Jacob likes to watchMickeyMouse Clubhouse! One day, he decides to create his own MickeyMouse head shown below, with two circles $\omega_1$ and $\omega_2$ and a circle $\omega$, and centers $O_1$, $O_2$, and $O$, respectively. Let $\omega_1$ and $\omega$ meet at points $P_1$ and $Q_1$, and let $\omega_2$ and $\omega$ meet at points $P_2$ and $Q_2$. Point $P_1$ is closer to $O_2$ than $Q_1$, and point $P_2$ is closer to $O_1$ than $Q_2$. Given that $P_1$ and $P_2$ lie on $O_1O_2$ such that $O_1P_1 = P_1P_2 = P_2O_2 = 2$, and $Q_1O_1 \parallel Q_2O_2$, the area of $\omega$ can be written as $n \pi$. Find $n$. [img]https://cdn.artofproblemsolving.com/attachments/6/d/d98a05ee2218e80fd84d299d47201669736d99.png[/img]

[b]8.1[/b] Find all primes $p,q$ and $r$ such that $$pqr= 5(p + q + r).$$ [b]8.2 [/b] Prove that if $\overline{ab}/\overline{bc} = a/c$, then $$\overline{abb...bb}/\overline{bb...bbc} = a/c$$ (each number has $n$ digits). [b]8.3 / 9.1[/b] Construct a triangle with perimeter, altitude and angle at the base. [b]8.4. / 9.4[/b] Prove that the square of the sum of N distinct non-zero squares of integers is also the sum of $N $squares of non-zero integers. [b]8.5.[/b] In the quadrilateral $ABCD$ the diagonals $AC$ and $BD$ are drawn. Prove that if the circles inscribed in $ABC$ and $ ADC$ touch each other each other, then the circles inscribed in $BAD$ and in $BCD$ also touch each other. [b]8.6 / 9.6[/b] If the numbers $A$ and $n$ are coprime, then there are integers $X$ and $Y$ such that $ |X| <\sqrt{n}$, $|Y| <\sqrt{n} $ and $AX-Y$ is divided by $n$. Prove it. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here[/url].

Let $ABC$ be a triangle, and let $\ell$ be the line passing through the circumcenter of $ABC$ and parallel to the bisector of the angle $\angle A$. Prove that the line $\ell$ passes through the orthocenter of $ABC$ if and only if $AB = AC$ or $\angle BAC = 120^o$

In a triangle $ABC$, right­ angled at $A$, the altitude through $A$ and the internal bisector of $\angle A$ have lengths $3$ and $4$, respectively. Find the length of the median through $A$.

Let $ABC$ be a triangle with incentre $I$. The angle bisectors $AI$, $BI$ and $CI$ meet $[BC]$, $[CA]$ and $[AB]$ at $D$, $E$ and $F$, respectively. The perpendicular bisector of $[AD]$ intersects the lines $BI$ and $CI$ at $M$ and $N$, respectively. Show that $A$, $I$, $M$ and $N$ lie on a circle.

Consider a triangle $ABC$. The tangent line to its circumcircle at point $C$ meets line $AB$ at point $D$. The tangent lines to the circumcircle of triangle $ACD$ at points $A$ and $C$ meet at point $K$. Prove that line $DK$ bisects segment $BC$. (F.Ivlev)

Call a subset of the plane a circular set iff there exists a point such that for every ray starting from it, the ray intersects the subset once. show that the plane is a countable union of circular sets. [hide=idea] let H be a transcendence basis of R over Q. let $\{h_1,h_2,...\}$ be a subset of H. let $K_n$ be the field of real numbers that are algebraic over $H\setminus\{h_n\}$. $K_n\times K_n$ can be covered by a circular set $J_n$. $R\times R\subseteq \cup (K_n\times K_n) \subseteq \cup J_n \subseteq R\times R$ the first inclusion proof: x,y algebraically depend on H, so they depend on H', where H' is a finite subset of H. $\exists n$ st $h_n\notin H'$ $(x,y)\in K_n\times K_n$[/hide]

Three spheres each of unit radius have centers $P,Q,R$ with the property that the center of each sphere lies on the surface of the other two spheres. Let $C$ denote the cylinder with cross-section $PQR$ (the triangular lamina with vertices $P,Q,R$) and axis perpendicular to $PQR$. Let $M$ denote the space which is common to the three spheres and the cylinder $C$, and suppose the mass density of $M$ at a given point is the distance of the point from $PQR$. Determine the mass of $M$.

Points $A_1,A_2,...,A_n$ are selected from the equilateral triangle with a side that is equal to $1$. Denote by $d_k$ the least distance from $A_k$ to all other selected points. Prove that $d_1^2+...+d_n^2 \le 3,5$.

Let $\vartriangle ABC$ be a triangle with circumcenter $O$. Let $ P$ and $Q$ be internal points on the sides $AB$ and $AC$ respectively such that $\angle POB = \angle ABC$ and $\angle QOC = \angle ACB$. Show that the reflection of line $BC$ over line $PQ$ is tangent to the circumcircle of triangle $\vartriangle APQ$.

Let $c(O, R)$ be a circle, $AB$ a diameter and $C$ an arbitrary point on the circle different than $A$ and $B$ such that $\angle AOC > 90^o$. On the radius $OC$ we consider point $K$ and the circle $c_1(K, KC)$. The extension of the segment $KB$ meets the circle $(c)$ at point $E$. From $E$ we consider the tangents $ES$ and $ET$ to the circle $(c_1)$. Prove that the lines $BE, ST$ and $AC$ are concurrent.

In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}\equal{}\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.

[b]p1.[/b] Isosceles trapezoid $ABCD$ has $AB = 2$, $BC = DA =\sqrt{17}$, and $CD = 4$. Point $E$ lies on $\overline{CD}$ such that $\overline{AE}$ splits $ABCD$ into two polygons of equal area. What is $DE$? [b]p2.[/b] At the Berkeley Sandwich Parlor, the famous BMT sandwich consists of up to five ingredients between the bread slices. These ingredients can be either bacon, mayo, or tomato, and ingredients of the same type are indistiguishable. If there must be at least one of each ingredient in the sandwich, and the order in which the ingredients are placed in the sandwich matters, how many possible ways are there to prepare a BMT sandwich? [b]p3.[/b] Three mutually externally tangent circles have radii $2$, $3$, and $3$. A fourth circle, distinct from the other three circles, is tangent to all three other circles. The sum of all possible radii of the fourth circle can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Points $A$ and $B$ move on circles centered at $O_A$ and $O_B$ such that $O_AA$ and $O_BB$ rotate at the same speed. Prove that vertex $C$ of the equilateral triangle $ABC$ moves along a certain circle at the same angular velocity. (The vertices of $ABC$ are oriented clockwise.)

A triangular piece of paper of area $1$ is folded along a line parallel to one of the sides and pressed flat. What is the minimum possible area of the resulting figure?

Let $ C_1, C_2 $ be two circles of equal radius, disjoint, of centers $ O_1, O_2 $, such that $ C_1 $ is to the left of $ C_2 $. Let $ l $ be a line parallel to the line $ O_1O_2 $, secant to both circles. Let $ P_1 $ be a point of $ l $, to the left of $ C_1 $ and $ P_2 $ a point of $ l $, to the right of $ C_2 $ such that the tangents of $ P_1 $ to $ C_1 $ and of $ P_2 $ a $ C_2 $ form a quadrilateral. Show that there is a circle tangent to the four sides of said quadrilateral.

Points $Y,X$ lies on $AB,BC$ of $\triangle ABC$ and $X,Y,A,C$ are concyclic. $AX$ and $CY$ intersect in $O$. Points $M,N$ are midpoints of $AC$ and $XY$. Prove, that $BO$ is tangent to circumcircle of $\triangle MON$

Let $O$ be a point in the plane and let $F$ be a (not necessary convex) polygon. Let $P$ be the perimeter of $F$, let $D$ be sum of the distances of the point $O$ from the vertices of $F$, and let $H$ be sum of the distances of the point $O$ from the lines that pass through the vertices of $F$. Show that \[D^2-H^2 \geq \frac{P^2}{4}.\]

In a metallic disk, a circular sector is removed, so that with the remaining can form a conical glass of maximum volume. Calculate, in radians, the angle of the sector that is removed. [hide=original wording]En un disco metalico se quita un sector circular, de modo que con la parte restante se pueda formar un vaso c´onico de volumen maximo. Calcular, en radianes, el angulo del sector que se quita.[/hide]

A starship is located in a halfspace at the distance $a$ from its boundary. The crew knows this but does not know which direction to move to reach the boundary plane. The starship may travel through the space by any path, may measure the way it has already travelled and has a sensor that signals when the boundary is reached. Is it possible to reach the boundary for sure, having passed no more than: $a)14a$ $b)13a$?

Let $ABCD$ be a cyclic quadrilateral whose side $AB$ is at the same time the diameter of the circle. The lines $AC$ and $BD$ intersect at point $E$ and the extensions of lines $AD$ and $BC$ intersect at point $F$. Segment $EF$ intersects the circle at $G$ and the extension of segment $EF$ intersects $AB$ at $H$. Show that if $G$ is the midpoint of $FH$, then $E$ is the midpoint of $GH$.

A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle? $ \textbf{(A) }2\sqrt{3}\qquad \textbf{(B) }4\qquad \textbf{(C) }3\sqrt{2}\qquad \textbf{(D) }2\sqrt{5}\qquad \textbf{(E) }5\qquad $

An acute-angle non-isosceles triangle $ ABC $ is given. The point $ H $ is its orthocenter, the points $ O $ and $ I $ are the centers of its circumscribed and inscribed circles, respectively. The circumcircle of the triangle $ OIH $ passes through the vertex $ A $. Prove that one of the angles of the triangle is $ 60^\circ $.

A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F$, in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k$. Given that $k=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

$ABCDE$ is a convex pentagon such that the triangles $ABC, BCD, CDE, DEA$ and $EAB$ have equal areas. Show that $(1/4)$ area $(ABCDE) <$ area $(ABC) < (1/3)$ area $(ABCDE)$.