If you multiply the number of faces that a pyramid has with the number of edges of the pyramid, you get $5.100$. Determine the number of faces of the pyramid.

Two equilateral triangles are inscribed in a circle with radius $r$. Let $A$ be the area of the set consisting of all points interior to both triangles. Prove that $2A \geq r^2 \sqrt 3.$

The quadrilateral $ABCD$ is inscribed in a circle centered at $O$, and $\{P\} = AC \cap BD, \{Q\} = AB \cap CD$. Let $R$ be the second intersection point of the circumcircles of the triangles $ABP$ and $CDP$. a) Prove that the points $P, Q$, and $R$ are collinear. b) If $U$ and $V$ are the circumcenters of the triangles $ABP$, and $CDP$, respectively, prove that the points $U, R, O, V$ are concyclic.

Consider a ball that moves inside an acute-angled triangle along a straight line, unit it hits the boundary, which is when it changes direction according to the mirror law, just like a ray of light (angle of incidence = angle of reflection). Prove that there exists a triangular periodic path for the ball, as pictured below. [asy] size(10cm); pen thickbrown = rgb(0.6, 0.2, 0); pen thickdark = rgb(0.2, 0, 0); pen dashedarrow = linetype("6 6"); pair A = (-1.14, 4.36), B = (-4.46, -1.28), C = (3.32, -2.78); pair D = (-1.479, -1.855), E = (0.727, 1.372), F = (-3.014, 1.176); draw(A--B--C--cycle, thickbrown); draw(A--B, thickdark); draw(B--C, thickdark); draw(C--A, thickdark); draw(D--F, dashedarrow, EndArrow(6)); draw(F--E, dashedarrow, EndArrow(6)); draw(E--D, dashedarrow, EndArrow(6)); dot(A); label("$A$", A, N); dot(B); label("$B$", B, dir(180)); dot(C); label("$C$", C, dir(330)); dot(D); label("$D$", D, S); dot(E); label("$E$", E, NE); dot(F); label("$F$", F, W); [/asy]

The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20. It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k.$

Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Let $D$ be an arbitrary point inside $BC$ such that $BD\neq DC$. Ray $AD$ intersects $\omega$ again at $E$ (other than $A$). Point $F$ (other than $E$) is chosen on $\omega$ such that $\angle DFE = 90^\circ$. Line $FE$ intersects rays $AB$ and $AC$ at points $X$ and $Y$, respectively. Prove that $\angle XDE = \angle EDY$. [i]Proposed by Anton Trygub[/i]

Assume that the two triangles $ABC$ and $A'B'C'$ have the common incircle and the common circumcircle. Let a point $P$ lie inside both the triangles. Prove that the sum of the distances from $P$ to the sidelines of triangle $ABC$ is equal to the sum of distances from $P$ to the sidelines of triangle $A'B'C'$.

Suppose $\triangle ABC$ has $D$ as the midpoint of $BC$ and orthocenter $H$. Let $P$ be an arbitrary point on the nine point circle of $ABC$. The line through $P$ perpendicular to $AP$ intersects $BC$ at $Q$. The line through $A$ perpendicular to $AQ$ intersects $PQ$ at $X$. If $M$ is the midpoint of $AQ$, show that $HX \perp DM$.

Let $O$ be the circumcenter of an acute triangle $ABC$. Suppose that the circumradius of the triangle is $R = 2p$, where $p$ is a prime number. The lines $AO,BO,CO$ meet the sides $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Given that the lengths of $OA_1,OB_1,OC_1$ are positive integers, find the side lengths of the triangle.

Let $\Omega$ be the circumcircle of the triangle $ABC$. The circle $\omega$ is tangent to the sides $AC$ and $BC$, and it is internally tangent to the circle $\Omega$ at the point $P$. A line parallel to $AB$ intersecting the interior of triangle $ABC$ is tangent to $\omega$ at $Q$. Prove that $\angle ACP = \angle QCB$.

Two sides of a triangle have lengths $10$ and $15$. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side? $\textbf{(A) }6\qquad \textbf{(B) }8\qquad \textbf{(C) }9\qquad \textbf{(D) }12\qquad \textbf{(E) }18\qquad$

The hexagon has five $90^o$ angles and one $270^o$ angle (see picture). Use a straight-line ruler to divide it into two equal-sized polygons. [img]https://cdn.artofproblemsolving.com/attachments/d/8/cdd4df68644bb8e04adbe1b265039b82a5382b.png[/img]

Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle. Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$

In the hexagon $ABCDEF$, all angles are equally large. The side lengths satisfy $AB = CD = EF = 3$ and $BC = DE = F A = 2$. The diagonals $AD$ and $CF$ intersect each other in the point $G$. The point $H$ lies on the side $CD$ so that $DH = 1$. Prove that triangle $EGH$ is equilateral.

Let ­ $\Omega$ be a circle of radius $8$ centered at point $O$, and let $M$ be a point on ­$\Omega$. Let $S$ be the set of points $P$ such that $P$ is contained within $\Omega$ ­, or such that there exists some rectangle $ABCD$ containing $P$ whose center is on ­ $\Omega$ with$ AB = 4$, $BC = 5$, and $BC \parallel OM$. Find the area of $S$.

A convex quadrilateral $ABCD$ is said to be [i]dividable[/i] if for every internal point $P$, the area of $\triangle PAB$ plus the area of $\triangle PCD$ is equal to the area of $\triangle PBC$ plus the area of $\triangle PDA$. Characterize all quadrilaterals which are dividable.

A triangle $ABC$ with $\beta= 45^o$ is given. There is a point $P$ on side $BC$, where $BP : PC =1 : 2$ (inner division) and $\angle APC = 60^o$. Someone claims that you can do it with elementary geometric theorems alone without using the plane trigonometry, the size of the angle $\gamma$ determine.

Let $a\leqslant b\leqslant c$ be non-negative integers. A triangle on a checkered plane with vertices in the nodes of the grid is called an $(a,b,c)$[i]-triangle[/i] if there are exactly $a{}$ nodes on one side of it (not counting vertices), exactly $b{}$ nodes on the second side, and exactly $c{}$ nodes on the third side. [list] [*]Does there exist a $(9,10,11)$-triangle? [*]Find all triples of non-negative integers $a\leqslant b\leqslant c$ for which there exists an $(a,b,c)$-triangle. [*]For each such triple, find the minimum possible area of the $(a,b,c)$-triangle. [/list] [i]Proposed by P. Kozhevnikov[/i]

Let $D$ be the foot of the altitude towards $BC$ in the triangle $ABC$. Let $E$ be the intersection of $AB$ with the bisector of angle $\angle C$. Assume that the angle $\angle AEC = 45^o$ . Determine the angle $\angle EDB$.

Given triangle $ABC$ and point $D$ on the $AC$ side. Let $r_1, r_2$ and $r$ denote the radii of the incircle of the triangles $ABD, BCD$, and $ABC$, respectively. Prove that $r_1 + r_2> r$.

In a circle, let $AB$ and $BC$ be chords , with $AB =\sqrt3, BC =3\sqrt3, \angle ABC =60^o$. Find the length of the circle chord that divides angle $ \angle ABC$ in half.

Let $ ABC $ be a right angle in $ A, $ and $ M $ be the mid of $ BC. $ On the perpendicular of $ AM $ through $ A $ choose a point $ D $ so that $ DM $ meets $ AB $ at a point, namely $ P. $ Let $ E $ be the projection of $ D $ on $ BC. $ Show that $ \angle BPM =\angle EAC. $

Circles $w_1$ and $w_2$ with centers at points $O_1$ and $O_2$ respectively, intersect at points $A$ and $B$. A line passing through point $B$, intersects the circles $w_1$ and $w_2$ at points $C$ and $D$ other than $B$. Tangents to the circles $w_1$ and $w_2$ at points $C$ and $D$ intersect at point $E$. Line $EA$ intersects the circumscribed circle $w$ of triangle $AO_1O_2$ at point $F$. Prove that the length of the segment is $EF$ is equal to the diameter of the circle $w$. (Vovchenko V., Plotnikov M.)

$X$ and $Y$ are two points lying on or on the extensions of side $BC$ of $\triangle{ABC}$ such that $\widehat{XAY} = 90$. Let $H$ be the orthocenter of $\triangle{ABC}$. Take $X'$ and $Y'$ as the intersection points of $(BH,AX)$ and $(CH,AY)$ respectively. Prove that circumcircle of $\triangle{CYY'}$,circumcircle of $\triangle{BXX'}$ and $X'Y'$ are concurrent.

A square with side length $2$ cm is placed next to a square with side length $6$ cm, as shown in the diagram. Find the shaded area, in cm$^2$. [img]https://cdn.artofproblemsolving.com/attachments/5/7/ceb4912a6e73ca751113b2b5c92cbfdbb6e0d1.png[/img]