $ST$ is parallel to $CD$ and is tangent to $\alpha$

An octahedron (a solid with 8 triangular faces) has a volume of $1040$. Two of the spatial diagonals intersect, and their plane of intersection contains four edges that form a cyclic quadrilateral. The third spatial diagonal is perpendicularly bisected by this plane and intersects the plane at the circumcenter of the cyclic quadrilateral. Given that the side lengths of the cyclic quadrilateral are $7, 15, 24, 20$, in counterclockwise order, the sum of the edge lengths of the entire octahedron can be written in simplest form as $a/b$. Find $a + b$.

The incircle of triangle $ABC$ touches $BC$ at $D$ and $AB$ at $F$, intersects the line $AD$ again at $H$ and the line $CF$ again at $K$. Prove that $\frac{FD\times HK}{FH\times DK}=3$

Let $ABCD$ be a cyclic quadrilateral such that $|AD| =|BD|$ and let $M$ be the intersection of its diagonals. Furthermore, let $N$ be the second intersection of the diagonal $AC$ with the circle passing through points $B, M$ and the center of the circle inscribed in triangle $BCM$. Prove that $AN \cdot NC = CD \cdot BN$

Points $E, F$ are selected on sides $AC, AB$ respectively of triangle $ABC$ with $AC=AB$ so that $AE = BF$. Point $D$ is chosen so that $D, A$ are in the same halfplane with respect to line $EF$, and $\triangle DFE \sim \triangle ABC$. Lines $EF, BC$ intersect at point $K$. Prove that the line $DK$ is tangent to the circumscribed circle of $\triangle ABC$. [i]Proposed by Fedir Yudin[/i]

For $A\subset\mathbb Z$ and $a,b\in\mathbb Z$. We define $aA+b: =\{ax+b|x\in A\}$. If $a\neq0$ then we calll $aA+b$ and $A$ to similar sets. In this question the Cantor set $C$ is the number of non-negative integers that in their base-3 representation there is no $1$ digit. You see \[C=(3C)\dot\cup(3C+2)\ \ \ \ \ \ (1)\] (i.e. $C$ is partitioned to sets $3C$ and $3C+2$). We give another example $C=(3C)\dot\cup(9C+6)\dot\cup(3C+2)$. A representation of $C$ is a partition of $C$ to some similiar sets. i.e. \[C=\bigcup_{i=1}^{n}C_{i}\ \ \ \ \ \ (2)\] and $C_{i}=a_{i}C+b_{i}$ are similar to $C$. We call a representation of $C$ a primitive representation iff union of some of $C_{i}$ is not a set similar and not equal to $C$. Consider a primitive representation of Cantor set. Prove that a) $a_{i}>1$. b) $a_{i}$ are powers of 3. c) $a_{i}>b_{i}$ d) (1) is the only primitive representation of $C$.

Let $ABCD$ be an isosceles trapezium inscribed in circle $\omega$, such that $AB||CD$. Let $P$ be a point on the circle $\omega$. Let $H_1$ and $H_2$ be the orthocenters of triangles $PAD$ and $PBC$ respectively. Prove that the length of $H_1H_2$ remains constant, when $P$ varies on the circle.