In convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ meet at the point $P$. We know that $\angle DAC = 90^o$ and $2 \angle ADB = \angle ACB$. If we have $ \angle DBC + 2 \angle ADC = 180^o$ prove that $2AP = BP$. Proposed by Iman Maghsoudi

On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·

Two circles with radii $1$ and $2$ have a common center at the point $O$. The vertex $A$ of the regular triangle $ABC$ lies on the larger circle, and the middpoint of the base $CD$ lies on the smaller one. What can the angle $BOC$ be equal to?

Two quarter-circles touch as shown. Find the angle $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/4/e70d5d69e46d6d40368f143cb83cf10b7d6d98.png[/img]

How many different (non similar) triangles are there whose angles have an integer number of degrees?

Find all values of $n$ for which there exists a convex cyclic non-regular polygon with $n$ vertices such that the measures of all its internal angles are equal.

Let $A'$ be the result of reflection of vertex $A$ of triangle ABC through line $BC$ and let $B'$ be the result of reflection of vertex $B$ through line $AC$. Given that $\angle BA' C = \angle BB'C$, can the largest angle of triangle $ABC$ be located: a) At vertex $A$, b) At vertex $B$, c) At vertex $C$?

Five congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two neighbors. A sixth circle (shaded in the diagram below) congruent to the other five is placed tangent to two of the five. If this sixth circle is allowed to roll without slipping around the exterior of the figure formed by the other five circles, then it will turn through an angle of $k$ degrees before it returns to its starting position. Find $k$. [asy] import graph; size(6cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); filldraw(circle((2.96,2.58), 1),grey); draw(circle((-1,3), 1)); draw(circle((1,3), 1)); draw(circle((1.62,1.1), 1)); draw(circle((0,-0.08), 1)); draw(circle((-1.62,1.1), 1)); [/asy]

We have a pool table $8$ meters long and $2$ meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling $29$ meters, it stops at a corner of the table. How many times did the ball hit the edges of the table? Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.

Let $180^o < \theta_1 < \theta_2 <...< \theta_n = 360^o$. For $i = 1,2,..., n$, $P_i = (\cos \theta_i^o, \sin \theta_i^o)$ is a point on the circle $C$ with centre $(0,0)$ and radius $1$. Let $P$ be any point on the upper half of $C$. Find the coordinates of $P$ such that the sum of areas $[PP_1P_2] + [PP_2P_3] + ...+ [PP_{n-1}P_n]$ attains its maximum.

In a quadrilateral $ABCD$, angle $B$ is equal to $150^o$, angle $C$ is right, and sides $AB$ and $CD$ are equal. Determine the angle between $BC$ and the line connecting the midpoints of sides $BC$ and $AD$.

In triangle $ABC$, we have $\angle ABC=\angle ACB=44^o$. Point $M$ is in its interior such that $\angle MBC=16^o$ and $\angle MCB=30^o$. Prove that $\angle MAC=\angle MBC$. [i] (Andra Elena Mircea)[/i]

In an isosceles triangle $ABC$ with base $AC$, on side $BC$ is selected point $K$ so that $\angle BAK = 24^o$. On the segment $AK$ the point $M$ is chosen so that $\angle ABM = 90^o$, $AM=2BK$. Find the values ​​of all angles of triangle $ABC$.

Let $[AD]$ be a median of the triangle $[ABC]$. Knowing that $\angle ADB = 45^o$ and $\angle A CB = 30^o$, prove that $\angle BAD = 30^o$.

Let $ABC$ be an equilateral $ABC$. Points $M, N, P$ are located on the sides $AC, AB, BC$, respectively, such that $\angle CBM= \frac{1}{2} \angle AMN = \frac{1}{3} \angle BNP$ and $\angle CMP = 90 ^o$. a) Show that $\vartriangle NMB$ is isosceles. b) Determine $\angle CBM$.

In a convex quadrilateral $ABCD$ we are given that $\angle CBD = 10^o$, $\angle CAD = 20^o$, $\angle ABD = 40^o$, $\angle BAC = 50^o$. Determine the angles $\angle BCD$ and $\angle ADC$.

Let $ABCD$ be a circumscribed quadrilateral and $M$ the centre of the incircle. There are points $P$ and $Q$ on the lines $MA$ and $MC$ such that $\angle CBA= 2\angle QBP.$ Prove that $\angle ADC = 2 \angle PDQ.$

A square is drawn on a sheet of grid paper on the sides of the cells $ABCD$ with side $8$. Point $E$ is the midpoint of side $BC$, $Q$ is such a point on the diagonal $AC$ such that $AQ: QC = 3: 1$. Find the angle between straight lines $AE$ and $DQ$.

In the plane are $10$ lines in general position, which means that no $2$ are parallel and no $3$ are concurrent. Where $2$ lines intersect, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?

$(YUG 3)$ Let four points $A_i (i = 1, 2, 3, 4)$ in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by $S.$ What is the exact placement of the points $A_i$ if $S = 180^{\circ}$?

Let $ABC$ be an isosceles triangle ($AB = AC$) with its circumcenter $O$. Point $N$ is the midpoint of the segment $BC$ and point $M$ is the reflection of the point $N$ with respect to the side $AC$. Suppose that $T$ is a point so that $ANBT$ is a rectangle. Prove that $\angle OMT = \frac{1}{2} \angle BAC$. [i]Proposed by Ali Zamani[/i]

Let $ABC$ be a triangle with $\angle BAC > 90^o$ and with $AB < AC$. Let $r$ be the internal bisector of $\angle ACB$ and let $s$ be the perpendicular, through $A$, on $r$. Denote by $F$ the intersection of $r$ and $ s$, and denote by $E$ the intersection of $s$ with the segment $BC$. Let also $D$ be the symmetric of $A$ with respect to the line $BF$. Assuming that the circumcircle of triangle $EAC$ is tangent to line $AB$ and $ D$ lies on $r$, determine the value of $\angle CDB$.

An acute-angle non-isosceles triangle $ ABC $ is given. The point $ H $ is its orthocenter, the points $ O $ and $ I $ are the centers of its circumscribed and inscribed circles, respectively. The circumcircle of the triangle $ OIH $ passes through the vertex $ A $. Prove that one of the angles of the triangle is $ 60^\circ $.

Let $ABC$ be a triangle and $I$ is your incenter, let $P$ be a point in $AC$ such that $PI$ is perpendicular to $AC$, and let $D$ be the reflection of $B$ wrt circumcenter of $\triangle ABC$. The line $DI$ intersects again the circumcircle of $\triangle ABC$ in the point $Q$. Prove that $QP$ is the angle bisector of the angle $\angle AQC$.

Let $ABCD$ be a square, $E$ a point on the side $CD$, and $F$ a point inside the square such that that triangle $BFE$ is isosceles and $\angle BFE = 90^o$ . If $DF=DE$, find the measure of angle $\angle FDE$.