Right triangular prism $ABCDEF$ with triangular faces $\vartriangle ABC$ and $\vartriangle DEF$ and edges $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ has $\angle ABC = 90^o$ and $\angle EAB = \angle CAB = 60^o$ . Given that $AE = 2$, the volume of $ABCDEF$ can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/4/7/25fbe2ce2df50270b48cc503a8af4e0c013025.png[/img]

It is allowed to perform the following transformations in the plane with any integers $a$: (1) Transform every point $(x, y)$ to the corresponding point $(x + ay, y)$, (2) Transform every point $(x, y)$ to the corresponding point $(x, y + ax)$. Does there exist a non-square rhombus whose all vertices have integer coordinates and which can be transformed to: a) Vertices of a square, b) Vertices of a rectangle with unequal side lengths?

Let $ABCD$ a cyclic convex quadrilateral. There is a line $l$ parallel to $DC$ containing $A$. Let $P$ a point on $l$ closer to $A$ than to $B$. Let $B'$ the reflection of $B$ over the midpoint of $AD$. Prove that $\angle B'AP = \angle BAC$

Given an acute-angled triangle $ABC$. A straight line parallel to $BC$ intersects sides $AB$ and $AC$ at points $M$ and $P$, respectively. At what location of the points $M$ and $P$ will the radius of the circle circumscribed about the triangle $BMP$ be the smallest? (I. Sharygin)

Let $\omega_1$ be the circumcircle of triangle $ABC$ and $O$ be its circumcenter. A circle $\omega_2$ touches the sides $AB, AC$, and touches the arc $BC$ of $\omega_1$ at point $K$. Let $I$ be the incenter of $ABC$. Prove that the line $OI$ contains the symmedian of triangle $AIK$.

A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.

Circles $\omega$ and $\gamma$, both centered at $O$, have radii $20$ and $17$, respectively. Equilateral triangle $ABC$, whose interior lies in the interior of $\omega$ but in the exterior of $\gamma$, has vertex $A$ on $\omega$, and the line containing side $\overline{BC}$ is tangent to $\gamma$. Segments $\overline{AO}$ and $\overline{BC}$ intersect at $P$, and $\dfrac{BP}{CP} = 3$. Then $AB$ can be written in the form $\dfrac{m}{\sqrt{n}} - \dfrac{p}{\sqrt{q}}$ for positive integers $m$, $n$, $p$, $q$ with $\gcd(m,n) = \gcd(p,q) = 1$. What is $m+n+p+q$? $\phantom{}$ $\textbf{(A) } 42 \qquad \textbf{(B) }86 \qquad \textbf{(C) } 92 \qquad \textbf{(D) } 114 \qquad \textbf{(E) } 130$

* Given an infinite cone. The measure of its unfolding’s angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the number of the curve’s self-intersections.

With putting the four shapes drawn in the following figure together make a shape with at least two reflection symmetries. [img]https://cdn.artofproblemsolving.com/attachments/6/0/8ace983d3d9b5c7f93b03c505430e1d2d189fd.png[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

Let $k_1$ and $k_2$ be two circles that are externally tangent at point $C$. We have a point $A$ on $k_1$ and a point $B$ on $k_2$ such that $C$ is an interior point of segment $AB$. Let $k_3$ be a circle that passes through points $A$ and $B$ and intersects circles $k_1$ and $k_2$ another time at points $M$ and $N$ respectively. Let $k_4$ be the circumscribed circle of triangle $CMN$. Prove that the centres of circles $k_1, k_2, k_3$ and $k_4$ all lie on the same circle.

Two equal rectangles of area $10$ are arranged as follows. Find the area of the gray rectangle. [img]https://cdn.artofproblemsolving.com/attachments/7/1/112b07530a2ef42e5b2cf83a2cb9fb11dfc9e6.png[/img]

The sum $\sqrt[3] {5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}$ equals $\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac{\sqrt[3]{65}}{4} \qquad \text{(C)} \ \frac{1+\sqrt[6]{13}}{2} \qquad \text{(D)} \ \sqrt[3]{2} \qquad \text{(E)} \ \text{none of these}$

Let $O$ be the center and $R$ be the radius of circumcircle $\omega$ of triangle $ABC$. Circle $\omega_1$ with center $O_1$ and radius $R$ pass through points $A, O$ and intersects the side $AC$ at point $K$. Let $AF$ be the diameter of circle $\omega$ and points $F, K, O_1$ are collinear. Determine $\angle ABC$:

Let $ABC$ be an acute triangle with ${AB\neq AC}$. The incircle ${\omega}$ of the triangle touches the sides ${BC, CA}$ and ${AB}$ at ${D, E}$ and ${F}$, respectively. The perpendicular line erected at ${C}$ onto ${BC}$ meets ${EF}$ at ${M}$, and similarly the perpendicular line erected at ${B}$ onto ${BC}$ meets ${EF}$ at ${N}$. The line ${DM}$ meets ${\omega}$ again in ${P}$, and the line ${DN}$ meets ${\omega}$ again at ${Q}$. Prove that ${DP=DQ}$. Ruben Dario & Leo Giugiuc (Romania)

Let $ABC$ be a triangle, let $B_{1}$ be the midpoint of the side $AB$ and $C_{1}$ the midpoint of the side $AC$. Let $P$ be the point of intersection, other than $A$, of the circumscribed circles around the triangles $ABC_{1}$ and $AB_{1}C$. Let $P_{1}$ be the point of intersection, other than $A$, of the line $AP$ with the circumscribed circle around the triangle $AB_{1}C_{1}$. Prove that $2AP=3AP_{1}$.

Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear. [i]Proposed by Michael Kural[/i]

Given an inscribed quadrilateral $ABCD$, which marked the midpoints of the points $M, N, P, Q$ in this order. Let diagonals $AC$ and $BD$ intersect at point $O$. Prove that the triangle $OMN, ONP, OPQ, OQM$ have the same radius of the circles

We are given a polygon, a line $l$ and a point $P$ on $l$ in general position: all lines containing a side of the polygon meet $l$ at distinct points diering from $P$. We mark each vertex of the polygon the sides meeting which, extended away from the vertex, meet the line $l$ on opposite sides of $P$. Show that $P$ lies inside the polygon if and only if on each side of $l$ there are an odd number of marked vertices. [i]O. Musin[/i]

Given is $\triangle ABC$ such that $\angle A = 57^o, \angle B = 61^o$ and $\angle C = 62^o$. Which segment is longer: the angle bisector through $A$ or the median through $B$? (N.Beluhov)

Let $ABCD$ be a quadrilateral, such that $AB = BC = CD.$ There are points $X, Y$ on rays $CA, BD,$ respectively, such that $BX = CY.$ Let $P, Q, R, S$ be the midpoints of segments $BX, CY ,$ $XD, YA,$ respectively. Prove that points $P, Q, R, S$ lie on a circle.

In convex quadrilateral $ABCD$, $\angle BAC=\angle CAD$. $E$ lies on segment $CD$, $BE$ and $AC$ intersect at $F,$ $DF$ and $BC$ intersect at $G.$ Prove that $\angle GAC=\angle EAC$.

Let $ ABC$ be a triangle. $ W_a$ is a circle with center on $ BC$ passing through $ A$ and perpendicular to circumcircle of $ ABC$. $ W_b,W_c$ are defined similarly. Prove that center of $ W_a,W_b,W_c$ are collinear.

Let $ABC$ be an acute triangle with circumcenter $O$, on the sides $BC, CA$ and $AB$ they take the points $D, E$ and $F$, respectively, in such a way that $BDEF$ is a parallelogram. Supposing that $DF^2 = AE\cdot EC <\frac{AC^2}{4}$ show that the circles circumscribed to the triangles $FBD$ and $AOC$ are tangent.

Let $P$ be a point on the circumcircle of triangle $ABC$. Construct an arbitrary triangle $A_1B_1C_1$ whose sides $A_1B_1$, $B_1C_1$ and $C_1A_1$ are parallel to the segments $PC$, $PA$ and $PB$ respectively and draw lines through the vertices $A_1$, $B_1$ and $C_1$ and parallel to the sides $BC$, $CA$ and $AB$ respectively. Prove that these three lines have a common point lying on the circumcircle of triangle $A_1B_1C_1$. (V. Prasolov)

Let $O$ be the center of circumscribed circle $\Omega$ of acute triangle $\Delta ABC$. The line $AC$ intersects the circumscribed circle of triangle $\Delta ABO$ for the second time in $X$. Prove that $BC$ and $XO$ are perpendicular.