In rectangle $ABCD$, $AB = 40$ and $AD = 30$. Let $C' $ be the reflection of $C$ over $BD$. Find the length of $AC'$.

Two circles have a common internal tangent of length $17$ and a common external tangent of length $25$. Find the product of the radii of the two circles.

In the arrow-shaped polygon [see figure], the angles at vertices $A$, $C$, $D$, $E$ and $F$ are right angles, $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$. The area of the polygon is closest to [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(10,10), C=(10,5), D=(30,5), E=(30,-5), F=(10,-5), G=(10,-10); draw(A--B--C--D--E--F--G--A); label("$A$", A, W); label("$B$", B, NE); label("$C$", C, S); label("$D$", D, NE); label("$E$", E, SE); label("$F$", F, N); label("$G$", G, SE); label("$5$", (11,7.5)); label("$5$", (11,-7.5)); label("$20$", (C+D)/2, N); label("$20$", (F+E)/2, S); label("$10$", (31,0)); [/asy] $ \textbf{(A)}\ 288\qquad\textbf{(B)}\ 291\qquad\textbf{(C)}\ 294\qquad\textbf{(D)}\ 297\qquad\textbf{(E)}\ 300 $

In the figure, there is a circle of radius $1$ such that the segment $AG$ is diameter and that line $AF$ is perpendicular to line $DC$. There are also two squares $ABDC$ and $DEGF$, where $B$ and $E$ are points on the circle, and the points $A$, $D$ and $E$ are collinear. What is the area of square $DEGF$? [img]https://cdn.artofproblemsolving.com/attachments/1/e/794da3bc38096ef5d5daaa01d9c0f8c41a6f84.png[/img]

In triangle $ABC$ it is known that $\angle ACB = 2\angle ABC$. Furthermore $P$ is an interior point of the triangle $ABC$ such that $AP = AC$ and $PB = PC$. Prove that $\angle BAC = 3 \angle BAP$.

Let $ABC$ be an acute triangle. Let $J$ be the $A$-excenter. The $A$-excircle is tangent to $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. The line $DE$ intersects $CJ$ and $BJ$ at $P$ and $Q$, respectively. $M$ is the midpoint of $AD$. Prove that $PM=QM$.

In a triangle $ ABC$, the angle bisector at $ B$ intersects $ AC$ at $ D$ and the circumcircle again at $ E$. The circumcircle of the triangle $ DAE$ meets the segment $ AB$ again at $ F$. Prove that the triangles $ DBC$ and $ DBF$ are congruent.

Let $K{}$ be an equilateral triangle of unit area, and choose $n{}$ independent random points uniformly from $K{}$. Let $K_n$ be the intersection of all translations of $K{}$ that contain all the selected points. Determine the expected value of the area of $K_n.$

Let $H$ be orthocenter of the triangle $ABC$. Let $d_1, d_2$ be lines perpendicular to each-another at $H$. The line $d_1$ intersects $AB, AC$ at $D, E$ and the line d_2 intersects $B C$ at $F$. Prove that $H$ is the midpoint of segment $DE$ if and only if $F$ is the midpoint of segment $BC$.

Let $ABC$ be a acute triangle where $\angle BAC =60$. Prove that if the Euler's line of $ABC$ intersects $AB,AC$ in $D,E$, then $ADE$ is equilateral.

Let $ a$ and $ b$ be positive real numbers with $ a\ge b$. Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations \[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$. Then $ \rho^2$ can be expressed as a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

Let $ABC$ be a non-isosceles triangle. The incircle $(I)$ of the triangle touches sides $BC,CA,AB$ at $A_0,B_0$, and $C_0$. Points $A_1,B_1$, and $C_1$ are on $BC,CA,AB$ such that $BA1 = CA_0, CB_1 = AB_0, AC_1 = BC_0$. Prove that the circumcircles $(IAA1), (IBB_1), (ICC_1)$ pass all through a common point, distinct from $I$. Nguyễn Minh Hà

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

In the quadrilateral $ABCD $, $AB = BC $, the point $K $ is the midpoint of the side $CD $, the rays $BK $ and $AD $ intersect at the point $M $ , the circumscribed circle $ \Delta ABM $ intersects the line $AC $ for the second time at the point $P $. Prove that $\angle BKP = 90 {} ^ \circ $. (Anton Trygub)

The circumcircle of $\triangle ABC$ has radius $1$ and centre $O$ and $P$ is a point inside the triangle such that $OP=x$. Prove that \[AP\cdot BP\cdot CP\le(1+x)^2(1-x),\] with equality if and only if $P=O$.

Inside a rectangle is inscribed a quadrilateral, which has a vertex on each side of the rectangle. Prove that the perimeter of the inscribed quadrilateral is not smaller than double the length of a diagonal of the rectangle. (V. V . Proizvolov , Moscow)

Let $ABC$ be an acute scalene triangle with circumcenter $O$, and let $T$ be on line $BC$ such that $\angle TAO = 90^{\circ}$. The circle with diameter $\overline{AT}$ intersects the circumcircle of $\triangle BOC$ at two points $A_1$ and $A_2$, where $OA_1 < OA_2$. Points $B_1$, $B_2$, $C_1$, $C_2$ are defined analogously. [list=a][*] Prove that $\overline{AA_1}$, $\overline{BB_1}$, $\overline{CC_1}$ are concurrent. [*] Prove that $\overline{AA_2}$, $\overline{BB_2}$, $\overline{CC_2}$ are concurrent on the Euler line of triangle $ABC$. [/list][i]Evan Chen[/i]

Prove that if $h_1,h_2,h_3,h_4$ are the altitudes of a tetrahedron and $d_1,d_2,d_3$ the distances between the pairs of opposite edges of the tetrahedron, then $$\frac{1}{h_1^2} +\frac{1}{h_2^2} +\frac{1}{h_3^2} +\frac{1}{h_4^2} =\frac{1}{d_1^2} +\frac{1}{d_2^2} +\frac{1}{d_3^2}.$$

On the standard unit circle, draw 4 unit circles with centers [0,1],[1,0],[0,-1],[-1,0]. You get a figure as below, find the area of the colored part. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=277[/img]

A parallelogram is inscribed in a quadrilateral, two opposite vertices of which are the midpoints of the opposite sides of the quadrilateral. Determine the area of ​​such a parallelogram if the area of ​​the quadrilateral is equal to $S_o$.

[b]Q[/b]. $ABC$ is an acute - angled triangle with $\odot(ABC)$ and $\Omega$ as the circumcircle and incircle respectively. Let $D, E, F$ to be the respective intouch points on $\overline{BC}, \overline{CA}$ and $\overline{AB}$. Circle $\gamma_A$ is drawn internally tangent to sides $\overline{AC}, \overline{AB}$ and $\odot(ABC)$ at $X, Y$ and $Z$ respectively. Another circle $(\omega)$ is constructed tangent to $\overline{BC}$ at $\mathcal{T}_1$ and internally tangent to $\odot(ABC)$ at $\mathcal{T}_2$. A tangent is drawn from $A$ such that it touches $\omega$ at $W$ and meets $BC$ at $V$, with $V$ lying inside $\odot(ABC)$. Now if $\overline{EF}$ meets $\odot(BC)$ at $\mathcal{X}_1$ and $\mathcal{X}_2$, opposite to vertex $B$ and $C$ respectively, where $\odot(BC)$ denotes the circle with $BC$ as diameter, prove that the set of lines $\{\overline{B\mathcal{X}_1}, \overline{ZS}, \overline{C\mathcal{X}_2}, \overline{DU}, \overline{YX}, \overline{\mathcal{T}_1W} \}$ are concurrent where $S$ is the mid-point of $\widehat{BC}$ containing $A$ and $U$ is the anti-pode of $D$ with respect to $\Omega$. If the line joining that concurrency point and $A$ meets $\odot(ABC)$ at $N\not = A$ prove that $\overline{AD}, \overline{ZN}$ and $\gamma_A$ pass through a common point. [i] Proposed by srijonrick[/i]

We are given a line $ g$ with four successive points $ P$, $ Q$, $ R$, $ S$, reading from left to right. Describe a straightedge and compass construction yielding a square $ ABCD$ such that $ P$ lies on the line $ AD$, $ Q$ on the line $ BC$, $ R$ on the line $ AB$ and $ S$ on the line $ CD$.

Find three isosceles triangles, no two of which are congruent, with integer sides, such that each triangle's area is numerically equal to $6$ times its perimeter.

Given a triangle $ABC$ ($AB> AC$) and a circle circumscribed around it. Construct with a compass and a ruler the midpoint of the arc $BC$ (not containing vertex $A$), with no more than two lines (straight or circles).

Let $\Omega$ be the incircle of a triangle $ABC$. Suppose that there exists a circle passing through $B$ and $C$ and tangent to $\Omega$ in $A'$. Suppose the similar points $B'$, $C'$ exist. Prove that the lines $AA', BB'$ and $CC'$ are concurrent.