Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? $ \textbf{(A) }60 \qquad \textbf{(B) }65 \qquad \textbf{(C) }70 \qquad \textbf{(D) }75 \qquad \textbf{(E) }80 \qquad $

Triangle $\vartriangle ABC$ is inscribed in circle $O$ and has sides $AB = 47$, $BC = 69$, and $CA = 34$. Let $E$ be the point on $O$ such that $\overline{AE}$ and $\overline{BC}$ intersect inside $O$, $8$ units away from $B$. Let $P$ and $Q$ be the points on $\overleftrightarrow{BE}$ and $\overleftrightarrow{CE}$, respectively, such that $\angle EPA$ and $\angle EQA$ are right angles. Suppose lines $\overleftrightarrow{AP}$ and $\overleftrightarrow{AQ}$ respectively intersect $O$ again at $X$ and $Y$ . Compute the distance $XY$.

Let $\vartriangle ABC$ be equilateral. Two points $D$ and $E$ are on side $BC$ (with order $B, D, E, C$), and satisfy $\angle DAE = 30^o$ . If $BD = 2$ and $CE = 3$, what is $BC$? [img]https://cdn.artofproblemsolving.com/attachments/c/8/27b756f84e086fe31b5ea695f51fb6c78b63d0.png[/img]

It is known that circle $\Gamma_1(O_1)$ has center at $O_1$, circle $\Gamma_2(O_2)$ has center at $O_2$, and both intersect at points $C$ and $D$. It is also known that points $P$ and $Q$ lie on circles $\Gamma_1(O_1)$ and $\Gamma_2(O_2)$, respectively. ). A line $\ell$ passes through point $D$ and intersects $\Gamma_1(O_1)$ and $\Gamma_2(O_2)$ at points $A$ and $B$, respectively. The lines $PD$ and $AC$ meet at point $M$, and the lines $QD$ and $BC$ meet at point $N$. Let $O$ be center outer circle of triangle $ABC$. Prove that $OD$ is perpendicular to $MN$ if and only if a circle can be found which passes through the points $P, Q, M$ and $N$.

It is known that a circle can be inscribed in the quadrilateral $ABCD$, in addition $\angle A = \angle C$. Prove that $AB = BC$, $CD = DA$. (Olena Artemchuk)

Let $O$ be the circumcenter of triangle $ABC$ and let $AD$ be the height from $A$ ($D\in BC$). Let $M,N,P$ and $Q$ be the midpoints of $AB,AC,BD$ and $CD$ respectively. Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be the circumcircles of triangles $AMN$ and $POQ$. Prove that $\mathcal{C}_1\cap \mathcal{C}_2\cap AD\neq \emptyset$.

The vertical mast located on the tower can be seen at the greatest angle from a point on the ground whose distance from the mast axis is $ a $; this angle equals the given angle $ \alpha $. Calculate the height of the tower and the height of the mast.

Let $ n$ be a natural number. A cube of edge $ n$ may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for $ n$.

[hide=E stands for Euclid, L stands for Lobachevsky]they had two problem sets under those two names[/hide] [b]E1.[/b] What is the perimeter of a rectangle if its area is $24$ and one side length is $6$? [b]E2.[/b] John moves 3 miles south, then $2$ miles west, then $7$ miles north, and then $5$ miles east. What is the length of the shortest path, in miles, from John's current position to his original position? [b]E3.[/b] An equilateral triangle $ABC$ is drawn with side length $2$. The midpoints of sides $AB$, $BC$, and $CA$ are constructed, and are connected to form a triangle. What is the perimeter of the newly formed triangle? [b]E4.[/b] Let triangle $ABC$ have sides $AB = 74$ and $AC = 5$. What is the sum of all possible integral side lengths of BC? [b]E5.[/b] What is the area of quadrilateral $ABCD$ on the coordinate plane with $A(1, 0)$, $B(0, 1)$, $C(1, 3)$, and $D(5, 2)$? [b]E6 / L1.[/b] Let $ABCD$ be a square with side length $30$. A circle centered at the center of $ABCD$ with diameter $34$ is drawn. Let $E$ and $F$ be the points at which the circle intersects side $AB$. What is $EF$? [b]E7 / L2.[/b] What is the area of the quadrilateral bounded by $|2x| + |3y| = 6$? [b]E8.[/b] A circle $O$ with radius $2$ has a regular hexagon inscribed in it. Upon the sides of the hexagon, equilateral triangles of side length $2$ are erected outwards. Find the area of the union of these triangles and circle $O$. [b]L3.[/b] Right triangle $ABC$ has hypotenuse $AB$. Altitude $CD$ divides $AB$ into segments $AD$ and $DB$, with $AD = 20$ and $DB = 16$. What is the area of triangle $ABC$? [b]L4.[/b] Circle $O$ has chord $AB$. Extend $AB$ past $B$ to a point $C$. A ray from $C$ is drawn, and this ray intersects circle $O$. Let point $D$ be the point of intersection of the ray and the circle that is closest to point $C$. Given $AB = 20$, $BC = 16$, and $OA = \frac{201}{6}$ , find the longest possible length of $CD$. [b]L5.[/b] Consider a circular cone with vertex $A$. The cone's height is $4$ and the radius of its base is $3$. Inscribe a sphere inside the cone. Find the ratio of the volume of the cone to the volume of the sphere. [b]L6.[/b] A disk of radius $\frac12$ is randomly placed on the coordinate plane. What is the probability that it contains a lattice point (point with integer coordinates)? [b]L7.[/b] Let $ABC$ be an equilateral triangle of side length $2$. Let $D$ be the midpoint of $BC$, and let $P$ be a variable point on $AC$. By moving $P$ along $AC$, what is the minimum perimeter of triangle $BDP$? [b]L8.[/b] Let $ABCD$ be a rectangle with $AB = 8$ and $BC = 9$. Let $DEFG$ be a rhombus, where $G$ is on line $BC$ and $A$ is on line $EF$. If $m\angle EFG = 30^o, what is $DE$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle, and let $P$ be a point on side $BC$ such that $\frac{BP}{PC}=\frac{1}{2}$. If $\measuredangle ABC$ $=$ $45^{\circ}$ and $\measuredangle APC$ $=$ $60^{\circ}$, determine $\measuredangle ACB$ without trigonometry.

In triangle $ABC$, point $M$ is the midpoint of $BC$, $P$ the point of intersection of the tangents at points $B$ and $C$ of the circumscribed circle of $ABC$, $N$ is the midpoint of the segment $MP$. The segment $AN$ meets the circumcircle $ABC$ at the point $Q$. Prove that $\angle PMQ = \angle MAQ$.

Let $ \mathcal P$ be a parabola, and let $ V_1$ and $ F_1$ be its vertex and focus, respectively. Let $ A$ and $ B$ be points on $ \mathcal P$ so that $ \angle AV_1 B \equal{} 90^\circ$. Let $ \mathcal Q$ be the locus of the midpoint of $ AB$. It turns out that $ \mathcal Q$ is also a parabola, and let $ V_2$ and $ F_2$ denote its vertex and focus, respectively. Determine the ratio $ F_1F_2/V_1V_2$.

Let $ABC$ be a right-angled triangle $\left(\angle A=90^{\circ}\right)$ and $M$ be the midpoint of $BC$. $\omega_1$ is a circle which passes through $B,M$ and touchs $AC$ at $X$. $\omega_2$ is a circle which passes through $C,M$ and touchs $AB$ at $Y$ ($X,Y$ and $A$ are in the same side of $BC$). Prove that $XY$ passes through the midpoint of arc $BC$ (does not contain $A$) of the circumcircle of $ABC$.

Let $ ABCD $ be a parallelogram with $ BC = 17 $. Let $ M $ be the midpoint of $ \overline{BC} $ and let $ N $ be the point such that $ DANM $ is a parallelogram. What is the length of segment $ \overline{NC} $?

In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $E$. If $AB=BE=5$, $EC=CD=7$, and $BC=11$, compute $AE$.

Let $A, B$ and $C$ three points on a line $l$, in that order .Let $D$ a point outside $l$ and $\Gamma$ the circumcircle of $\triangle BCD$, the tangents from $A$ to $\Gamma$ touch $\Gamma$ on $S$ and $T$. Let $P$ the intersection of $ST$ and $AC$. Prove that $P$ does not depend of the choice of $D$.

Let $\triangle ABC$ be an acute triangle and $AD$ the bisector of the angle $\angle BAC$ with $D\in(BC)$. Let $E$ and $F$ denote feet of perpendiculars from $D$ to $AB$ and $AC$ respectively. If $BF\cap CE=K$ and $\odot AKE\cap BF=L$ prove that $DL\perp BF$.

All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$. [b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length; [b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.

Let $P$ be a point outside a circle $\omega$. The tangents from $P$ to $\omega$ are drawn touching $\omega$ at points $A$ and $B$. Let $M$ and $N$ be the midpoints of $AP$ and $AB$, respectively. Line $MN$ is extended to cut $\omega$ at $C$ so that $N$ lies between $M$ and $C$. Line $PC$ intersects $\omega$ again at $D$, and lines $ND$ and $PB$ intersect at $O$. Prove that $MNOP$ is a rhombus.

In triangle $\vartriangle ABC$, $\angle A = 50^o, \angle B = 60^o$ and $\angle C = 70^o$. The point $P$ is on the side $[AB]$ (with $P \ne A$ and $P \ne B$). The inscribed circle of $\vartriangle ABC$ intersects the inscribed circle of $\vartriangle ACP$ at points $U$ and $V$ and intersects the inscribed circle of $\vartriangle BCP$ at points $X$ and $Y$. The rights $UV$ and $XY$ intersect in $K$. Calculate the $\angle UKX$.

Given that two circles $\sigma_1$ and $\sigma_2$ internally tangent at $N$ so that $\sigma_2$ is inside $\sigma_1$. The points $Q$ and $R$ lies at $\sigma_1$ and $\sigma_2$, respectively, such that $N,R,Q$ are collinear. A line through $Q$ intersects $\sigma_2$ at $S$ and intersects $\sigma_1$ at $O$. The line through $N$ and $S$ intersects $\sigma_1$ at $P$. Prove that $$\frac{PQ^3}{PN^2} = \frac{PS \cdot RS}{NS}.$$

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

Circles of radii 5, 5, 8, and $m/n$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

$ M$ is the midpoint of base $ BC$ in a trapezoid $ ABCD$. A point $ P$ is chosen on the base $ AD$. The line $ PM$ meets the line $ CD$ at a point $ Q$ such that $ C$ lies between $ Q$ and $ D$. The perpendicular to the bases drawn through $ P$ meets the line $ BQ$ at $ K$. Prove that $ \angle QBC \equal{} \angle KDA$. [i]Proposed by S. Berlov[/i]

A solid cube has side length $ 3$ inches. A $ 2$-inch by $ 2$-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid? $ \textbf{(A)}\ 7\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 15$