A chord $AB = a$ is drawn in a circle of radius $B$. A circle with center on line $AB$ passes through $A$ and intersects this circle a second time at point $C$. Let $M$ be an arbitrary point of the second circle. Straight lines $MA$ and $MC$ intersect the first circle a second time at points $P$ and $Q$. Find $PQ$.

Let $ABC$ be an acute-angled triangle with $AB =4$ and $CD$ be the altitude through $C$ with $CD = 3$. Find the distance between the midpoints of $AD$ and $BC$

Consider two triangular pyramids $ABCD$ and $A'BCD$, with a common base $BCD$, and such that $A'$ is inside $ABCD$. Prove that the sum of planar angles at vertex $A'$ of pyramid $A'BCD$ is greater than the sum of planar angles at vertex $A$ of pyramid $ABCD$.

On the planet Brick, which has the shape of a rectangular parallelepiped with edges of $1$ km,$ 2$ km and $4$ km, the Little Prince built a house in the center of the largest face. What is the distance from the house to the most remote point on the planet? (The distance between two points on the surface of a planet is defined as the length of the shortest path along the surface connecting these points.)

[u]Round 1[/u] [b]p1.[/b] The alphabet of the Aau-Bau language consists of two letters: A and B. Two words have the same meaning if one of them can be constructed from the other by replacing any AA with A, replacing any BB with B, or by replacing any ABA with BAB. For example, the word AABA means the same thing as ABA, and AABA also means the same thing as ABAB. In this language, is it possible to name all seven days of the week? [b]p2.[/b] A museum has a $4\times 4$ grid of rooms. Every two rooms that share a wall are connected by a door. Each room contains some paintings. The total number of paintings along any path of $7$ rooms from the lower left to the upper right room is always the same. Furthermore, the total number of paintings along any path of $7$ rooms from the lower right to the upper left room is always the same. The guide states that the museum has exactly $500$ paintings. Show that the guide is mistaken. [img]https://cdn.artofproblemsolving.com/attachments/7/6/0fd93a0deaa71a5bb1599d2488f8b4eac5d0eb.jpg[/img] [b]p3.[/b] A playground has a swing-set with exactly three swings. When 3rd and 4th graders from Dr. Anna’s math class play during recess, she has a rule that if a $3^{rd}$ grader is in the middle swing there must be $4^{th}$ graders on that person’s left and right. And if there is a $4^{th}$ grader in the middle, there must be $3^{rd}$ graders on that person’s left and right. Dr. Anna calculates that there are $350$ different ways her students can arrange themselves on the three swings with no empty seats. How many students are in her class? [img]https://cdn.artofproblemsolving.com/attachments/5/9/4c402d143646582376d09ebbe54816b8799311.jpg[/img] [b]p4.[/b] The archipelago Artinagos has $19$ islands. Each island has toll bridges to at least $3$ other islands. An unsuspecting driver used a bad mapping app to plan a route from North Noether Island to South Noether Island, which involved crossing $12$ bridges. Show that there must be a route with fewer bridges. [img]https://cdn.artofproblemsolving.com/attachments/e/3/4eea2c16b201ff2ac732788fe9b78025004853.jpg[/img] [b]p5.[/b] Is it possible to place the numbers from $1$ to $121$ in an $11\times 11$ table so that numbers that differ by $1$ are in horizontally or vertically adjacent cells and all the perfect squares $(1, 4, 9, ... , 121)$ are in one column? [u]Round 2[/u] [b]p6.[/b] Hungry and Sneaky have opened a rectangular box of chocolates and are going to take turns eating them. The chocolates are arranged in a $2m \times 2n$ grid. Hungry can take any two chocolates that are side-by-side, but Sneaky can take only one at a time. If there are no more chocolates located side-by-side, all remaining chocolates go to Sneaky. Hungry goes first. Each player wants to eat as many chocolates as possible. What is the maximum number of chocolates Sneaky can get, no matter how Hungry picks his? [img]https://cdn.artofproblemsolving.com/attachments/b/4/26d7156ca6248385cb46c6e8054773592b24a3.jpg[/img] [b]p7.[/b] There is a thief hiding in the sultan’s palace. The palace contains $2019$ rooms connected by doors. One can walk from any room to any other room, possibly through other rooms, and there is only one way to do this. That is, one cannot walk in a loop in the palace. To catch the thief, a guard must be in the same room as the thief at the same time. Prove that $11$ guards can always find and catch the thief, no matter how the thief moves around during the search. [img]https://cdn.artofproblemsolving.com/attachments/a/b/9728ac271e84c4954935553c4d58b3ff4b194d.jpg[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two circles have a common internal tangent of length $17$ and a common external tangent of length $25$. Find the product of the radii of the two circles.

The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle? [asy] size(5cm); filldraw((4,0)--(8,3)--(8-3/4,4)--(1,4)--cycle,mediumgray); draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.1)); draw((1,0)--(1,4)--(4,0)--(8,3)--(5,7)--(1,4),linewidth(1.1)); label("$4$", (8,2), E); label("$8$", (4,0), S); label("$5$", (3,11/2), NW); draw((1,.35)--(1.35,.35)--(1.35,0),linewidth(.4)); draw((5,7)--(5+21/100,7-28/100)--(5-7/100,7-49/100)--(5-28/100,7-21/100)--cycle,linewidth(.4)); [/asy] $\textbf{(A) } 15\dfrac{1}{8} \qquad \textbf{(B) } 15\dfrac{3}{8} \qquad \textbf{(C) } 15\dfrac{1}{2} \qquad \textbf{(D) } 15\dfrac{5}{8} \qquad \textbf{(E) } 15\dfrac{7}{8}$

Triangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11.$ Suppose that there is a point $D$ on $\overline{AM}$ with $AD=10$ and $\angle BDC=3\angle BAC.$ Then the perimeter of $\triangle ABC$ may be written in the form $a+\sqrt{b},$ where $a$ and $b$ are integers. Find $a+b.$ [asy] import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3; draw((1,3)--(0,0)); draw((0,0)--(2,0)); draw((2,0)--(1,3)); draw((1,3)--(1,0)); draw((1,0.7)--(0,0)); draw((1,0.7)--(2,0)); label("$11$",(0.75,1.63),SE*lsf); dot((1,3),ds); label("$A$",(0.96,3.14),NE*lsf); dot((0,0),ds); label("$B$",(-0.15,-0.18),NE*lsf); dot((2,0),ds); label("$C$",(2.06,-0.18),NE*lsf); dot((1,0),ds); label("$M$",(0.97,-0.27),NE*lsf); dot((1,0.7),ds); label("$D$",(1.05,0.77),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

A line $l$ is drawn through the intersection point $H$ of altitudes of acute-angle triangles. Prove that symmetric images $l_a, l_b, l_c$ of $l$ with respect to the sides $BC,CA,AB$ have one point in common, which lies on the circumcircle of $ABC.$

In an acute-angled triangle $ABC$, the points $M$ and $N$ are the midpoints of the sides $AB$ and $AC$, respectively. For an arbitrary point $S$ lying on the side of $BC$ prove that the condition holds $(MB- MS)(NC-NS) \le 0$

In the triangle $ABC$ , the angle bisector $AD$ divides the side $BC$ into the ratio $BD: DC = 2: 1$. In what ratio, does the median $CE$ divide this bisector?

Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. The point $F$ belongs to the side $AB$, and $DE \perp EF$. The point $G$ lies inside the square, and $GF = FE$ and $GF \perp FE$. Prove that: a) $DE$ is the bisector of the $\angle FDC$ b) $FG$ is the bisector of the $\angle AFD$ c) the point $G$ is the center of the circle inscribed in the triangle $ADF$. (Ercole Suppa, Italy)

Sides $AB,~ BC, ~CD$ and $DA$, respectively, of convex quadrilateral $ABCD$ are extended past $B,~ C ,~ D$ and $A$ to points $B',~C',~ D'$ and $A'$. Also, $AB = BB' = 6,~ BC = CC' = 7, ~CD = DD' = 8$ and $DA = AA' = 9$; and the area of $ABCD$ is 10. The area of $A 'B 'C'D'$ is $\textbf{(A) }20\qquad\textbf{(B) }40\qquad\textbf{(C) }45\qquad\textbf{(D) }50\qquad \textbf{(E) }60$

Let $x$ be a complex number such that $x^{2011}=1$ and $x\neq 1$. Compute the sum \[\dfrac{x^2}{x-1}+\dfrac{x^4}{x^2-1}+\dfrac{x^6}{x^3-1}+\cdots+\dfrac{x^{4020}}{x^{2010}-1}.\]

Let $D,E,F$ denote the tangent points of the incircle of $ABC$ with sides $BC,AC,AB$ respectively. Let $M$ be the midpoint of the segment $EF$. Let $L$ be the intersection point of the circle passing through $D,M,F$ and the segment $AB$, $K$ be the intersection point of the circle passing through $D,M,E$ and the segment $AC$. Prove that the circle passing through $A,K,L$ touches the line $BC$

A closed line on a plane is such that any quadrangle inscribed in it has the sum of opposite angles equal to $180^o$. Prove that this line is a circle.

Given the circle $c(O,R)$ (with center $O$ and radius $R$), one diameter $AB$ and midpoint $C$ of the arc $AB$. Consider circle $c_1(K,KO)$, where center $K$ lies on the segment $OA$, and consider the tangents $CD,CO$ from the point $C$ to circle $c_1(K,KO)$. Line $KD$ intersects circle $c(O,R)$ at points $E$ and $Z$ (point $E$ lies on the semicircle that lies also point $C$). Lines $EC$ and $CZ$ intersects $AB$ at points $N$ and $M$ respectively. Prove that quadrilateral $EMZN$ is an isosceles trapezoid, inscribed in a circle whose center lie on circle $c(O,R)$.

Let $ABC$ be an acute, non isosceles triangle with $M, N, P$ are midpoints of $BC, CA, AB$, respectively. Denote $d_1$ as the line passes through $M$ and perpendicular to the angle bisector of $\angle BAC$, similarly define for $d_2, d_3$. Suppose that $d_2 \cap d_3 = D$, $d_3 \cap d_1 = E,$ $d_1 \cap d_2 = F$. Let $I, H$ be the incenter and orthocenter of triangle $ABC$. Prove that the circumcenter of triangle $DEF$ is the midpoint of segment $IH$.

Let $ABC$ be a triangle and $D$ is a point inside of $\triangle ABC$. The point $A'$ is the midpoint of the arc $BDC$, in the circle which passes by $B,C,D$. Analogously define $B'$ and $C'$, being the midpoints of the arc $ADC$ and $ADB$ respectively. Prove that the four points $D,A',B',C'$ are concyclic.

In a triangle $ABC,$ the internal and external bisectors of the angle $A$ intersect the line $BC$ at $D$ and $E$ respectively. The line $AC$ meets the circle with diameter $DE$ again at $F.$ The tangent line to the circle $ABF$ at $A$ meets the circle with diameter $DE$ again at $G.$ Show that $AF = AG.$

Let $S$ be a point inside $\angle pOq$, and let $k$ be a circle which contains $S$ and touches the legs $Op$ and $Oq$ in points $P$ and $Q$ respectively. Straight line $s$ parallel to $Op$ from $S$ intersects $Oq$ in a point $R$. Let $T$ be the intersection point of the ray $PS$ and circumscribed circle of $\vartriangle SQR$ and $T \ne S$. Prove that $OT // SQ$ and $OT$ is a tangent of the circumscribed circle of $\vartriangle SQR$.

An acute triangle $ABC$ is inscribed in a circle of radius 1 with centre $O;$ all the angles of $ABC$ are greater than $45^\circ.$ $B_{1}$ is the foot of perpendicular from $B$ to $CO,$ $B_{2}$ is the foot of perpendicular from $B_{1}$ to $AC.$ Similarly, $C_{1}$ is the foot of perpendicular from $C$ to $BO,$ $C_{2}$ is the foot of perpendicular from $C_{1}$ to $AB.$ The lines $B_{1}B_{2}$ and $C_{1}C_{2}$ intersect at $A_{3}.$ The points $B_{3}$ and $C_{3}$ are defined in the same way. Find the circumradius of triangle $A_{3}B_{3}C_{3}.$ [i]Proposed by F.Bakharev, F.Petrov[/i]

Let $P,Q,R$ be three points on a circle $k_1$ with $|PQ|=|PR|$ and $|PQ|>|QR|$. Let $k_2$ be the circle with center in $P$ that goes through $Q$ and $R$. The circle with center $Q$ through $R$ intersects $k_1$ in another point $X\ne R$ and intersects $k_2$ in another point $Y\ne R$. The two points $X$ and $R$ lie on different sides of the line through $PQ$. Show that the three points $P$, $X$, $Y$ lie on a common line.

In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$

A circle $k$ touches a larger circle $K$ from inside in a point $P$. Let $Q$ be point on $k$ different from $P$. The line tangent to $k$ at $Q$ intersects $K$ in $A$ and $B$. Show that the line $PQ$ bisects $\angle APB$.