Let $ABC$ be an acute triangle. Let $H$ denote its orthocenter and $D, E$ and $F$ the feet of its altitudes from $A, B$ and $C$, respectively. Let the common point of $DF$ and the altitude through $B$ be $P$. The line perpendicular to $BC$ through $P$ intersects $AB$ in $Q$. Furthermore, $EQ$ intersects the altitude through $A$ in $N$. Prove that $N$ is the midpoint of $AH$. Proposed by Karl Czakler

In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.

One hundred concentric circles with radii $1, 2, 3, \dots, 100$ are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $ABC$ be a scalene triangle and $\omega$ is your incircle. The sides $BC,CA$ and $AB$ are tangents to $\omega$ in $X,Y,Z$ respectively. Let $M$ be the midpoint of $BC$ and $D$ is the intersection point of $BC$ with the angle bisector of $\angle BAC$. Prove that $\angle BAX=\angle MAC$ if and only if $YZ$ passes by the midpoint of $AD$.

Find the radius of the inscribed circle of a triangle with sides of length $40$, $42$, and $58$.

$ABCD$ is a regular tetrahedron. The points $P,Q,R$ and $S$ lie outside this tetrahedron in such a way that $ABCP$, $ABDQ$, $ACDR$ and $BCDS$ are regular tetrahedra. Prove that the volume of the tetrahedron $PQRS$ is less than the sum of the volumes of $ABCP$,$ABDQ$,$ACDR$, $BCDS$ and $ABCD$.

Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that: $a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area. $b) a\cdot AP=b\cdot BP=c\cdot PC.$

Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$. Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.

Let $ABC$ be a triangle with $AB=34,BC=25,$ and $CA=39$. Let $O,H,$ and $ \omega$ be the circumcenter, orthocenter, and circumcircle of $\triangle ABC$, respectively. Let line $AH$ meet $\omega$ a second time at $A_1$ and let the reflection of $H$ over the perpendicular bisector of $BC$ be $H_1$. Suppose the line through $O$ perpendicular to $A_1O$ meets $\omega$ at two points $Q$ and $R$ with $Q$ on minor arc $AC$ and $R$ on minor arc $AB$. Denote $\mathcal H$ as the hyperbola passing through $A,B,C,H,H_1$, and suppose $HO$ meets $\mathcal H$ again at $P$. Let $X,Y$ be points with $XH \parallel AR \parallel YP, XP \parallel AQ \parallel YH$. Let $P_1,P_2$ be points on the tangent to $\mathcal H$ at $P$ with $XP_1 \parallel OH \parallel YP_2$ and let $P_3,P_4$ be points on the tangent to $\mathcal H$ at $H$ with $XP_3 \parallel OH \parallel YP_4$. If $P_1P_4$ and $P_2P_3$ meet at $N$, and $ON$ may be written in the form $\frac{a}{b}$ where $a,b$ are positive coprime integers, find $100a+b$. [i]Proposed by Vincent Huang[/i]

The quadrilateral $ABCD$ has $AD = DC = CB < AB$ and $AB \parallel CD$. Points $E$ and $F$ lie on the sides $CD$ and $BC$ such that $\angle ADE = \angle AEF$. Prove that: (a) $4CF \le CB$. (b) If $4CF = CB$, then $AE$ is the angle bisector of $\angle DAF$.

Characterize all triangles $ABC$ s.t. \[ AI_a : BI_b : CI_c = BC: CA : AB \] where $I_a$ etc. are the corresponding excentres to the vertices $A, B , C$

In the triangle $ABC$ it is known that $2AC=AB$ and $\angle A = 2\angle B$. In this triangle draw the angle bisector $AL$, and mark point $M$, the midpoint of the side $AB$. It turned out that $CL = ML$. Prove that $\angle B= 30^o$. (Hilko Danilo)

$ABCD$ is a cyclic quadrilateral with $\angle BAD=90^\circ$ and $\angle ABC>90^\circ$. $AB$ is extended to a point $E$ such that $\angle AEC=90^\circ$.If $AB=7,BE=9,$ and $EC=12$,calculate $AD$.

Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that [b](a)[/b] $\triangle ABC$ is acute. [b](b)[/b] $a+b+c > r+r_a+r_b+r_c$.

Let $ABC$ be a triangle with $\angle A = 90$ and let $D$ be the orthogonal projection of $A$ onto $BC$. The midpoints of $AD$ and $AC$ are called $E$ and $F$, respectively. Let $M$ be the circumcentre of $BEF$. Prove that $AC$ and $ BM$ are parallel.

The inscribed circle of $\triangle ABC$ touches $BC, AC$ and $AB$ at $D,E$ and $F$ respectively. Denote the perpendicular foots from $F, E$ to $BC$ by $K, L$ respectively. Let the second intersection of these perpendiculars with the incircle be $M, N$ respectively. Show that $\frac{{{S}_{\triangle BMD}}}{{{S}_{\triangle CND}}}=\frac{DK}{DL}$ by Mahdi Etesami Fard

The vertices of a convex pentagon $ABCDE$ lie on a circle $\gamma_1$. The diagonals $AC , CE, EB, BD$, and $DA$ are tangents to another circle $\gamma_2$ with the same centre as $\gamma_1$. (a) Show that all angles of the pentagon $ABCDE$ have the same size and that all edges of the pentagon have the same length. (b) What is the ratio of the radii of the circles $\gamma_1$ and $\gamma_2$? (The answer should be given in terms of integers, the four basic arithmetic operations and extraction of roots only.)

Consider an octogon with equal angles and rational side lengths. Prove that it has a symmetry center.

Find the angles of the triangle $ABC$, if we know that its center $O$ of the circumscribed circle and the center $I_A$ of the exscribed circle (tangent to $BC$) are symmetric wrt $BC$. (Bogdan Rublev)

Let $X$ be a point inside convex quadrilateral $ABCD$ with $\angle AXB+\angle CXD=180^{\circ}$. If $AX=14$, $BX=11$, $CX=5$, $DX=10$, and $AB=CD$, find the sum of the areas of $\triangle AXB$ and $\triangle CXD$. [i]Proposed by Michael Kural[/i]

$\vartriangle ABC$ is right with $\angle C = 90^o$. The internal angle bisectors of $\angle A$ and $\angle B$ meet at point $D$, while the external angle bisectors of $\angle A$ and $\angle B$ meet at point $E$. Suppose that $AD = 1$ and $BD = 2$. The value of $DE^2$ can be expressed as $x+y \sqrt{z}$ for integers $x$, $y$, and $z$, where $z$ is greater than $1$ and not divisible by the square of any prime. Compute $100x + 10y + z$. Note: For a generic triangle $\vartriangle PQR$, if we let $Q'$ be the reflection of $Q$ over $P$, then the external angle bisector of $\angle P$ is the line that contains the internal angle bisector of $\angle Q'PR$.

Triangle $ABC$ has side lengths $AC = 3$, $BC = 4$, and $AB = 5$. Let $I$ be the incenter of $\triangle{ABC}$, and let $\mathcal{P}$ be the parabola with focus $I$ and directrix $\overleftrightarrow{AC}$. Suppose that $\mathcal{P}$ intersects $\overline{AB}$ and $\overline{BC}$ at points $D$ and $E$, respectively. Find $DI+EI$.

Given acute triangle $\vartriangle ABC$ with orthocenter $H$ and circumcenter $O$ ($O \ne H$) . Let $\Gamma$ be the circumcircle of $\vartriangle BOC$ . Segment $OH$ untersects $\Gamma$ at point $P$. Extension of $AO$ intersects $\Gamma$ at point $K$. If $AP \perp OH$, prove that $PK$ bisects $BC$. [img]https://cdn.artofproblemsolving.com/attachments/a/b/267053569c41692f47d8f4faf2a31ebb4f4efd.png[/img]

Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other

In angle $\angle AOB=60^{\circ}$ are two circle which circumscribed and tangjent to each other . If we write with $r$ and $R$ the radius of smaller and bigger circle respectively and if $r=1$ find $R$ .