Let a [i]triplet [/i] be some set of three distinct pairwise parallel lines. $20$ triplets are drawn on a plane. Find the maximum number of regions these $60$ lines can divide the plane into.

The diagram below shows rectangle $ABDE$ where $C$ is the midpoint of side $\overline{BD}$, and $F$ is the midpoint of side $\overline{AE}$. If $AB=10$ and $BD=24$, find the area of the shaded region. [asy] size(300); defaultpen(linewidth(0.8)); pair A = (0,10),B=origin,C=(12,0),D=(24,0),E=(24,10),F=(12,10),G=extension(C,E,D,F); filldraw(A--C--G--F--cycle,gray(0.7)); draw(A--B--D--E--F^^E--G--D); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,S); label("$D$",D,SE); label("$E$",E,NE); label("$F$",F,N); [/asy]

Let $P$ be a point inside a convex quadrilateral $ABCD$. Points $K,L,M,N$ are given on the sides $AB,BC,CD,DA$ respectively such that $PKBL$ and $PMDN$ are parallelograms. Let $S,S_1$, and $S_2$ be the areas of $ABCD, PNAK$, and $PLCM$. Prove that $\sqrt{S}\ge \sqrt{S_1} +\sqrt{S_2}$.

Let $ k$ and $ k'$ two concentric circles centered at $ O$, with $ k'$ being larger than $ k$. A line through $ O$ intersects $ k$ at $ A$ and $ k'$ at $ B$ such that $ O$ seperates $ A$ and $ B$. Another line through $ O$ intersects $ k$ at $ E$ and $ k'$ at $ F$ such that $ E$ separates $ O$ and $ F$. Show that the circumcircle of $ \triangle{OAE}$ and the circles with diametres $ AB$ and $ EF$ have a common point.

In quadrilateral $ABCD$, $AB = CD$, $M$ and $K$ are the midpoints of $BC$ and $AD$.Prove that the angle between $MK$ and $AC$ is equal to the half-sum of angles $BAC$ and $DCA$ [i](Proposed by M.Volchkevich)[/i]

Consider a convex quadrilateral $ABCD$ with $AB=CB$ and $\angle ABC +2 \angle CDA = \pi$ and let $E$ be the midpoint of $AC$. Show that $\angle CDE =\angle BDA$. Paolo Leonetti

The following spiral sequence of squares is drawn on an infinite blackboard: The $1$st square $(1 \times 1)$ has a common vertical side with the $2$nd square (also $1\times 1$) drawn on the right side of it; the 3rd square $(2 \times 2)$ is drawn on the upper side of the $1$st and 2nd ones; the $4$th square $(3 \times 3)$ is drawn on the left side of the $1$st and $3$rd ones; the $5$th square $(5 \times 5)$ is drawn on the bottom side of the $4$th, 1st and $2$nd ones; the $6$th square $(8 \times 8)$ is drawn on the right side, and so on. Each of the squares has a common side with the rectangle consisting of squares constructed earlier. Prove that the centres of all the squares except the $1$st lie on two straight lines. (A Andjans, Riga)

The square was cut into $n^2$ rectangles with sides $a_i \times b_j, i , j= 1,..., n$. For what is the smallest $n$ in the set $\{a_1, b_1, ..., a_n, b_n\}$ all the numbers can be different?

The rectangle is cut into 6 squares, as shown on the figure below. The gray square in the middle has a side equal to 1. What is the area of the rectangle? [img]https://i.ibb.co/gg1tBTN/Kyiv-MO-2023-7-1.png[/img]

Adam is walking in the city. In order to get around a large building, he walks $12$ miles east and then $5$ miles north, then stop. His friend Neutrino, who can go through buildings, starts in the same place as Adam but walks in a straight line to where Adam stops. How much farther than Neutrino does Adam walk? $$ \mathrm a. ~ 1~\mathrm{mile}\qquad \mathrm b.~2 ~\mathrm{miles}\qquad \mathrm c. ~3~\mathrm{miles} \qquad \mathrm d. ~4~\mathrm{miles} \qquad \mathrm e. ~5~\mathrm{miles} $$

In triangle $ABC$, $D$ is the midpoint of side $AB$. $E$ and $F$ are points arbitrarily chosen on segments $AC$ and $BC$, respectively. Show that $[DEF] < [ADE] + [BDF]$.

Let $ABCD$ by a cyclic quadrilateral with circumcenter $O$. Let $P$ be the point of intersection of the diagonals $AC$ and $BD$, and $K, L, M, N$ the circumcenters of triangles $AOP, BOP$, $COP, DOP$, respectively. Prove that $KL = MN$.

(F.Nilov) Two arcs with equal angular measure are constructed on the medians $ AA'$ and $ BB'$ of triangle $ ABC$ towards vertex $ C$. Prove that the common chord of the respective circles passes through $ C$.

In acute triangle $ABC$, points $D$ and $E$ are the feet of the angle bisector and altitude from $A$ respectively. Suppose that $AC - AB = 36$ and $DC - DB = 24$. Compute $EC - EB$.

[u]Round 1 [/u] [b]p1.[/b] Pirate Jim had $8$ boxes with gun powder weighing $1, 2, 3, 4, 5, 6, 7$, and $8$ pounds (the weight is printed on top of every box). Pirate Bob hid a $1$-pound gold bar in one of these boxes. Pirate Jim has a balance scale that he can use, but he cannot open any of the boxes. Help him find the box with the gold bar using two weighings on the balance scale. [b]p2.[/b] James Bond will spend one day at Dr. Evil's mansion to try to determine the answers to two questions: a) Is Dr. Evil at home? b) Does Dr. Evil have an army of ninjas? The parlor in Dr. Evil's mansion has three windows. At noon, Mr. Bond will sneak into the parlor and use open or closed windows to signal his answers. When he enters the parlor, some windows may already be opened, and Mr. Bond will only have time to open or close one window (or leave them all as they are). Help Mr. Bond and Moneypenny design a code that will tell Moneypenny the answers to both questions when she drives by later that night and looks at the windows. Note that Moneypenny will not have any way to know which window Mr. Bond opened or closed. [b]p3.[/b] Suppose that you have a triangle in which all three side lengths and all three heights are integers. Prove that if these six lengths are all different, there cannot be four prime numbers among them. p4. Fred and George have designed the Amazing Maze, a $5\times 5$ grid of rooms, with Adorable Doors in each wall between rooms. If you pass through a door in one direction, you gain a gold coin. If you pass through the same door in the opposite direction, you lose a gold coin. The brothers designed the maze so that if you ever come back to the room in which you started, you will find that your money has not changed. Ron entered the northwest corner of the maze with no money. After walking through the maze for a while, he had $8$ shiny gold coins in his pocket, at which point he magically teleported himself out of the maze. Knowing this, can you determine whether you will gain or lose a coin when you leave the central room through the north door? [b]p5.[/b] Bill and Charlie are playing a game on an infinite strip of graph paper. On Bill’s turn, he marks two empty squares of his choice (not necessarily adjacent) with crosses. Charlie, on his turn, can erase any number of crosses, as long as they are all adjacent to each other. Bill wants to create a line of $2013$ crosses in a row. Can Charlie stop him? [u]Round 2 [/u] [b]p6.[/b] $1000$ non-zero numbers are written around a circle and every other number is underlined. It happens that each underlined number is equal to the sum of its two neighbors and that each non-underlined number is equal to the product of its two neighbors. What could the sum of all the numbers written on the circle be? [b]p7.[/b] A grasshopper is sitting at the edge of a circle of radius $3$ inches. He can hop exactly $4$ inches in any direction, as long as he stays within the circle. Which points inside the circle can the grasshopper reach if he can make as many jumps as he likes? [img]https://cdn.artofproblemsolving.com/attachments/1/d/39b34b2b4afe607c1232f4ce9dec040a34b0c8.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $ B$ be the total area of the blue triangles, $ W$ the total area of the white squares, and $ R$ the area of the red square. Which of the following is correct? [asy]unitsize(3mm); fill((-4,-4)--(-4,4)--(4,4)--(4,-4)--cycle,blue); fill((-2,-2)--(-2,2)--(2,2)--(2,-2)--cycle,red); path onewhite=(-3,3)--(-2,4)--(-1,3)--(-2,2)--(-3,3)--(-1,3)--(0,4)--(1,3)--(0,2)--(-1,3)--(1,3)--(2,4)--(3,3)--(2,2)--(1,3)--cycle; path divider=(-2,2)--(-3,3)--cycle; fill(onewhite,white); fill(rotate(90)*onewhite,white); fill(rotate(180)*onewhite,white); fill(rotate(270)*onewhite,white);[/asy] $ \textbf{(A)}\ B \equal{} W \qquad \textbf{(B)}\ W \equal{} R \qquad \textbf{(C)}\ B \equal{} R \qquad \textbf{(D)}\ 3B \equal{} 2R \qquad \textbf{(E)}\ 2R \equal{} W$

Show that among all quadrilaterals of a given perimeter the square has the largest area.

Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$. Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$. If $\angle MPN = 40^\circ$, find the degree measure of $\angle BPC$. [i]Ray Li.[/i]

Let $ABC$ be an acute triangle. A line, parallel to $BC$, intersects sides $AB$ and $AC$ at points $M$ and $P$, respectively. At which placement of points $M$ and $P$, is the radius of the circumcircle of the triangle $BMP$ is the smallest?

Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$),\linebreak $\omega_1$ and $\omega_2$ be the circumcircles of triangles $AEM$ and $CDM.$ It is known that the circles $\omega_1$ and $\omega_2$ are tangent. Find the ratio in which the circle $\omega_1$ divides $AB.$

Find all triples $ (x,y,z)$ of integers which satisfy $ x(y \plus{} z) \equal{} y^2 \plus{} z^2 \minus{} 2$ $ y(z \plus{} x) \equal{} z^2 \plus{} x^2 \minus{} 2$ $ z(x \plus{} y) \equal{} x^2 \plus{} y^2 \minus{} 2$.

Let $ F$ be the midpoint of the side $ BC$ of a triangle $ ABC$. Isosceles right-angled triangles $ ABD$ and $ ACE$ are constructed externally on $ AB$ and $ AC$ with the right angles at $ D$ and $ E$. Prove that the triangle $ DEF$ is right-angled and isosceles.

In a triangle $ABC$, two angle trisectors of $A$ intersect with $BC$ at $D$ and $E$ respectively so that $B,D,E,C$ comes in order. If we have $BD = 3$, $DE = 1$ and $EC = 2$, find $\angle DAE$.

Let $ABC$ be an acute triangle and $D$ an interior point of segment $BC$. Points $E$ and $F$ lie in the half-plane determined by the line $BC$ containing $A$ such that $DE$ is perpendicular to $BE$ and $DE$ is tangent to the circumcircle of $ACD$, while $DF$ is perpendicular to $CF$ and $DF$ is tangent to the circumcircle of $ABD$. Prove that the points $A, D, E$ and $F$ are concyclic.

We are given an acute triangle $ABC$ with $AB > AC$ and orthocenter $H$. The point $E$ lies symmetric to $C$ with respect to the altitude $AH$. Let $F$ be the intersection of the lines $EH$ and $AC$. Prove that the circumcenter of the triangle $AEF$ lies on the line $AB$. (Karl Czakler)