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

[b]p1.[/b] Consider a triangular grid: nodes of the grid are painted black and white. At a single step you are allowed to change colors of all nodes situated on any straight line (with the slope $0^o$ ,$60^o$, or $120^o$ ) going through the nodes of the grid. Can you transform the combination in the left picture into the one in the right picture in a finite number of steps? [img]https://cdn.artofproblemsolving.com/attachments/3/a/957b199149269ce1d0f66b91a1ac0737cf3f89.png[/img] [b]p2.[/b] Find $x$ satisfying $\sqrt{x\sqrt{x \sqrt{x ...}}} = \sqrt{2022}$ where it is an infinite expression on the left side. [b]p3.[/b] $179$ glasses are placed upside down on a table. You are allowed to do the following moves. An integer number $k$ is fixed. In one move you are allowed to turn any $k$ glasses . (a) Is it possible in a finite number of moves to turn all $179$ glasses into “bottom-down” positions if $k=3$? (b) Is it possible to do it if $k=4$? [b]p4.[/b] An interval of length $1$ is drawn on a paper. Using a compass and a simple ruler construct an interval of length $\sqrt{93}$. [b]p5.[/b] Show that $5^{2n+1} + 3^{n+2} 2^{n-1} $ is divisible by $19$ for any positive integer $n$. [b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=1-z \\ \dfrac{yz}{y+z}=2-x \\ \dfrac{xz}{x+z}=2-y \end{cases}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

You have an equilateral paper triangle of area $9$ and fold it in two, following a straight line that passes through the center of the triangle and does not contain any vertex of the triangle. Thus there remains a quadrilateral in which the two pieces overlap, and three triangles without overlaps. Determine the smallest possible value of the quadrilateral area of the overlay.

Let $V$ be a rectangular prism with integer side lengths. The largest face has area $240$ and the smallest face has area $48$. A third face has area $x$, where $x$ is not equal to $48$ or $240$. What is the sum of all possible values of $x$?

The tangent at $C$ to the circumcircle of triangle $ABC$ intersects the line through $A$ and $B$ in a point $D$. Two distinct points $E$ and $F$ on the line through $B$ and $C$ satisfy $|BE| = |BF | =\frac{||CD|^2 - |BD|^2|}{|BC|}$. Show that either $|ED| = |CD|$ or $|FD| = |CD|$.

All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$.

The squares $OABC$ and $OA_1B_1C_1$ are situated in the same plane and are directly oriented. Prove that the lines $AA_1$ , $BB_1$, and $CC_1$ are concurrent.

A semicircle of radius $2$ is inscribed inside of a rectangle, as shown in the diagram below. The diameter of the semicircle coincides with the bottom side of the rectangle, and the semicircle is tangent to the rectangle at all points of intersection. Compute the length of the diagonal of the rectangle. [img]https://cdn.artofproblemsolving.com/attachments/c/7/81fcfb759188eae7dcb82fa5d58fb9525d85de.png[/img]

Over the sides of an equilateral triangle with area $ 1$ are triangles with the opposite angle $ 60^{\circ}$ to each side drawn outside of the triangle. The new corners are $ P$, $ Q$ and $ R$. (and the new triangles $ APB$, $ BQC$ and $ ARC$) 1)What is the highest possible area of the triangle $ PQR$? 2)What is the highest possible area of the triangle whose vertexes are the midpoints of the inscribed circles of the triangles $ APB$, $ BQC$ and $ ARC$?

A set of points $P_i$ [i]covers[/i] a polygon if for every point in the polygon, a line can be drawn inside the polygon to at least one $P_i$. Points $A_1, A_2, \cdots, A_n$ in the plane form a $2007$-gon, not necessarily convex. Find the minimum value of $n$ such that for any such polygon, we can pick $n$ points inside it that cover the polygon.

Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$

Taotao wants to buy a bracelet. The bracelets have 7 different beads on them, arranged in a circle. Two bracelets are the same if one can be rotated or flipped to get the other. If she can choose the colors and placement of the beads, and the beads come in orange, white, and black, how many possible bracelets can she buy?

The insphere of any tetrahedron has radius $r$. The four tangential planes parallel to the side faces of the tetrahedron cut from the tetrahedron four smaller tetrahedrons whose in-sphere radii are $r_1, r_2, r_3$ and $r_4$. Prove that $$r_1 + r_2 + r_3 + r_4 = 2r$$

The altitude drawn to the base of an isosceles triangle is $ 8$, and the perimeter $ 32$. The area of the triangle is: $ \textbf{(A)}\ 56\qquad \textbf{(B)}\ 48\qquad \textbf{(C)}\ 40\qquad \textbf{(D)}\ 32\qquad \textbf{(E)}\ 24$

Let $ABC$ be an acute non isosceles triangle. The angle bisector of angle $A$ meets again the circumcircle of the triangle $ABC$ in $D$. Let $O$ be the circumcenter of the triangle $ABC$. The angle bisectors of $\angle AOB$, and $\angle AOC$ meet the circle $\gamma$ of diameter $AD$ in $P$ and $Q$ respectively. The line $PQ$ meets the perpendicular bisector of $AD$ in $R$. Prove that $AR // BC$.

The inscribed circle $\omega$ of the triangle $ABC$ touches its sides $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$, respectively. Let $S$ be the intersection point of lines passing through points $B$ and $C$ and parallel to $A_1C_1$ and $A_1B_1$ respectively, $A_0$ be the foot of the perpendicular drawn from point $A_1$ on $B_1C_1$, $G_1$ be the centroid of triangle $A_1B_1C_1$, $P$ be the intersection point of the ray $G_1A_0$ with $\omega$. Prove that points $S, A_1$, and $P$ lie on a straight line.

Let $k$ be a circle with centre $M$ and let $AB$ be a diameter of $k$. Furthermore, let $C$ be a point on $k$ such that $AC = AM$. Let $D$ be the point on the line $AC$ such that $CD = AB$ and $C$ lies between $A$ and $D$. Let $E$ be the second intersection of the circumcircle of $BCD$ with line $AB$ and $F$ be the intersection of the lines $ED$ and $BC$. The line $AF$ cuts the segment $BD$ in $X$. Determine the ratio $BX/XD$.

The center $O$ of the circle $\omega$ passing through the vertex $C$ of the isosceles triangle $ABC$ ($AB = AC$) is the interior point of the triangle $ABC$. This circle intersects segments $BC$ and $AC$ at points $D \ne C$ and $E \ne C$, respectively, and the circumscribed circle $\Omega$ of the triangle $AEO$ at the point $F \ne E$. Prove that the center of the circumcircle of the triangle $BDF$ lies on the circle $\Omega$.

Line $\ell$ in the coordinate plane has the equation $3x - 5y + 40 = 0$. This line is rotated $45^{\circ}$ counterclockwise about the point $(20, 20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k?$ $\textbf{(A) } 10 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 25 \qquad \textbf{(E) } 30$

Let $\triangle ABC$ be an acute-angled triangle. Let $P$ be the midpoint of $BC$, and $K$ the foot of the altitude from $A$ to side $BC$. Let $D$ be a point on segment $AP$ such that $\angle BDC = 90^\circ$. Let $E$ be the second point of intersection of line $BC$ with the circumcircle of $\triangle ADK$. Let $F$ be the second point of intersection of line $AE$ with the circumcircle of $\triangle ABC$. Prove that $\angle AFD = 90^\circ$.

Given an angle $ \angle QBP$ and a point $ L$ outside the angle $ \angle QBP$. Draw a straight line through $ L$ meeting $ BQ$ in $ A$ and $ BP$ in $ C$ such that the triangle $ \triangle ABC$ has a given perimeter.

A set $A_1 , A_2 , A_3 , A_4$ of 4 points in the plane is said to be [i]Athenian[/i] set if there is a point $P$ of the plane satsifying (*) $P$ does not lie on any of the lines $A_i A_j$ for $1 \leq i < j \leq 4$; (**) the line joining $P$ to the mid-point of the line $A_i A_j$ is perpendicular to the line joining $P$ to the mid-point of $A_k A_l$, $i,j,k,l$ being distinct. (a) Find all [i]Athenian[/i] sets in the plane. (b) For a given [i]Athenian[/i] set, find the set of all points $P$ in the plane satisfying (*) and (**)

In triangle $ABC$, the angle at $C$ is $30^o$, side $BC$ has length $4$, and side $AC$ has length $5$. Let $ P$ be the point such that triangle $ABP$ is equilateral and non-overlapping with triangle $ABC$. Find the distance from $C$ to $ P$.

Let $ABCD$ be an arbitrary quadrilateral. Let squares $ABB_1A_2, BCC_1B_2, CDD_1C_2, DAA_1D_2$ be constructed in the exterior of the quadrilateral. Furthermore, let $AA_1PA_2$ and $CC_1QC_2$ be parallelograms. For any arbitrary point $P$ in the interior of $ABCD$, parallelograms $RASC$ and $RPTQ$ are constructed. Prove that these two parallelograms have two vertices in common.

If $AG_a,BG_b$, and $CG_c$ are symmedians in $\Delta ABC$ ($G_a\in BC,G_b\in AC,G_c\in AB$), is it possible for $\Delta G_a G_b G_c$ to be equilateral when $\Delta ABC$ is not equilateral?