A pyramid $MABCD$ with the top-vertex $M$ is circumscribed about a sphere with center $O$ so that $O$ lies on the altitude of the pyramid. Each of the planes $ACM,BDM,ABO$ divides the lateral surface of the pyramid into two parts of equal areas. The areas of the sections of the planes $ACM$ and $ABO$ inside the pyramid are in ratio $(\sqrt2+2):4$. Determine the angle $\delta$ between the planes $ACM$ and $ABO$, and the dihedral angle of the pyramid at the edge $AB$.

In a convex quadrilateral all sidelengths and all angles are pairwise different. a) Can the greatest angle be adjacent to the greatest side and at the same time the smallest angle be adjacent to the smallest side? b) Can the greatest angle be non-adjacent to the smallest side and at the same time the smallest angle be non-adjacent to the greatest side?

[b]a)[/b] Let be two nonzero complex numbers $ a,b. $ Show that the area of the triangle formed by the representations of the affixes $ 0,a,b $ in the complex plane is $ \frac{1}{4}\left| \overline{a} b-a\overline{b} \right| . $ [b]b)[/b] Let be an equilateral triangle $ ABC, $ its circumcircle $ \mathcal{C} , $ its circumcenter $ O, $ and two distinct points $ P_1,P_2 $ in the interior of $ \mathcal{C} . $ Prove that we can form two triangles with sides $ P_1A,P_1B,P_1C, $ respectively, $ P_2A,P_2B,P_2C, $ whose areas are equal if and only if $ OP_1=OP_2. $

Two parallelograms $ABCD$ and $A'B'C'D'$ are given in space. Points $A'',B'',C'',D''$ divide the segments $AA',BB',CC',DD'$ in the same ratio. What can be said about the quadrilateral $A''B''C''D''$?

Let the set $S = \{P_1, P_2, \cdots, P_{12}\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called communal if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.)

A cubical box with sides of length $7$ has vertices at $(0,0,0)$, $(7,0,0)$, $(0,7,0)$, $(7,7,0)$, $(0,0,7)$, $(7,0,7)$, $(0,7,7)$, $(7,7,7)$. The inside of the box is lined with mirrors and from the point $(0,1,2)$, a beam of light is directed to the point $(1,3,4)$. The light then reflects repeatedly off the mirrors on the inside of the box. Determine how far the beam of light travels before it first returns to its starting point at $(0,1,2)$.

[b]p1.[/b] Find any integer solution of the puzzle: $WE+ST+RO+NG=128$ (different letters mean different digits between $1$ and $9$). [b]p2.[/b] A $5\times 6$ rectangle is drawn on the piece of graph paper (see the figure below). The side of each square on the graph paper is $1$ cm long. Cut the rectangle along the sides of the graph squares in two parts whose areas are equal but perimeters are different by $2$ cm. $\begin{tabular}{|l|l|l|l|l|l|} \hline & & & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline \end{tabular}$ [b]p3.[/b] Three runners started simultaneously on a $1$ km long track. Each of them runs the whole distance at a constant speed. Runner $A$ is the fastest. When he runs $400$ meters then the total distance run by runners $B$ and $C$ together is $680$ meters. What is the total combined distance remaining for runners $B$ and $C$ when runner $A$ has $100$ meters left? [b]p4.[/b] There are three people in a room. Each person is either a knight who always tells the truth or a liar who always tells lies. The first person said «We are all liars». The second replied «Only you are a liar». Is the third person a liar or a knight? [b]p5.[/b] A $5\times 8$ rectangle is divided into forty $1\times 1$ square boxes (see the figure below). Choose 24 such boxes and one diagonal in each chosen box so that these diagonals don't have common points. $\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline & & & & & & & \\ \hline \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a circle $K$, find the locus of vertices $A$ of parallelograms $ABCD$ with diagonals $AC \leq BD$, such that $BD$ is inside $K$.

The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \ \text{cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d \ \text{cm}$. The square has perimeter greater than 0. How many positive integers are NOT possible value for $d$? $\text{(A)} \ 0 \qquad \text{(B)} \ 9 \qquad \text{(C)} \ 221 \qquad \text{(D)} \ 663 \qquad \text{(E)} \ \text{infinitely many}$

The triangles $AB'C, CA'B$ and $BC'A$ are constructed on the sides of the equilateral triangle $ABC.$ In the resulting hexagon $AB'CA'BC'$ each of the angles $\angle A'BC',\angle C'AB'$ and $\angle B'CA'$ is greater than $120^\circ$ and the sides satisfy the equalities $AB' = AC',BC' = BA'$ and $CA' = CB'.$ Prove that the segments $AB',BC'$ and $CA'$ can form a triangle. [i]David Brodsky[/i]

A closed non-self-intersecting polygonal chain is drawn through the centers of some squares on the $8\times 8$ chess board. Every link of the chain connects the centers of adjacent squares either horizontally, vertically or diagonally, where the two squares are adjacent if they share an edge or a corner. For the interior polygon bounded by the chain, prove that the total area of black pieces equals the total area of white pieces. (Author: D. Khramtsov)

In triangle $ABC$, $BC = 23$, $CA = 27$, and $AB = 30$. Points $V$ and $W$ are on $\overline{AC}$ with $V$ on $\overline{AW}$, points $X$ and $Y$ are on $\overline{BC}$ with $X$ on $\overline{CY}$, and points $Z$ and $U$ are on $\overline{AB}$ with $Z$ on $\overline{BU}$. In addition, the points are positioned so that $\overline{UV} \parallel \overline{BC}$, $\overline{WX} \parallel \overline{AB}$, and $\overline{YZ} \parallel \overline{CA}$. Right angle folds are then made along $\overline{UV}$, $\overline{WX}$, and $\overline{YZ}$. The resulting figure is placed on a level floor to make a table with triangular legs. Let $h$ be the maximum possible height of a table constructed from triangle $ABC$ whose top is parallel to the floor. Then $h$ can be written in the form $\tfrac{k \sqrt{m}}{n}$, where $k$ and $n$ are relatively prime positive integers and $m$ is a positive integer that is not divisible by the square of any prime. Find $k + m + n$. [asy] unitsize(1 cm); pair translate; pair[] A, B, C, U, V, W, X, Y, Z; A[0] = (1.5,2.8); B[0] = (3.2,0); C[0] = (0,0); U[0] = (0.69*A[0] + 0.31*B[0]); V[0] = (0.69*A[0] + 0.31*C[0]); W[0] = (0.69*C[0] + 0.31*A[0]); X[0] = (0.69*C[0] + 0.31*B[0]); Y[0] = (0.69*B[0] + 0.31*C[0]); Z[0] = (0.69*B[0] + 0.31*A[0]); translate = (7,0); A[1] = (1.3,1.1) + translate; B[1] = (2.4,-0.7) + translate; C[1] = (0.6,-0.7) + translate; U[1] = U[0] + translate; V[1] = V[0] + translate; W[1] = W[0] + translate; X[1] = X[0] + translate; Y[1] = Y[0] + translate; Z[1] = Z[0] + translate; draw (A[0]--B[0]--C[0]--cycle); draw (U[0]--V[0],dashed); draw (W[0]--X[0],dashed); draw (Y[0]--Z[0],dashed); draw (U[1]--V[1]--W[1]--X[1]--Y[1]--Z[1]--cycle); draw (U[1]--A[1]--V[1],dashed); draw (W[1]--C[1]--X[1]); draw (Y[1]--B[1]--Z[1]); dot("$A$",A[0],N); dot("$B$",B[0],SE); dot("$C$",C[0],SW); dot("$U$",U[0],NE); dot("$V$",V[0],NW); dot("$W$",W[0],NW); dot("$X$",X[0],S); dot("$Y$",Y[0],S); dot("$Z$",Z[0],NE); dot(A[1]); dot(B[1]); dot(C[1]); dot("$U$",U[1],NE); dot("$V$",V[1],NW); dot("$W$",W[1],NW); dot("$X$",X[1],dir(-70)); dot("$Y$",Y[1],dir(250)); dot("$Z$",Z[1],NE); [/asy]

How many integer triangles are there with inradius $1$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{Infinite} $

[b]p16[/b] Madelyn is being paid $\$50$/hour to find useful [i]Non-Functional Trios[/i], where a Non-Functional Trio is defined as an ordered triple of distinct real numbers $(a, b, c)$, and a Non- Functional Trio is [i]useful [/i] if $(a, b)$, $(b, c)$, and $(c, a)$ are collinear in the Cartesian plane. Currently, she’s working on the case $a+b+c = 2022$. Find the number of useful Non-Functional Trios $(a, b, c)$ such that $a + b + c = 2022$. [b]p17[/b] Let $p(x) = x^2 - k$, where $k$ is an integer strictly less than $250$. Find the largest possible value of k such that there exist distinct integers $a, b$ with $p(a) = b$ and $p(b) = a$. [b]p18[/b] Let $ABC$ be a triangle with orthocenter $H$ and circumcircle $\Gamma$ such that $AB = 13$, $BC = 14$, and $CA = 15$. $BH$ and $CH$ meet $\Gamma$ again at points $D$ and $E$, respectively, and $DE$ meets $AB$ and $AC$ at $F$ and $G$, respectively. The circumcircles of triangles $ABG$ and $ACF$ meet BC again at points $P$ and $Q$. If $PQ$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd (a, b) = 1$, find $a + b$.

Let $ABC$ be an acute triangle. Let $D$ and $E$ be the feet of the altitudes of the triangle $ABC$ from $A$ and $B$, respectively. Let $F$ be the reflection of the point $A$ over $BC$. Let $G$ be a point such that the quadrilateral $ABCG$ is a parallelogram. Show that the circumcircles of triangles $BCF$ , $ACG$ and $CDE$ are concurrent on a point different from $C$.

Prove that it is possible to place $2n(2n + 1)$ parallelepipedic (rectangular) pieces of soap of dimensions $1 \times 2 \times (n + 1)$ in a cubic box with edge $2n + 1$ if and only if $n$ is even or $n = 1$. [i]Remark[/i]. It is assumed that the edges of the pieces of soap are parallel to the edges of the box.

Can you draw some diagonals in a convex $2014$-gon so that they do not intersect, the whole $2014$-gon is divided into triangles and each vertex belongs to an odd number of these triangles?

You are given a quadrilateral $ABCD$. It is known that $\angle BAC = 30^o$, $\angle D = 150^o$ and, in addition, $AB = BD$. Prove that $AC$ is the bisector of angle $C$.

Through a random point $C_1$ from the edge $DC$ of the regular tetrahedron $ABCD$ is drawn a plane, parallel to the plane $ABC$. The plane constructed intersects the edges $DA$ and $DB$ at the points $A_1,B_1$ respectively. Let the point $H$ is the midpoint of the altitude through the vertex $D$ of the tetrahedron $DA_1B_1C_1$ and $M$ is the center of gravity (barycenter) of the triangle $ABC_1$. Prove that the measure of the angle $HMC$ doesn’t depend on the position of the point $C_1$. [i](Ivan Tonov)[/i]

Two chords are drawn in a circle of radius equal to unit, $AB$ and $AC$ of equal length. a) Describe how you can construct a third chord $DE$ that is divided into three equal parts by the intersections with $AB$ and $AC$. b) If $AB = AC =\sqrt2$, what are the lengths of the two segments that the chord $DE$ determines in $AB$?

[b]8.1[/b] On the median drawn from the vertex of the triangle to the base, point $A$ is taken. The sum of the distances from $A$ to the sides of the triangle is equal to $s$. Find the distances from $A$ to the sides if the lengths of the sides are equal to $x$ and $y$. [b]8.2[/b] Fraction $0, abc...$ is composed according to the following rule: $a$ and $c$ are arbitrary digits, and each next digit is equal to the remainder of the sum of the previous two digits when divided by $10$. Prove that this fraction is purely periodic. [b]8.3[/b] Two convex polygons with $m$ and $n$ sides are drawn on the plane ($m>n$). What is the greatest possible number of parts, they can break the plane? [b]8.4 [/b]The sum of three integers that are perfect squares is divisible by $9$. Prove that among them, there are two numbers whose difference is divisible by $9$. [b]8.5 / 9.5[/b] Given $k+2$ integers. Prove that among them there are two integers such that either their sum or their difference is divisible by $2k$. [b]8.6[/b] A right angle rotates around its vertex. Find the locus of the midpoints of the segments connecting the intersection points sides of an angle and a given circle. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].

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]

Let $P$ be a non-degenerate polygon with $n$ sides, where $n > 4$. Prove that there exist three distinct vertices $A, B, C$ of $P$ with the following property:If $\ell_1,\ell_2,\ell_3$ are the lengths of the three polygonal chains into which $A, B, C$ break the perimeter of $P$, then there is a triangle with side lengths $\ell_1,\ell_2$ and $\ell_3$. [size=150]Remark[/size]: By a non-degenerate polygon we mean a polygon in which every two sides are disjoint, apart from consecutive ones, which share only the common endpoint.(Poland)

Quadrilateral $ABCD$ is inscribed in circle. Points $P$ and $Q$ lie respectively on rays $AB^{\rightarrow}$ and $AD^{\rightarrow}$ such that $AP = CD$, $AQ = BC$. Show that middle point of line segment $PQ$ lies on the line $AC$.

Let the inscribed circle $\omega$ of the triangle $ABC$ have a center $I{}$ and touch the sides $BC, CA$ and $AB$ at points $D, E$ and $F{}$ respectively. Let $M{}$ and $N{}$ be points on the straight line $EF$ such that $BM \parallel AC$ and $CN \parallel AB$. Let $P{}$ and $Q{}$ be points on the segments $DM{}$ and $DN{}$, respectively, such that $BP \parallel CQ$. Prove that the intersection point of the lines $PF$ and $QE$ lies on $\omega$. [i]Proposed by Don Luu (Vietnam)[/i]