Incircle of triangle $ABC$ tangent to $AB$ and $AC$ on $E$ and $F$ respectively. If $X$ is the midpoint of $EF$, prove $\angle BXC > 90^{\circ}$

Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that \[ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}. \]

Consider the square grid with $A=(0,0)$ and $C=(n,n)$ at its diagonal ends. Paths from $A$ to $C$ are composed of moves one unit to the right or one unit up. Let $C_n$ (n-th catalan number) be the number of paths from $A$ to $C$ which stay on or below the diagonal $AC$. Show that the number of paths from $A$ to $C$ which cross $AC$ from below at most twice is equal to $C_{n+2}-2C_{n+1}+C_n$

Let $N$ be the midpoint of arc $ABC$ of the circumcircle of $\Delta ABC$, and $NP$, $NT$ be the tangents to the incircle of this triangle. The lines $BP$ and $BT$ meet the circumcircle for the second time at points $P_1$ and $T_1$ respectively. Prove that $PP_1 = TT_1$.

Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.

Let $ABC$ be a scalene triangle and $I$ be its incenter. Suppose the incircle $\omega$ touches $BC$ at a point $D$, and $N$ lies on $\omega$ such that $ND$ is a diameter of $\omega$. Let $X$ and $Y$ be points on lines $AC$ and $AB$ respectively such that $\angle BIX = \angle CIY = 90^\circ$. Let $V$ be the feet of perpendicular from $I$ onto line $XY$. Prove that the points $I$, $V$, $A$, $N$ are concyclic. [i](Proposed by Ivan Chan Guan Yu)[/i]

Cyclic quadrilateral $ABCD$ with circumcenter $O$ is given. Point $P$ is the intersection of diagonals $AC$ and $BD$. Let $M$ and $N$ be the midpoint of the sides $AD$ and $BC$, respectively. Suppose that $\omega_1$, $\omega_2$ and $\omega_3$ be the circumcircle of triangles $ADP$, $BCP$ and $OMN$, respectively. The intersection point of $\omega_1$ and $\omega_3$, which is not on the arc $APD$ of $\omega_1$, is $E$ and the intersection point of $\omega_2$ and $\omega_3$, which is not on the arc $BPC$ of $\omega_2$, is $F$. Prove that $OF=OE$. Proposed by Seyed Amirparsa Hosseini Nayeri

Prove that for all $X > 1$, there exists a triangle whose sides have lengths $P_1(X) = X^4+X^3+2X^2+X+1, P_2(X) = 2X^3+X^2+2X+1$, and $P_3(X) = X^4-1$. Prove that all these triangles have the same greatest angle and calculate it.

Let $ABC$ be an acute triangle. Tangents on the circumscribed circle of triangle $ABC$ at points $B$ and $C$ intersect at point $T$. Let $D$ and $E$ be a foot of the altitudes from $T$ onto $AB$ and $AC$ and let $M$ be the midpoint of $BC$. Prove: A) Prove that $M$ is the orthocenter of the triangle $ADE$. B) Prove that $TM$ cuts $DE$ in half.

Let $ ABC$ be a triangle with $ \left|AB\right| \equal{} 14$, $ \left|BC\right| \equal{} 12$, $ \left|AC\right| \equal{} 10$. Let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[BC\right]$ such that $ \left|AD\right| \equal{} 4$ and $ Area\left(ABC\right) \equal{} 2Area\left(CDE\right)$. Find $ Area\left(ABE\right)$. $\textbf{(A)}\ 4\sqrt {6} \qquad\textbf{(B)}\ 6\sqrt {2} \qquad\textbf{(C)}\ 3\sqrt {6} \qquad\textbf{(D)}\ 4\sqrt {2} \qquad\textbf{(E)}\ 4\sqrt {5}$

The bisector of angle $\angle BAC$ of triangle $ABC$ intersects arc $BC$ (not containing point $A$) of the circle circumscribed around this triangle at point $P$. Segment $AP$ is divided by side $BC$ in ratio $k$ (counting from vertex $A$). Find the perimeter of triangle $ABC$ if $BC = a$.

In a $\triangle ABC$, $\angle C=2 \angle B$. $P$ is a point in the interior of $\triangle ABC$ satisfying that $AP=AC$ and $PB=PC$. Show that $AP$ trisects the angle $\angle A$.

Let $n$ be a positive integer and let $x_1,\ldots,x_n$ be $n$ nonzero points in $\mathbb R^2$. Suppose $\langle x_i,x_j\rangle$ (scalar or dot product) is a rational number for all $i,j$ ($1\le i,j\le n$). Let $S$ denote all points of $\mathbb R^2$ of the form $\sum_{i=1}^na_ix_i$ where the $a_i$ are integers. A closed disk of radius $R$ and center $P$ is the set of points at distance at most $R$ from $P$ (includes the points distance $R$ from $P$). Prove that there exists a positive number $R$ and closed disks $D_1,D_2,\ldots$ of radius $R$ such that (a) Each disk contains exactly two points of $S$; (b) Every point of $S$ lies in at least one disk; (c) Two distinct disks intersect in at most one point.

Inside the circle are given three points that do not belong to one line. In one step it is allowed to replace one of the points with a symmetric one wrt the line containing the other two points. Is it always possible for a finite number of these steps to ensure that all three points are outside the circle?

On the circle with center 0 and radius 1 the point $A_0$ is fixed and points $A_1, A_2, \ldots, A_{999}, A_{1000}$ are distributed in such a way that the angle $\angle A_00A_k = k$ (in radians). Cut the circle at points $A_0, A_1, \ldots, A_{1000}.$ How many arcs with different lengths are obtained. ?

Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$. [i]Proposed by C.R. Pranesachar, India [/i]

Given a quadrilateral $ABCD$. $A ', B', C'$ and $D'$ are the midpoints of the sides $BC, CB, BA$ and $AB$, respectively. It is known that $AA'= CC'$, $BB'= DD'$. Is it true that $ABCD$ is a parallelogram? (M. Volchkevich)

Let $P$ be a point inside a triangle $ABC$, and $A_1B_2,B_1C_2,C_1A_2$ be segments passing through $P$ and parallel to $AB, BC, CA$ respectively, where points $A_1, A_2$ lie on $BC, B_1, B_2$ on $CA$, and $C_1,C_2$ on $AB$. Prove that \[ \text{Area}(A_1A_2B_1B_2C_1C_2) \ge \frac{1}{2}\text{Area}(ABC)\]

Triangle $ABC$ is assumed. The point $T$ is the second intersection of the symmedian of vertex $A$ with the circumcircle of the triangle $ABC$ and the point $D \neq A$ lies on the line $AC$ such that $BA=BD$. The line that at $D$ tangents to the circumcircle of the triangle $ADT$, intersects the circumcircle of the triangle $DCT$ for the second time at $K$. Prove that $\angle BKC = 90^{\circ}$(The symmedian of the vertex $A$, is the reflection of the median of the vertex $A$ through the angle bisector of this vertex).

Chord $D E$ of the circumcircle of the triangle $A B C$ intersects sides $A B$ and $B C$ in points $P$ and $Q$ respectively. Point $P$ lies between $D$ and $Q$. Angle bisectors $D F$ and $E G$ are drawn in triangles $A D P$ and $Q E C$. It turned out that the points $D$, $F, G, E$ are concyclic. Prove that the points $A, P, Q, C$ are concyclic. Azamat Mardanov

Determine the set $S$ with minimal number of points defining $7$ distinct lines

In a convex quadrilateral $ABCD$ it is given that $\angle{CAB} = 40^{\circ}, \angle{CAD} = 30^{\circ}, \angle{DBA} = 75^{\circ}$, and $\angle{DBC}=25^{\circ}$. Find $\angle{BDC}$.

In triangle $\triangle ABC$, $AB<AC$, $M,D,E$ are the midpoints of $BC$, the arcs $BAC$ and $BC$ of the circumcircle of $\triangle ABC$ respectively. The incircle of $\triangle ABC$ touches $AB$ at $F$, $AE$ meets $BC$ at $G$, and the perpendicular to $AB$ at $B$ meets segment $EF$ at $N$. If $BN=EM$, prove that $DF$ is perpendicular to $FG$.

The triangle $ABC$ has angle $A = 30^o$ and $AB = \frac{3}{4} AC$. Find the point $P$ inside the triangle which minimizes $5 PA + 4 PB + 3 PC$.

In a scalene triangle $ABC$, the incircle touches $BC, AC$ and $AB$ at $D, E, F$ respectively. Let $K$ be the foot of the perpendicular from $A$ onto $BC$, and $M$ the midpoint of $BC$. Let $AD$ intersect the incircle again at $X$, and $BE$ at $Y$. Given that $E,F,K,M$ are concyclic, prove that $AX=XY=YD$.