In acute, scalene Triangle $ABC$, $H$ is orthocenter,$ BD$ and $CE$ are heights. $X,Y$ are reflection of $A$ from $D$,$E$ respectively such that the points$ X,Y$ are on segments $DC$ and $EB$. The intersection of circles $ HXY$ and $ADE$ is $F.$ ( $F \neq H$). Prove that$ AF$ intersects middle point of $BC$. ( $M$ in the diagram is Midpoint of $BC$)

Let $ABCDEF$ be a regular hexagon with side length $a$. At point $A$, the perpendicular $AS$, with length $2a\sqrt{3}$, is erected on the hexagon's plane. The points $M, N, P, Q,$ and $R$ are the projections of point $A$ on the lines $SB, SC, SD, SE,$ and $SF$, respectively. [list=a] [*]Prove that the points $M, N, P, Q, R$ lie on the same plane. [*]Find the measure of the angle between the planes $(MNP)$ and $(ABC)$.[/list]

Let $ABC$ be a triangle and let $O$ be the centre of its circumscribed circle. Points $X, Y$ which are neither of the points $A, B$ or $C$, lie on the circumscribed circle and are so that the angles $XOY$ and $BAC$ are equal (with the same orientation). Show that the orthocentre of the triangle that is formed by the lines $BY, CX$ and $XY$ is a fixed point.

Let $ABC$ be a non isosceles triangle inscribed in a circle $(O)$ and $BE, CF$ are two angle bisectors intersect at $I$ with $E$ belongs to segment $AC$ and $F$ belongs to segment $AB$. Suppose that $BE, CF$ intersect $(O)$ at $M,N$ respectively. The line $d_1$ passes through $M$ and perpendicular to $BM$ intersects $(O)$ at the second point $P,$ the line $d_2$ passes through $N$ and perpendicular to $CN$ intersect $(O)$ at the second point $Q$. Denote $H, K$ are two midpoints of $MP$ and $NQ$ respectively. 1. Prove that triangles $IEF$ and $OKH$ are similar. 2. Suppose that S is the intersection of two lines $d_1$ and $d_2$. Prove that $SO$ is perpendicular to $EF$.

This figure is cut out from a sheet of paper. Folding the sides upwards along the dashed lines, one gets a (non-equilateral) pyramid with a square base. Calculate the area of the base. [img]https://1.bp.blogspot.com/-lPpfHqfMMRY/XzcBIiF-n2I/AAAAAAAAMW8/nPs_mLe5C8srcxNz45Wg-_SqHlRAsAmigCLcBGAsYHQ/s0/2005%2BMohr%2Bp1.png[/img]

What is the area of a triangle with side lengths $17$, $25$, and $26$? [i]2019 CCA Math Bonanza Lightning Round #3.2[/i]

$(BEL 2) (a)$ Find the equations of regular hyperbolas passing through the points $A(\alpha, 0), B(\beta, 0),$ and $C(0, \gamma).$ $(b)$ Prove that all such hyperbolas pass through the orthocenter $H$ of the triangle $ABC.$ $(c)$ Find the locus of the centers of these hyperbolas. $(d)$ Check whether this locus coincides with the nine-point circle of the triangle $ABC.$

Let $ABC$ be a triangle such that the angular bisector of $\angle BAC$, the $B$-median and the perpendicular bisector of $AB$ intersect at a single point $X$. Let $H$ be the orthocenter of $ABC$. Show that $\angle BXH = 90^{\circ}$. [i]Proposed by usjl[/i]

(V.Protasov, 10--11) In the space, given two intersecting spheres of different radii and a point $ A$ belonging to both spheres. Prove that there is a point $ B$ in the space with the following property: if an arbitrary circle passes through points $ A$ and $ B$ then the second points of its meet with the given spheres are equidistant from $ B$.

For a convex hexagon $ ABCDEF$ in which each pair of opposite sides is unequal, consider the following statements. ($ a_1$) $ AB$ is parallel to $ DE$. ($ a_2$)$ AE \equal{} BD$. ($ b_1$) $ BC$ is parallel to $ EF$. ($ b_2$)$ BF \equal{} CE$. ($ c_1$) $ CD$ is parallel to $ FA$. ($ c_2$) $ CA \equal{} DF$. $ (a)$ Show that if all six of these statements are true then the hexagon is cyclic. $ (b)$ Prove that, in fact, five of the six statements suffice.

Let $O$ be the circumcenter of triangle $ABC$ and the line $AO$ intersects segment $BC$ at point $T$ . Assume that lines $m$ and $\ell$ passing through point $T$ are perpendicular to $AB$ and $AC$ respectively. If $E$ is the point of intersection of $m$ and $OB$ and $F$ is the point of intersection of $\ell$ and $OC$, prove that $BE = CF$. (Oleksii Karliuchenko)

Let $E$ be a point lies inside the parallelogram $ABCD$ such that $\angle BCE = \angle BAE$. Prove that the circumcenters of triangles $ABE,BCE,CDE,DAE$ are concyclic.

A quadrilateral $ABCD$ is inscribed in a circle of center $O$. It is known that the diagonals $AC$ and $BD$ are perpendicular. On each side we build semicircles, externally, as shown in the figure. a) Show that the triangles $AOB$ and $COD$ have the equal areas. b) If $AC=8$ cm and $BD= 6$ cm, determine the area of the shaded region.

Let $f(x)=e^{-x}\ \forall\ x\geq 0$ and let $g$ be a function defined as for every integer $k \ge 0$, a straight line joining $(k,f(k))$ and $(k+1,f(k+1))$ . Find the area between the graphs of $f$ and $g$.

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 $\frac{BT}{BM}$.

Given a circumscribed quadrilateral $ABCD$. Prove that its inradius is smaller than the sum of the inradii of triangles $ABC$ and $ACD$.

Let $ABCD$ be a convex pentagon such that \[\angle DCB = \angle DEA = 90^\circ, \ \text{and} \ DC=DE.\] Let $F$ be a point on AB such that $AF:BF=AE:BC$. Show that \[\angle FEC= \angle BDC, \ \text{and} \ \angle FCE= \angle ADE.\]

Triangle $ABC$ with $AB=4$, $BC=5$, $CA=6$ has circumcircle $\Omega$ and incircle $\omega$. Let $\Gamma$ be the circle tangent to $\Omega$ and the sides $AB$, $BC$, and let $X=\Gamma \cap \Omega$. Let $Y$, $Z$ be distinct points on $\Omega$ such that $XY$, $YZ$ are tangent to $\omega$. Find $YZ^2$. [i]The following fact may be useful: if $\triangle{ABC}$ has incircle $w$ with incenter $I$ and radius $r$, and $\triangle{DEF}$ is the intouch triangle (i.e. $D$, $E$, $F$ are intersections of incircle with $BC$, $CA$, $AB$, respectively) and $H$ is the orthocenter of $\triangle{DEF}$, then the inversion of $X$ about $\omega$ (i.e. the point $X'$ on ray $IX$ such that $IX' \cdot IX=r^2$) is the midpoint of $DH$.[/i]

In acute scalene triangle $ABC$ the external angle bisector of $\angle BAC$ meet $BC$ at point $X$.Lines $l_b$ and $l_c$ which tangents of $B$ and $C$ with respect to $(ABC)$.The line pass through $X$ intersects $l_b$ and $l_c$ at points $Y$ and $Z$ respectively. Suppose $(AYB)\cap(AZC)=N$ and $l_b\cap l_c=D$. Show that $ND$ is angle bisector of $\angle YNZ$. Proposed by [i]Alireza Haghi[/i]

The line passing through $B$ is perpendicular to the side $AC$ at $E$. This line meets the circumcircle of $\triangle ABC$ at $D$. The foot of the perpendicular from $D$ to the side $BC$ is $F$. If $O$ is the center of the circumcircle of $\triangle ABC$, prove that $BO$ is perpendicular to $EF$.

$K$ is an arbitrary point inside the acute-angled triangle $ABC$, in which $\angle A = 30^o$. $F$ and $N$ are the points of intersection of the medians in the triangles $AKC$ and $AKB$, respectively . It is known that $FN = q$. Find the radius of the circle circumscribed around the triangle $ABC$. (Grigory Filippovsky)

Let $ABC$ be a triangle with $AB=c$ , $BC=a$ and $AC=b$. If $ x,y\in\mathbb{R}$ satisfy $ \frac{1}{x} +\frac{1}{y+z} = \frac{1}{a} $ , $ \frac{1}{y} +\frac{1}{x+z} = \frac{1}{b} $ , $ \frac{1}{z} +\frac{1}{y+x} = \frac{1}{c} $ . Prove that the following equality holds $ x(p-a) + y(p-b) + z(p-c) = 3r^2 + 12R*r , $ Where $p$ is semi-perimeter, $R$ is the circumradius and $r$ is the inradius.

In $\triangle ABC$, points $D$, $E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, such that each of the quadrilaterals $AFDE$, $BDEF$, and $CEFD$ has an incircle. Prove that the inradius of $\triangle ABC$ is twice the inradius of $\triangle DEF$.

The area of a rectangle remains unchanged when it is made $2 \frac{1}{2}$ inches longer and $\frac{2}{3}$ inch narrower, or when it is made $2 \frac{1}{2}$ inches shorter and $\frac{4}{3}$ inch wider. Its area, in square inches, is: $\textbf{(A)}\ 30\qquad \textbf{(B)}\ \frac{80}{3}\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ \frac{45}{2}\qquad \textbf{(E)}\ 20$

Let $O$ be the midpoint of the axis of a right circular cylinder. Let $A$ and $B$ be diametrically opposite points of one base, and $C$ a point of the other base circle that does not belong to the plane $OAB$. Prove that the sum of dihedral angles of the trihedral $OABC$ is equal to $2\pi$.