Each side face of a dodecahedron lies in a uniquely determined plane. Those planes cut the space in a finite number of disjoint [i]regions[/i]. Find the number of such regions.

Find the angle $BCA$ in the quadrilateral of the figure. [img]https://cdn.artofproblemsolving.com/attachments/0/2/974e23be54125cde8610a78254b59685833b5b.png[/img]

Convex quadrilateral $ABCD$ is described on a circle $\omega$, and is not a trapezius inscribed in a circle. Let the tangency points of $\omega$ and $AB, BC, CD, DA$ be $K, L, M, N$ respectively. A circle with a center $I_K$, different from $\omega$ is tangent to the segement $AB$ and lines $AD, BC$. A circle with center $I_L$, different from $\omega$ is tangent to segment $BC$ and lines $AB, CD$. A circle with center $I_M$, different from $\omega$ is tangent to segment $CD$ and lines $AD, BC$. A circle with center $I_N$, different from $\omega$ is tangent to segment $AD$ and lines $AB, CD$. Prove that the lines $I_KK, I_LL, I_MM, I_NN$ are concurrent.

Let $O$ be the circumcenter of $\triangle ABC$. A circle with center $P$ pass through $A$ and $O$ and $OP$//$BC$. $D$ is a point such that $\angle DBA = \angle DCA = \angle BAC$. Prove that: Circle $(P)$, circle $(BCD)$ and the circle with diameter $(AD)$ share a common point. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMS9jLzlmZjdlN2ExZDJjYjAwYWJlZTQzYWRkYzg3NDlhMTUyZjRlNGJjLmpwZw==&rn=c291dGhlYXN0UDYuanBn[/img]

As shown in the figure, it is known that $BC= AC$ in $\vartriangle ABC$, $M$ is the midpoint of $AB$, points $D$, $E$ lie on $AB$ such that $\angle DCE= \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F $(different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$, $O_2$ respectively. Prove that $O_1O_2 \perp CF$. [img]https://cdn.artofproblemsolving.com/attachments/8/c/62d4ecbc18458fb4f2bf88258d5024cddbc3b0.jpg[/img]

In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of [b]a.)[/b] all vertices $A$ of such triangles; [b]b.)[/b] all vertices $B$ of such triangles; [b]c.)[/b] all vertices $C$ of such triangles.

Let be a triangle $ ABC $, the midpoint of the $ AC $ and $ N $ side, and the midpoint of the $ AB $ side. Let $ r $ and $ s $ reflect the straight lines $ BM $ and $ CN $ on the straight $ BC $, respectively. Also define $ D $ and $ E $ as the intersection of the lines $ r $ and $ s $ and the line $ MN $, respectively. Let $ X $ and $ Y $ be the intersection points between the circumcircles of the triangles $ BDM $ and $ CEN $, $ Z $ the intersection of the lines $ BE $ and $ CD $ and $ W $ the intersection between the lines $ r $ and $ s $. Prove that $ XY, WZ $, and $ BC $ are concurrents.

Let $ABCD$ be a square with side length $2$, and let a semicircle with flat side $CD$ be drawn inside the square. Of the remaining area inside the square outside the semi-circle, the largest circle is drawn. What is the radius of this circle?

Rectangle $ABCD$ is folded in half so that the vertices $D$ and $B$ coincide, creating the crease $\overline{EF}$, with $E$ on $\overline{AD}$ and $F$ on $\overline{BC}$. Let $O$ be the midpoint of $\overline{EF}$. If triangles $DOC$ and $DCF$ are congruent, what is the ratio $BC : CD$?

Triangle $ABC$ has circumcircle $\omega$ where $B'$ is the point diametrically opposite $B$ and $C'$ is the point diametrically opposite $C$. Given $B'C'$ passes through the midpoint of $AB$, if $AC' = 3$ and $BC = 7$, find $AB'^2$..

$ABCD$ is a cyclic quadrilateral and $M$ a point on the side $CD$ such that $ADM$ and $ABCM$ have the same area and the same perimeter. Show that two sides of $ABCD$ have the same length.

Given a triangle $ABC$, in which the medians $BE$ and $CF$ are perpendicular. Let $M$ is the intersection point of the medians of this triangle, and $L$ is its Lemoine point (the intersection point of lines symmetrical to the medians with respect to the bisectors of the corresponding angles). Prove that $ML \perp BC$. (Mykhailo Sydorenko)

Let $\omega$ be a semicircle with diamater $AB$. Let $M$ be the midpoint of $AB$. Let $X,Y$ be points on the same semiplane with $\omega$ with respect to the line $AB$ such that $AMXY$ is a parallelogram. Let $XM\cap \omega = C$ and $YM \cap \omega = D$. Let $I$ be the incenter of $\triangle XYM$. Let $AC \cap BD= E$ and $ME$ intersects $XY$ at $T$. Let the intersection point of $TI$ and $AB$ be $Q$ and let the perpendicular projection of $T$ onto $AB$ be $P$. Prove that $M$ is midpoint of $PQ$

Two touching balls with radii $a$ and $b$ are enclosed in a cylindrical tin of diameter $d$ . Both balls hit the top surface and the shell of the cylinder. The largest ball also hits the bottom surface. Show that $\sqrt{d} =\sqrt{a} +\sqrt{b}$ [img]https://1.bp.blogspot.com/-O4B3P3bghFs/Xy1fDv9zGkI/AAAAAAAAMSQ/ePLVnsXsRi0mz3SWBpIzfGdsizWoLmGVACLcBGAsYHQ/s0/flanders%2B2019%2Bp1.png[/img]

Circles $k_1$ and $k_2$ with radii $r_1=6$ and $r_2=3$ are externally tangent and touch a circle $k$ with radius $r=9$ from inside. A common external tangent of $k_1$ and $k_2$ intersects $k$ at $P$ and $Q$. Determine the length of $PQ$.

Given an unit square and some circles inside. Radius of each circle is less than $0.001$, and there is no couple of points belonging to the different circles with the distance between them $0.001$ exactly. Prove that the area, covered by the circles is not greater than $0.34$.

$\textbf{Bake Sale}$ Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. $\circ$ Art's cookies are trapezoids: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label("5 in", (2.5,0), S); label("3 in", (5,1.5), E); label("3 in", (3.5,3), N);[/asy] $\circ$ Roger's cookies are rectangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label("4 in", (2,0), S); label("2 in", (4,1), E);[/asy] $\circ$ Paul's cookies are parallelograms: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label("3 in", (1.5,0), S); label("2 in", (2.5,1), W);[/asy] $\circ$ Trisha's cookies are triangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label("3 in", (1.5,0), S); label("4 in", (3,2), E);[/asy] Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Art's cookies sell for 60 cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents? $ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 75\qquad\textbf{(E)}\ 90$

(a) $K_1,K_2,..., K_n$ are the feet of the perpendiculars from an arbitrary point $M$ inside a given regular $n$-gon to its sides (or sides produced). Prove that the sum $\overrightarrow{MK_1} + \overrightarrow{MK_2} + ... + \overrightarrow{MK_n}$ equals $\frac{n}{2}\overrightarrow{MO}$, where $O$ is the centre of the $n$-gon. (b) Prove that the sum of the vectors whose origin is an arbitrary point $M$ inside a given regular tetrahedron and whose endpoints are the feet of the perpendiculars from $M$ to the faces of the tetrahedron equals $\frac43 \overrightarrow{MO}$, where $O$ is the centre of the tetrahedron. (VV Prasolov, Moscow)

Is there a closed self-intersecting broken line in space that intersects each of its links exactly once, and in its midpoint ?

In $\triangle ABC$, $P,Q,R$ divides the perimeter of $\triangle ABC$ into three equal parts. $P,Q\in AB$. Prove that $\frac{S_{\triangle PQR}}{S_{\triangle ABC}}>\frac{2}{9}$.

Find the minimum area bounded by the graphs of $ y\equal{}x^2$ and $ y\equal{}kx(x^2\minus{}k)\ (k>0)$.

Consider a cyclic quadrilateral $ABCD$. The extensions of its sides $AB,DC$ meet at the point $P$ and the extensions of its sides $AD,BC$ meet at the point $Q$. Suppose $\quad QE,QF$ are tangents to the circumcircle of quadrilateral $ABCD$ at $E,F$ respectively. Show that $P,E,F$ are collinear.

Let $ D$, $ E$, $ F$ be the feet of the altitudes wrt sides $ BC$, $ CA$, $ AB$ of acute-angled triangle $ \triangle ABC$, respectively. In triangle $ \triangle CFB$, let $ P$ be the foot of the altitude wrt side $ BC$. Define $ Q$ and $ R$ wrt triangles $ \triangle ADC$ and $ \triangle BEA$ analogously. Prove that lines $ AP$, $ BQ$, $ CR$ don't intersect in one common point.

[u]Round 1[/u] [b]p1.[/b] Alice and Bob compiled a list of movies that exactly one of them saw, then Cindy and Dale did the same. To their surprise, these two lists were identical. Prove that if Alice and Cindy list all movies that exactly one of them saw, this list will be identical to the one for Bob and Dale. [b]p2.[/b] Several whole rounds of cheese were stored in a pantry. One night some rats sneaked in and consumed $10$ of the rounds, each rat eating an equal portion. Some were satisfied, but $7$ greedy rats returned the next night to finish the remaining rounds. Their portions on the second night happened to be half as large as on the first night. How many rounds of cheese were initially in the pantry? [b]p3.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake, and call that number N. Pick up the stack of the top N pancakes, and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack. [b]p4.[/b] There are two lemonade stands along the $4$-mile-long circular road that surrounds Sour Lake. $100$ children live in houses along the road. Every day, each child buys a glass of lemonade from the stand that is closest to her house, as long as she does not have to walk more than one mile along the road to get there. A stand's [u]advantage [/u] is the difference between the number of glasses it sells and the number of glasses its competitor sells. The stands are positioned such that neither stand can increase its advantage by moving to a new location, if the other stand stays still. What is the maximum number of kids who can't buy lemonade (because both stands are too far away)? [b]p5.[/b] Merlin uses several spells to move around his $64$-room castle. When Merlin casts a spell in a room, he ends up in a different room of the castle. Where he ends up only depends on the room where he cast the spell and which spell he cast. The castle has the following magic property: if a sequence of spells brings Merlin from some room $A$ back to room $A$, then from any other room $B$ in the castle, that same sequence brings Merlin back to room $B$. Prove that there are two different rooms $X$ and $Y$ and a sequence of spells that both takes Merlin from $X$ to $Y$ and from $Y$ to $X$. [u]Round 2[/u] [b]p6.[/b] Captains Hook, Line, and Sinker are deciding where to hide their treasure. It is currently buried at the $X$ in the map below, near the lairs of the three pirates. Each pirate would prefer that the treasure be located as close to his own lair as possible. You are allowed to propose a new location for the treasure to the pirates. If at least two out of the three pirates prefer the new location (because it moves closer to their own lairs), then the treasure will be moved there. Assuming the pirates’ lairs form an acute triangle, is it always possible to propose a sequence of new locations so that the treasure eventually ends up in your backyard (wherever that is)? [img]https://cdn.artofproblemsolving.com/attachments/c/c/a9e65624d97dec612ef06f8b30be5540cfc362.png[/img] [b]p7.[/b] Homer went on a Donut Diet for the month of May ($31$ days). He ate at least one donut every day of the month. However, over any stretch of $7$ consecutive days, he did not eat more than $13$ donuts. Prove that there was some stretch of consecutive days over which Homer ate exactly $30$ donuts. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The isosceles triangle $\triangle ABC$, with $AB=AC$, is inscribed in the circle $\omega$. Let $P$ be a variable point on the arc $\stackrel{\frown}{BC}$ that does not contain $A$, and let $I_B$ and $I_C$ denote the incenters of triangles $\triangle ABP$ and $\triangle ACP$, respectively. Prove that as $P$ varies, the circumcircle of triangle $\triangle PI_BI_C$ passes through a fixed point.