Let $E$ be a point on side $[AD]$ of rhombus $ABCD$. Lines $AB$ and $CE$ meet at $F$, lines $BE$ and $DF$ meet at $G$. If $m(\widehat{DAB}) = 60^\circ $, what is$m(\widehat{DGB})$? $ \textbf{a)}\ 45^\circ \qquad\textbf{b)}\ 50^\circ \qquad\textbf{c)}\ 60^\circ \qquad\textbf{d)}\ 65^\circ \qquad\textbf{e)}\ 75^\circ $

3 segments $AA_1$, $BB_1$, $CC_1$ in space share a common midpoint $M$. Turns out, the sphere circumscribed about the tetrahedron $MA_1B_1C_1$ is tangent to plane $ABC$ at point $D$. Point $O$ is the circumcenter of triangle $ABC$. Prove that $MO = MD$.

Circles $\Gamma_1$ and $\Gamma_2$ meet at points $X$ and $Y$. A circle $S_1$ touches internally $\Gamma_1$ at $A$ and $\Gamma_2$ externally at $B$. A circle $S_2$ touches $\Gamma_2$ internally at $C$ and $\Gamma_1$ externally at $D$. Prove that the points $A, B, C, D$ are either collinear or concyclic. (A. Voidelevich)

Matías has a rectangular sheet of paper $ABCD$, with $AB<AD$.Initially, he folds the sheet along a straight line $AE$, where $E$ is a point on the side $DC$ , so that vertex $D$ is located on side $BC$, as shown in the figure. Then folds the sheet again along a straight line $AF$, where $F$ is a point on side $BC$, so that vertex $B$ lies on the line $AE$; and finally folds the sheet along the line $EF$. Matías observed that the vertices $B$ and $C$ were located on the same point of segment $AE$ after making the folds. Calculate the measure of the angle $\angle DAE$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/b9ab717e1806c6503a9310ee923f20109da31a.png[/img]

$M$ and $N$ are the midpoints of sides $BC$ and $AD$, respectively, of a square $ABCD$. $K$ is an arbitrary point on the extension of the diagonal $AC$ beyond $A$. The segment $KM$ intersects the side $AB$ at some point $L$. Prove that $\angle KNA = \angle LNA$. [i](5 points)[/i]

p1. Find all real numbers $x$ that satisfy the inequality $$\frac{x^2-3}{x^2-1}+ \frac{x^2 + 5}{x^2 + 3} \ge \frac{x^2-5}{x^2-3}+\frac{x^2 + 3}{x^2 + 1}$$ p2. It is known that $m$ is a four-digit natural number with the same units and thousands digits. If $m$ is a square of an integer, find all possible numbers $m$. p3. In the following figure, $\vartriangle ABP$ is an isosceles triangle, with $AB = BP$ and point $C$ on $BP$. Calculate the volume of the object obtained by rotating $ \vartriangle ABC$ around the line $AP$ [img]https://cdn.artofproblemsolving.com/attachments/c/a/65157e2d49d0d4f0f087f3732c75d96a49036d.png[/img] p4. A class farewell event is attended by $10$ male students and $ 12$ female students. Homeroom teacher from the class provides six prizes to randomly selected students. Gifts that provided are one school bag, two novels, and three calculators. If the total students The number of male students who received prizes was equal to the total number of female students who received prizes. How many possible arrangements are there of the student who gets the prize? p5. It is known that $S =\{1945, 1946, 1947, ..., 2016, 2017\}$. If $A = \{a, b, c, d, e\}$ a subset of $S$ where $a + b + c + d + e$ is divisible by $5$, find the number of possible $A$'s.

The center of the circle circumscribing $\vartriangle ABC$ is mirrored through each side of the triangle and three points are obtained: $O_1, O_2, O_3$. Reconstruct $\vartriangle ABC$ from $O_1, O_2, O_3$ if everything else is erased.

In a given pentagon $ABCDE$, triangles $ABC$, $BCD$, $CDE$, $DEA$ and $EAB$ all have the same area. The lines $AC$ and $AD$ intersect $BE$ at points $M$ and $N$. Prove that $BM = EN$.

Let $O$ be a circumcenter of acute triangle $ABC$ and let $O_1$ and $O_2$ be circumcenters of triangles $OAB$ and $OAC$, respectively. Circumcircles of triangles $OAB$ and $OAC$ intersect side $BC$ in points $D$ ($D \neq B$) and $E$ ($E \neq C$), respectively. Perpendicular bisector of side $BC$ intersects side $AC$ in point $F$($F \neq A$). Prove that circumcenter of triangle $ADE$ lies on $AC$ iff $F$ lies on line $O_1O_2$

In a triangle $ ABC,$ let $ D$ and $ E$ be the intersections of the bisectors of $ \angle ABC$ and $ \angle ACB$ with the sides $ AC,AB,$ respectively. Determine the angles $ \angle A,\angle B, \angle C$ if $ \angle BDE \equal{} 24 ^{\circ},$ $ \angle CED \equal{} 18 ^{\circ}.$

Let $ABC$ be a triangle such that in its interior there exists a point $D$ with $\angle DAC = \angle DCA = 30^o$ and $ \angle DBA = 60^o$. Denote $E$ the midpoint of the segment $BC$, and take $F$ on the segment $AC$ so that $AF = 2FC$. Prove that $DE \perp EF$.

Consider two segments of length $a, b \ (a > b)$ and a segment of length $c = \sqrt{ab}$. [b](a)[/b] For what values of $a/b$ can these segments be sides of a triangle ? [b](b)[/b] For what values of $a/b$ is this triangle right-angled, obtuse-angled, or acute-angled ?

Two equilateral triangles $ABC$ and $CDE$ have a common vertex (see fig). Find the angle between straight lines $AD$ and $BE$. [img]https://1.bp.blogspot.com/-OWpqpAqR7Zw/Xzj_fyqhbFI/AAAAAAAAMao/5y8vCfC7PegQLIUl9PARquaWypr8_luAgCLcBGAsYHQ/s0/2010%2Boral%2Bmoscow%2Bgeometru%2B8.1.gif[/img]

In triangle $ABC$, the bisectors of the internal angles $AA_1$ , $BB_1$ and $CC_1$ are drawn ($A_1, B_1$, $C_1$ - on the sides of the triangle). It is known that $\angle AA_1C = \angle AC_1B_1$. Find $\angle BCA$.

Let $ABCD$ be a convex quadrilateral and $P$ and $Q$ are the midpoints of the diagonals $AC$ and $BD$,and $O$ their intersection point.Point $M$ is the midpoint of $AB$ and $N$ is the midpoint of $CD$ such that $ MN \cap AC ={E},MN \cap BD={F}$.Prove that $OE \cdot QF= OF\cdot PE $

Let $ABC$ be an acute triangle and $H$ be its orthocenter. Let $E$ be the foot of the altitude from $C$ to $AB$, $F$ be the foot of the altitude from $B$ to $AC$. Let $G \neq H$ be the intersection of the circles $(AEF)$ and $(BHC)$. Prove that $AG$ bisects $BC$. [i]Proposed by Kang Taeyoung, South Korea[/i]

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

On the radius $OA$ of a certain circle, as on the diameter, a circle is constructed. A ray is drawn from the center $O$, intersecting the larger and smaller circles at points $B$ and $C$, respectively. Show that the lengths of arcs $AB$ and $AC$ are equal.

Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.

Let $P$ be the set of points in the plane, and let $f : P \to P$ be a function such that the image through $f$ of any triangle is a square (any polygon is considered to be formed by the reunion of the points on its sides). Prove that $f(P)$ is a square.

You are given positive integers $m, n>1$. Vasyl and Petryk play the following game: they take turns marking on the coordinate plane yet unmarked points of the form $(x, y)$, where $x, y$ are positive integers with $1 \leq x \leq m, 1 \leq y \leq n$. The player loses if after his move there are two marked points, the distance between which is not a positive integer. Who will win this game if Vasyl moves first and each player wants to win? [i]Proposed by Mykyta Kharin[/i]

Parallelogram $ABCD$ is such that angle $B < 90$ and $AB<BC$. Points E and F are on the circumference of $\omega$ inscribing triangle ABC, such that tangents to $\omega$ in those points pass through D. If $\angle EDA= \angle{FDC}$, find $\angle{ABC}$.

Consider the regular quadrilateral prism $ABCDA'B'C 'D'$, in which $AB = a,AA' = \frac{a \sqrt {2}}{2}$, and $M$ is the midpoint of $B' C'$. Let $F$ be the foot of the perpendicular from $B$ on line $MC$, Let determine the measure of the angle between the planes $(BDF)$ and $(HBS)$.

Let $\Gamma$ be the circumcircle of isosceles triangle $ABC$ with vertex $C$. An arbitrary point $M$ is chosen on the segment $BC$ and point $N$ lies on the ray $AM$ with $M$ between $A,N$ such that $AN=AC$. The circumcircle of $CMN$ cuts $\Gamma$ in $P$ other than $C$ and $AB,CP$ intersect at $Q$. Prove that $\angle BMQ = \angle QMN.$

What maximal area can have a triangle if its sides $a,b,c$ satisfy inequality $0\le a\le 1\le b\le 2\le c\le 3$ ?