In the triangle $ABC$, the angle  $\alpha = \angle BAC$ and the side $a = BC$ are given. Assume that $a = \sqrt{rR}$, where $r$ is the inradius and $R$ the circumradius. Compute all possible lengths of sides $AB$ and $AC.$

Let $ABCD$ be a rectangle. Point $E$ lies on side $AB$, and point $F$ lies on segment $CE$. Prove that if triangles $ADE$ and $CDF$ have equal areas, then triangles $BCE$ and $DEF$ also have equal areas.

Let $I$ be the incenter of triangle $ABC$; $H_B, H_C$ the orthocenters of triangles $ACI$ and $ABI$ respectively; $K$ the touching point of the incircle with the side $BC$. Prove that $H_B, H_C$ and K are collinear. [i]Proposed by M.Plotnikov[/i]

Given an $ a \times b$ rectangle with $ a > b > 0,$ determine the minimum side of a square that covers the rectangle. (A square covers the rectangle if each point in the rectangle lies inside the square.)

A moth starts at vertex $A$ of a certain cube and is trying to get to vertex $B$, which is opposite $A$, in five or fewer “steps,” where a step consists in traveling along an edge from one vertex to another. The moth will stop as soon as it reaches $B$. How many ways can the moth achieve its objective?

In triangle $ABC$, points $M_1, M_2$ and $M_3$ are midpoints of sides $AB$, $BC$ and $AC$, respectively, while points $H_1, H_2$ and $H_3$ are bases of altitudes drawn from $C$, $A$ and $B$, respectively. Prove that one can construct a triangle from segments $H_1M_2, H_2M_3$ and $H_3M_1$. [i](3 points)[/i]

Let $ABC$ and $AMN$ be two similar triangles with the same orientation, such that $AB=AC$, $AM=AN$ and having disjoint interiors. Let $O$ be the circumcenter of the triangle $MAB$. Prove that the points $O$, $C$, $N$, $A$ lie on the same circle if and only if the triangle $ABC$ is equilateral. [i]Valentin Vornicu[/i]

The incircle of a scalene triangle $ABC$ touches the sides $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Triangles $APE$ and $AQF$ are constructed outside the triangle so that \[AP =PE, AQ=QF, \angle APE=\angle ACB,\text{ and }\angle AQF =\angle ABC.\]Let $M$ be the midpoint of $BC$. Find $\angle QMP$ in terms of the angles of the triangle $ABC$. [i]Iran, Shayan Talaei[/i]

The triangle $ABC$ has $O$ as its circumcenter, and $H$ as its orthocenter. The line $AH$ and $BH$ intersect the circumcircle of $ABC$ for the second time at points $D$ and $E$, respectively. Let $A'$ and $B'$ be the circumcenters of triangle $AHE$ and $BHD$ respectively. If $A', B', O, H$ are [b]not[/b] collinear, prove that $OH$ intersects the midpoint of segment $A'B'$.

Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown. Which of these arrangements gives the dog the greater area to roam, and by how many square feet? [asy]defaultpen(linewidth(0.7)); size(7cm); D((0,0)--(16,0)--(16,-16)--(0,-16)--cycle, black); D((16,-8)--(24,-8), black); label('Dog', (24, -8), SE); label('I', (8,-8), (0,0)); MP('8', (16,-4), W); MP('8', (16,-12),W); MP('8', (20,-8), N); label('Rope', (20,-8),S); D((0,-20)--(16,-20)--(16,-36)--(0,-36)--cycle, black); D((16,-24)--(24,-24), black); label("II", (8,-28), (0,0)); MP('4', (16,-22), W); MP('8', (20,-24), N); label("Dog",(24,-24),SE); label("Rope", (20,-24), S); dot((24,-24)^^(24,-8));[/asy] $ \textbf{(A)}\text{ I, by }8\pi\qquad\textbf{(B)}\text{ I, by }6\pi\qquad\textbf{(C)}\text{ II, by }4\pi\qquad\textbf{(D) }\text{II, by }8\pi\qquad\textbf{(E)}\text{ II, by }10\pi $

Let $ABC$ be a acute-angled triangle with centroid $G$, the angle bisector of $\angle ABC$ intersects $AC$ in $D$. Let $P$ and $Q$ be points in $BD$ where $\angle PBA = \angle PAB$ and $\angle QBC = \angle QCB$. Let $M$ be the midpoint of $QP$, let $N$ be a point in the line $GM$ such that $GN = 2GM$(where $G$ is the segment $MN$), prove that: $\angle ANC + \angle ABC = 180$

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]

A regular dodecahedron is a convex polyhedra that its faces are regular pentagons. The regular dodecahedron has twenty vertices and there are three edges connected to each vertex. Suppose that we have marked ten vertices of the regular dodecahedron. [b]a)[/b] prove that we can rotate the dodecahedron in such a way that at most four marked vertices go to a place that there was a marked vertex before. [b]b)[/b] prove that the number four in previous part can't be replaced with three. [i]proposed by Kasra Alishahi[/i]

Ten distinct points are placed on a circle. All ten of the points are paired so that the line segments connecting the pairs do not intersect. In how many different ways can this pairing be done? [asy] import graph; size(12cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; draw((2.46,0.12)--(3.05,-0.69)); draw((2.46,1.12)--(4,-1)); draw((5.54,0.12)--(4.95,-0.69)); draw((3.05,1.93)--(5.54,1.12)); draw((4.95,1.93)--(4,2.24)); draw((8.05,1.93)--(7.46,1.12)); draw((7.46,0.12)--(8.05,-0.69)); draw((9,2.24)--(9,-1)); draw((9.95,-0.69)--(9.95,1.93)); draw((10.54,1.12)--(10.54,0.12)); draw((15.54,1.12)--(15.54,0.12)); draw((14.95,-0.69)--(12.46,0.12)); draw((13.05,-0.69)--(14,-1)); draw((12.46,1.12)--(14.95,1.93)); draw((14,2.24)--(13.05,1.93)); label("1",(-1.08,2.03),SE*labelscalefactor); label("2",(-0.3,1.7),SE*labelscalefactor); label("3",(0.05,1.15),SE*labelscalefactor); label("4",(0.00,0.38),SE*labelscalefactor); label("5",(-0.33,-0.12),SE*labelscalefactor); label("6",(-1.08,-0.4),SE*labelscalefactor); label("7",(-1.83,-0.19),SE*labelscalefactor); label("8",(-2.32,0.48),SE*labelscalefactor); label("9",(-2.3,1.21),SE*labelscalefactor); label("10",(-1.86,1.75),SE*labelscalefactor); dot((-1,-1),dotstyle); dot((-0.05,-0.69),dotstyle); dot((0.54,0.12),dotstyle); dot((0.54,1.12),dotstyle); dot((-0.05,1.93),dotstyle); dot((-1,2.24),dotstyle); dot((-1.95,1.93),dotstyle); dot((-2.54,1.12),dotstyle); dot((-2.54,0.12),dotstyle); dot((-1.95,-0.69),dotstyle); dot((4,-1),dotstyle); dot((4.95,-0.69),dotstyle); dot((5.54,0.12),dotstyle); dot((5.54,1.12),dotstyle); dot((4.95,1.93),dotstyle); dot((4,2.24),dotstyle); dot((3.05,1.93),dotstyle); dot((2.46,1.12),dotstyle); dot((2.46,0.12),dotstyle); dot((3.05,-0.69),dotstyle); dot((9,-1),dotstyle); dot((9.95,-0.69),dotstyle); dot((10.54,0.12),dotstyle); dot((10.54,1.12),dotstyle); dot((9.95,1.93),dotstyle); dot((9,2.24),dotstyle); dot((8.05,1.93),dotstyle); dot((7.46,1.12),dotstyle); dot((7.46,0.12),dotstyle); dot((8.05,-0.69),dotstyle); dot((14,-1),dotstyle); dot((14.95,-0.69),dotstyle); dot((15.54,0.12),dotstyle); dot((15.54,1.12),dotstyle); dot((14.95,1.93),dotstyle); dot((14,2.24),dotstyle); dot((13.05,1.93),dotstyle); dot((12.46,1.12),dotstyle); dot((12.46,0.12),dotstyle); dot((13.05,-0.69),dotstyle);[/asy]

A point $X$ is marked on the base $BC$ of an isosceles $\triangle ABC$, and points $P$ and $Q$ are marked on the sides $AB$ and $AC$ so that $APXQ$ is a parallelogram. Prove that the point $Y$ symmetrical to $X$ with respect to line $PQ$ lies on the circumcircle of the $\triangle ABC$. [i]($5$ points)[/i]

Let $\omega$ be the unit circle in the $xy$-plane in $3$-dimensional space. Find all points $P$ not on the $xy$-plane that satisfy the following condition: There exist points $A,B,C$ on $\omega$ such that $$ \angle APB = \angle APC = \angle BPC = 90^\circ.$$

Given is a trihedral angle with edges $ SA $, $ SB $, $ SC $, all plane angles of which are acute, and the dihedral angle at edge $ SA $ is right. Prove that the section of this triangle with any plane perpendicular to any edge, at a point different from the vertex $ S $, is a right triangle.

In the plane we have $n$ rectangles with parallel sides. The sides of distinct rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into connected regions. A region is [i]nice[/i] if it has at least one of the vertices of the $n$ rectangles on the boundary. Prove that the sum of the numbers of the vertices of all nice regions is less than $40n$. (There can be nonconvex regions as well as regions with more than one boundary curve.)

If the area of $\triangle ABC$ is $64$ square units and the geometric mean (mean proportional) between sides $AB$ and $AC$ is $12$ inches, then $\sin A$ is equal to $\textbf{(A) }\dfrac{\sqrt{3}}{2}\qquad\textbf{(B) }\frac{3}{5}\qquad\textbf{(C) }\frac{4}{5}\qquad\textbf{(D) }\frac{8}{9}\qquad \textbf{(E) }\frac{15}{17}$

The circle $C_1$ and $C_2$ are externally tangent to each other at $T$. A line passing through $T$ meets $C_1$ at $A$ and meets $C_2$ at $B$. The line which is tangent to $C_1$ at $A$ meets $C_2$ at $D$ and $E$. If $D \in [AE]$, $|TA|=a$, $|TB|=b$, what is $|BE|$? $ \textbf{(A)}\ \sqrt{a(a+b)} \qquad\textbf{(B)}\ \sqrt{a^2+b^2+ab} \qquad\textbf{(C)}\ \sqrt{a^2+b^2-ab} \qquad\textbf{(D)}\ \sqrt{a^2+b^2} \qquad\textbf{(E)}\ \sqrt{(a+b)b} $

Monica and Bogdan are playing a game, depending on given integers $n, k$. First, Monica writes some $k$ positive numbers. Bogdan wins, if he is able to find $n$ points on the plane with the following property: for any number $m$ written by Monica, there are some two points chosen by Bogdan with distance exactly $m$ between them. Otherwise, Monica wins. Determine who has a winning strategy depending on $n, k$. [i](Proposed by Fedir Yudin)[/i]

Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality \[abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.\] Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.

Prove that there are infinitely many positive integers $n$ such that $\Delta = nr^2$, where $\Delta$ and $r$ are respectively the area and the inradius of a triangle with integer sides.

In an acute-angled triangle $\triangle ABC$, construct $\triangle ACD$ and $\triangle BCE$ externally on sides $CA$ and $CB$ respectively, such that $AD=CD$. Let $M$ be the midpoint of $AB$, and connect $DM$ and $EM$. Given that $DM$ is perpendicular to $EM$, set $\frac{AC}{BC} =u$ and $\frac{DM}{EM}=v$. Express $\frac{DC}{EC}$ in terms of $u$ and $v$.

In isosceles triangle $ABC$, $A$ is located at the origin and $B$ is located at $(20, 0)$. Point $C$ is in the first quadrant with $AC = BC$ and $\angle BAC = 75^\circ$. If $\triangle ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive y-axis, the area of the region common to the original and the rotated triangle is in the form $p\sqrt{2}+q\sqrt{3}+r\sqrt{6}+s$ where $p$, $q$, $r$, $s$ are integers. Find $(p-q+r-s)/2$.