Let $ABC$ be an isosceles triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD=2DC$. Let $P$ be a point on the segment $AD$ such that $\angle BAC=\angle BPD$. Prove that $\angle BAC=2\angle DPC$.

\(ABCD\) is a inscribed quadrilateral. \(P\) is the intersection point of its diagonals. \(O\) is its circumcenter. \(\Gamma\) is the circumcircle of \(ABO\). \(\Delta\) is the circumcircle of \(CDO\). \(M\) is the midpoint of arc \(AB\) on \(\Gamma\) who doesn't contain \(O\). \(N\) is the midpoint of arc \(CD\) on \(\Delta\) who doesn't contain \(O\). Show that \(M,N,P\) are collinear.

Let $ABC$ be an acute angled triangle and $CD$ be the altitude through $C$. If $AB = 8$ and $CD = 6$, find the distance between the midpoints of $AD$ and $BC$.

In quadrilateral $ABCD$ holds $AB=6$, $AD=4$, $\angle DAB=\angle ABC = 60^{\circ}$ and $\angle ADC = 90^{\circ}$. Find length of diagonals and area of the quadrilateral

Three numbers have sum $k$ (where $k\in \mathbb{R}$) such that the numbers are arethmetic progression.If First of two numbers remain the same and to the third number we add $\frac{k}{6}$ than we have geometry progression. Find those numbers?

A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold: [list=1] [*] each triangle from $T$ is inscribed in $\omega$; [*] no two triangles from $T$ have a common interior point. [/list] Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.

The tetrahedron $OPQR$ has the $\angle POQ = \angle POR = \angle QOR = 90^o$. $X, Y, Z$ are the midpoints of $PQ, QR$ and $RP.$ Show that the four faces of the tetrahedron $OXYZ$ have equal area.

Circle $c$ passes through vertices $A$ and $B$ of an isosceles triangle $ABC$, whereby line $AC$ is tangent to it. Prove that circle $c$ passes through the circumcenter or the incenter or the orthocenter of triangle $ABC$.

Anthony the ant is at point $A$ of regular tetrahedron $ABCD$ with side length $4$. Anthony wishes to crawl on the surface of the tetrahedron to the midpoint of $\overline{BC}$. However, he does not want to touch the interior of face $\vartriangle ABC$, since it is covered with lava. What is the shortest distance Anthony must travel?

In triangle $ABC$ ($AB\not= AC$), the incircle is tangent to the sides of $BC$ ,$CA$ , $AB$ at $D$ ,$E$, $F$ respectively. Let $AD$ meet the incircle again at point $P$, let $EF$ and the line passing through the point $P$ and perpendicular to $AD$ intersect at $Q$. Let $AQ$ intersect $DE$ at $X$ and $DF$ at $Y$. Prove that $AX=AY$. [i](tatari/nightmare)[/i]

The lengths of a rectangle’s sides and of its diagonal are integers. Prove that the area of the rectangle is an integer multiple of $12$.

Suppose that $ABC$ is a triangle with $AB = 6, BC = 12$, and $\angle B = 90^{\circ}$. Point $D$ lies on side $BC$, and point $E$ is constructed on $AC$ such that $\angle ADE = 90^{\circ}$. Given that $DE = EC = \frac{a\sqrt{b}}{c}$ for positive integers $a, b,$ and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, find $a+ b+c$. [i]Proposed by Andrew Wu[/i]

In a triangle $ABC$, let $D$ and $E$ point in the sides $BC$ and $AC$ respectively. The segments $AD$ and $BE$ intersects in $O$, let $r$ be line (parallel to $AB$) such that $r$ intersects $DE$ in your midpoint, show that the triangle $ABO$ and the quadrilateral $ODCE$ have the same area.

Of the following statements, the one that is incorrect is: $ \textbf{(A)}\ \text{Doubling the base of a given rectangle doubles the area.}$ $ \textbf{(B)}\ \text{Doubling the altitude of a triangle doubles the area.}$ $ \textbf{(C)}\ \text{Doubling the radius of a given circle doubles the area.}$ $ \textbf{(D)}\ \text{Doubling the divisor of a fraction and dividing its numerator by 2 changes the quotient.}$ $ \textbf{(E)}\ \text{Doubling a given quantity may make it less than it originally was.}$

Given two circles ${\omega _1},{\omega _2}$, $S$ denotes all $\Delta ABC$ satisfies that ${\omega _1}$ is the circumcircle of $\Delta ABC$, ${\omega _2}$ is the $A$- excircle of $\Delta ABC$ , ${\omega _2}$ touches $BC,CA,AB$ at $D,E,F$. $S$ is not empty, prove that the centroid of $\Delta DEF$ is a fixed point.

Instead of walking along two adjacent sides of a rectangular field, a boy took a shortcut along the diagonal of the field and saved a distance equal to $ \frac{1}{2}$ the longer side. The ratio of the shorter side of the rectangle to the longer side was: $ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{2}{3} \qquad\textbf{(C)}\ \frac{1}{4} \qquad\textbf{(D)}\ \frac{3}{4} \qquad\textbf{(E)}\ \frac{2}{5}$

A sphere is centered at a point with integer coordinates and passes through the three points $(2,0,0), (0,4,0), (0,0,6),$ but not the origin $(0,0,0)$. If $r$ is the smallest possible radius of the sphere, compute $r^2$.

There are $n$ lattice points in a general position. (no three points are collinear) A convex polygon $P$ covers the said $n$ points. (the borders are included) Prove that, for large enough $n$ and a positive real $\epsilon$, the perimeter of $P$ is no less than $(\sqrt{2}+\epsilon)n$.

Let $ABC$ be an isosceles triangle with $AB = AC$. Suppose $P,Q,R$ are points on segments $AC, AB, BC$ respectively such that $AP = QB$, $\angle PBC = 90^\circ - \angle BAC$ and $RP = RQ$. Let $O_1, O_2$ be the circumcenters of $\triangle APQ$ and $\triangle CRP$. Prove that $BR = O_1O_2$. [i]Proposed by Atul Shatavart Nadig[/i]

Points $D$ and $E$ are chosen on the exterior of $\vartriangle ABC$ such that $\angle ADC = \angle BEC = 90^o$. If $\angle ACB = 40^o$, $AD = 7$, $CD = 24$, $CE = 15$, and $BE = 20$, what is the measure of $\angle ABC $ in,degrees?

For an acute triangle $ABC$, let $O$ be its circumcenter, and let $O_A,O_B,O_C$ be the circumcenter of $BCO,CAO,ABO$ respectively. Show that $AO_A,BO_B,CO_C$ are concurrent.

Given a circle $ k $ and a point inside it $ H $. Inscribe a triangle in the circle such that this point $ H $ is the point of intersection of the triangle's altitudes.

Let $A$ be a point in the plane, and $\ell$ a line not passing through $A$. Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is [i]not[/i] marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane. (i) Can Evan construct* the reflection of $A$ over $\ell$? (ii) Can Evan construct the foot of the altitude from $A$ to $\ell$? *To construct a point, Evan must have an algorithm which marks the point in finitely many steps. [i]Proposed by Zack Chroman[/i]

Let $ABC$ be a isosceles triangle, with $AB=AC$. A line $r$ that pass through the incenter $I$ of $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. Let $F$ and $G$ be points on $BC$ such that $BF=CE$ and $CG=BD$. Show that the angle $\angle FIG$ is constant when we vary the line $r$.

In the figure, $ABC$ and $CDE$ are right-angled and isosceles triangles. Segments $AD$ and $BC$ intersect at $P$, and segments $CD$ and $BE$ intersect at $Q$. (a) Show that segment$ PQ$ is parallel to segment $AE$. (b) If $BP = 4$ and $DQ = 9$, find the measure of segment $BD$. [img]https://cdn.artofproblemsolving.com/attachments/d/3/4c2c7514d71bbac68d58fc6de9ec2649e58957.png[/img]