A right cone in $xyz$-space has its apex at $(0,0,0)$, and the endpoints of a diameter on its base are $(12,13,-9)$ and $(12,-5,15)$. The volume of the cone can be expressed as $a\pi$. What is $a$?

Let $ABCDE$ be a convex pentagon with $AB\parallel CE$, $BC\parallel AD$, $AC\parallel DE$, $\angle ABC=120^\circ$, $AB=3$, $BC=5$, and $DE=15$. Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

A triangle of sides $\frac{3}{2}, \frac{\sqrt{5}}{2}, \sqrt{2}$ is folded along a variable line perpendicular to the side of $\frac{3}{2}.$ Find the maximum value of the coincident area.

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

The minimal value of $ f(x) \equal{} \sqrt{a^2 \plus{} x^2} \plus{} \sqrt{(x\minus{}b)^2 \plus{} c^2}$ is A. $ a\plus{}b\plus{}c$ B. $ \sqrt{a^2 \plus{} (b \plus{} c)^2}$ C. $ \sqrt{b^2 \plus{} (a\plus{}c)^2}$ D. $ \sqrt{(a\plus{}b)^2 \plus{} c^2}$ E. None of these

The circle $\omega$ with center at point $I$ inscribed in an triangle $ABC$ ($AB\neq AC$) touches the sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$, respectively. The circumcircles of triangles $ABC$ and $AEF$ intersect secondary at point $K.$ The lines $EF$ and $AK$ intersect at point $X$ and intersects the line $BC$ at points $Y$ and $Z$, respectively. The tangent lines to $\omega$, other than $BC$, passing through points $Y$ and $Z$ touch $\omega$ at points $P$ and $Q$, respectively. Let the lines $AP$ and $KQ$ intersect at the point $R$. Prove that if $M$ is a midpoint of segment $YZ,$ then $IR\perp XM$.

Given a triangle $ABC$ with the unit area. The first player chooses a point $X$ on the side $[AB]$, than the second -- $Y$ on $[BC]$ side, and, finally, the first chooses a point $Z$ on $[AC]$ side. The first tries to obtain the greatest possible area of the $XYZ$ triangle, the second -- the smallest. What area can obtain the first for sure and how?

Let \( O \) and \( H \) be the circumcenter and orthocenter of the acute triangle \( ABC \). On sides \( AC \) and \( AB \), points \( D \) and \( E \) are chosen respectively such that segment \( DE \) passes through point \( O \) and \( DE \parallel BC \). On side \( BC \), points \( X \) and \( Y \) are chosen such that \( BX = OD \) and \( CY = OE \). Prove that \( \angle XHY + 2\angle BAC = 180^\circ \). [i]Proposed by Matthew Kurskyi[/i]

We are given six points in space with distinct distances, no three of them collinear. Consider all triangles with vertices among these points. Show that among these triangles there is one such that its longest side is the shortest side in one of the other triangles.

Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.

Let $ABC$ be an acute triangle and let $\Gamma$ be its circumcircle. The bisector of $\angle{A}$ intersects $BC$ at $D$, $\Gamma$ at $K$ (different from $A$), and the line through $B$ tangent to $\Gamma$ at $X$. Show that $K$ is the midpoint of $AX$ if and only if $\frac{AD}{DC}=\sqrt{2}$.

Let $C_1$ and $C_2$ be two different circles , and let their radii be $r_1$ and $r_2$ , the two circles are passing through the two points $A$ and $B$ (i)Let $P_1$ on $C_1$ and $P_2$ on $C_2$ such that the line $P_1P_2$ passes through $A$. Prove that $P_1B \cdot r_2 = P_2B \cdot r_1$ (ii)Let $DEF$ be a triangle that it's inscribed in $C_1$ , and let $D'E'F'$ be a triangle that is inscribed in $C_2$ . The lines $EE'$,$DD'$ and $FF'$ all pass through $A$ . Prove that the triangles $DEF$ and $D'E'F'$ are similar (iii)The circle $C_3$ also passes through $A$ and $B$ . Let $l$ be a line that passes through $A$ and cuts circles $C_i$ in $M_i$ with $i = 1,2,3$ . Prove that the value of$$\frac{M_1M_2}{M_1M_3}$$is constant regardless of the position of $l$ Provided that $l$ is different from $AB$

Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\tfrac{m}{n}?$ $\textbf{(A) } \frac{\sqrt{2}}{4} \qquad \textbf{(B) } \frac{\sqrt{2}}{2} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{3\sqrt{2}}{5} \qquad \textbf{(E) } \frac{2\sqrt{2}}{3}$

$E$ and $F$ are interior points of convex quadrilateral $ABCD$ such that $AE = BE$, $CE = DE$, $\angle AEB = \angle CED$, $AF = DF$, $BF = CF$, $\angle AFD = \angle BFC$. Prove that $\angle AFD + \angle AEB = \pi$.

There are $20$ points in the plane and no three of them are collinear. Of these points $10$ are red while the other $10$ are blue. Prove that there exists a straight line such that there are $5$ red points and $5$ blue points on either side of this line. (A Kushnirenko, Moscow)

Let $ABC$ be an acute-angled scalene triangle and $\Gamma$ be its circumcircle. $K$ is the foot of the internal bisector of $\angle BAC$ on $BC$. Let $M$ be the midpoint of the arc $BC$ containing $A$. $MK$ intersect $\Gamma$ again at $A'$. $T$ is the intersection of the tangents at $A$ and $A'$. $R$ is the intersection of the perpendicular to $AK$ at $A$ and perpendicular to $A'K$ at $A'$. Show that $T, R$ and $K$ are collinear.

Let $P$ be a point inside a triangle $A_1A_2A_3$. For $i = 1, 2, 3$, line $PA_i$ intersects the side opposite to $A_i$ at $B_i$. Let $C_i$ and $D_i$ be the midpoints of $A_iB_i$ and $PB_i$, respectively. Prove that the areas of the triangles $C_1C_2C_3$ and $D_1D_2D_3$ are equal.

In a triangle $\Delta ABC$, $|AB|=|AC|$. Let $M$ be on the minor arc $AC$ of the circumcircle of $\Delta ABC$ different than $A$ and $C$. Let $BM$ and $AC$ meet at $E$ and the bisector of $\angle BMC$ and $BC$ meet at $F$ such that $\angle AFB=\angle CFE$. Prove that the triangle $\Delta ABC$ is equilateral.

Let $ABC$ be a triangle with $AB \neq AC$ and with incenter $I$. Let $M$ be the midpoint of $BC$, and let $L$ be the midpoint of the circular arc $BAC$. Lines through $M$ parallel to $BI,CI$ meet $AB,AC$ at $E$ and $F$, respectively, and meet $LB$ and $LC$ at $P$ and $Q$, respectively. Show that $I$ lies on the radical axis of the circumcircles of triangles $EMF$ and $PMQ$. Proposed by [i]Andrew Wu[/i]

Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the [i]square[/i] of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is $\frac{1734}{274}$, then you would submit 1734274).

Let $ABC$ be a triangle and let $M$ be the midpoint of the side $BC$ . The circle with radius $MA$ centered in $M$ meets the lines $AB$ and $AC$ again at $B^{'}$ and $C^{'}$, respectively , and the tangents to this circle at $B^{'}$ and $C^{'}$ meet at $D$ . Show that the perpendicular bisector of the segment $BC$ bisects the segment $AD$.

In the plane a circle $C$ of unit radius is given. For any line $l$, a number $s(l)$ is defined in the following way: If $l$ and $C$ intersect in two points, $s(l)$ is their distance; otherwise, $s(l) = 0$. Let $P$ be a point at distance $r$ from the center of $C$. One defines $M(r)$ to be the maximum value of the sum $s(m) + s(n)$, where $m$ and $n$ are variable mutually orthogonal lines through $P$. Determine the values of $r$ for which $M(r) > 2$.

Let $ABCD$ be a cyclic quadrilateral. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ such that $KLMN$ is a rhombus with $KL \parallel AC$ and $LM \parallel BD$. Let $\omega_A, \omega_B, \omega_C, \omega_D$ be the incircles of $\triangle ANK, \triangle BKL, \triangle CLM, \triangle DMN$. Prove that the common internal tangents to $\omega_A$, and $\omega_C$ and the common internal tangents to $\omega_B$ and $\omega_D$ are concurrent.

Let $ ABC$ be an acute-angled triangle. The lines $ L_{A}$, $ L_{B}$ and $ L_{C}$ are constructed through the vertices $ A$, $ B$ and $ C$ respectively according the following prescription: Let $ H$ be the foot of the altitude drawn from the vertex $ A$ to the side $ BC$; let $ S_{A}$ be the circle with diameter $ AH$; let $ S_{A}$ meet the sides $ AB$ and $ AC$ at $ M$ and $ N$ respectively, where $ M$ and $ N$ are distinct from $ A$; then let $ L_{A}$ be the line through $ A$ perpendicular to $ MN$. The lines $ L_{B}$ and $ L_{C}$ are constructed similarly. Prove that the lines $ L_{A}$, $ L_{B}$ and $ L_{C}$ are concurrent.

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$ lying in the interior. Let $E$ and $F$ be the midpoints of the segments $BC$ and $AD$, respectively. Let $X$ be the point lying on the same side of the line $EF$ as the vertex $C$ such that $\triangle EXF$ and $\triangle BOA$ are similar. Prove that $XC = XD$.