We have $3$ circles such that any $2$ of them are externally tangent. Let $a$ be length of the outer tangent common to a pair of them. The lengths $b$ and $c$ are defined similarly. If $T$ is the sum of the areas of such circles, show that $\pi (a + b + c)^2 \le 12T $. Note: In In the case of externally tangent circles, the common external tangent is the segment tangent to them that touches them at different points.

Carl has $5$ cubes each having side length $1$, and Kate has $5$ cubes each having side length $2$. What is the total volume of the $10$ cubes? $\textbf{(A) }24 \qquad \textbf{(B) }25 \qquad \textbf{(C) } 28\qquad \textbf{(D) } 40\qquad \textbf{(E) } 45$

In triangle $ABC$, $AB = AC = 3$ and $\angle A = 90^o$. Let $M$ be the midpoint of side $BC$. Points $D$ and $E$ lie on sides $AC$ and $AB$ respectively such that $AD > AE$ and $ADME$ is a cyclic quadrilateral. Given that triangle $EMD$ has area $2$, find the length of segment $CD$.

[u]Round 1[/u] [b]p1.[/b] Five children, Aisha, Baesha, Cosha, Dasha, and Erisha, competed in running, jumping, and throwing. In each event, first place was won by someone from Renton, second place by someone from Seattle, and third place by someone from Tacoma. Aisha was last in running, Cosha was last in jumping, and Erisha was last in throwing. Could Baesha and Dasha be from the same city? [b]p2.[/b] Fifty-five Brits and Italians met in a coffee shop, and each of them ordered either coffee or tea. Brits tell the truth when they drink tea and lie when they drink coffee; Italians do it the other way around. A reporter ran a quick survey: Forty-four people answered “yes” to the question, “Are you drinking coffee?” Thirty-three people answered “yes” to the question, “Are you Italian?” Twenty-two people agreed with the statement, “It is raining outside.” How many Brits in the coffee shop are drinking tea? [b]p3.[/b] Doctor Strange is lost in a strange house with a large number of identical rooms, connected to each other in a loop. Each room has a light and a switch that could be turned on and off. The lights might initially be on in some rooms and off in others. How can Dr. Strange determine the number of rooms in the house if he is only allowed to switch lights on and off? [b]p4.[/b] Fifty street artists are scheduled to give solo shows with three consecutive acts: juggling, drumming, and gymnastics, in that order. Each artist will spend equal time on each of the three activities, but the lengths may be different for different artists. At least one artist will be drumming at every moment from dawn to dusk. A new law was just passed that says two artists may not drum at the same time. Show that it is possible to cancel some of the artists' complete shows, without rescheduling the rest, so that at least one show is going on at every moment from dawn to dusk, and the schedule complies with the new law. [b]p5.[/b] Alice and Bob split the numbers from $1$ to $12$ into two piles with six numbers in each pile. Alice lists the numbers in the first pile in increasing order as $a_1 < a_2 < a_3 < a_4 < a_5 < a_6$ and Bob lists the numbers in the second pile in decreasing order $b_1 > b_1 > b_3 > b_4 > b_5 > b_6$. Show that no matter how they split the numbers, $$|a_1 -b_1| + |a_2 -b_2| + |a_3 -b_3| + |a_4 -b_4| + |a_5 -b_5| + |a_6 -b_6| = 36.$$ [u]Round 2[/u] [b]p6.[/b] The Martian alphabet has ? letters. Marvin writes down a word and notices that within every sub-word (a contiguous stretch of letters) at least one letter occurs an odd number of times. What is the length of the longest possible word he could have written? [b]p7.[/b] For a long space journey, two astronauts with compatible personalities are to be selected from $24$ candidates. To find a good fit, each candidate was asked $24$ questions that required a simple yes or no answer. Two astronauts are compatible if exactly $12$ of their answers matched (that is, both answered yes or both answered no). Miraculously, every pair of these $24$ astronauts was compatible! Show that there were exactly $12$ astronauts whose answer to the question “Can you repair a flux capacitor?” was exactly the same as their answer to the question “Are you afraid of heights?” (that is, yes to both or no to both). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two squares are arranged as shown in the picture. Prove that the areas of shaded quadrilaterals are equal. [img]https://3.bp.blogspot.com/-W50DOuizFvY/XT6wh3-L6sI/AAAAAAAAKaw/pIW2RKmttrwPAbrKK3bpahJz7hfIZwM8QCK4BGAYYCw/s400/Oral%2BSharygin%2B2016%2B10.11%2Bp3.png[/img]

Side $\overline{AB}$ of $\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$? $ \textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20 \qquad $

Let $ABC$ be a triangle and $D$ point on side $BC$ such that $AD$ is angle bisector of angle $\angle BAC$. Let $E$ be the intersection of the side $AB$ with circle $\omega_1$ which has diameter $CD$ and let $F$ be the intersection of the side $AC$ with circle $\omega_2$ which has diameter $BD$. Suppose that there exist points $P\in\omega_1$ and $Q\in\omega_2$ such that $E, P, Q$ and $F$ are collinear and on this order. Prove that $AD, BQ$ and $CP$ are concurrent. [i]Proposed by Dorlir Ahmeti, Kosovo and Noah Walsh, U.S.A.[/i]

Let $O$ be the circumcentre of triangle $ABC$. Lines $AO$ and $BC$ intersect at point $D$. Let $S$ be a point on line $BO$ such that $DS \parallel AB$ and lines $AS$ and $BC$ intersect at point $T$. Prove that if $O, D, S$ and $T$ lie on the same circle, then $ABC$ is an isosceles triangle.

The orthocenter of triangle $ABC$ lies on its circumcircle. One of the angles of $ABC$ must equal: (The orthocenter of a triangle is the point where all three altitudes intersect.) $$ \mathrm a. ~ 30^\circ\qquad \mathrm b.~60^\circ\qquad \mathrm c. ~90^\circ \qquad \mathrm d. ~120^\circ \qquad \mathrm e. ~\text{It cannot be deduced from the given information.} $$

Three edges of a parallelepiped lie on three intersecting diagonals of the lateral faces of a triangular prism. Find the ratio of the volumes of the parallelepiped and the prism.

A quadrilateral is cut from a piece of gift wrapping paper, which has equally wide white and gray stripes. The grey stripes in the quadrilateral have a combined area of $10$. Determine the area of the quadrilateral. [img]https://1.bp.blogspot.com/-ia13b4RsNs0/XzP0cepAcEI/AAAAAAAAMT8/0UuCogTRyj4yMJPhfSK3OQihRqfUT7uSgCLcBGAsYHQ/s0/2020%2Bmohr%2Bp2.png[/img]

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that \[ OD + DH = OE + EH = OF + FH\] and the lines $AD$, $BE$, and $CF$ are concurrent.

Let $ABC$ be triangle with incenter $I$. A point $P$ in the interior of the triangle satisfies \[\angle PBA+\angle PCA = \angle PBC+\angle PCB.\] Show that $AP \geq AI$, and that equality holds if and only if $P=I$.

Let $P$ and $Q$ be points on the plane $ABC$ such that $m(\widehat{BAC})=90^\circ$, $|AB|=1$, $|AC|=\sqrt 2$, $|PB|=1=|QB|$, $|PC|=2=|QC|$, and $|PA|>|QA|$. What is $|PA|/|QA|$? $ \textbf{(A)}\ \sqrt 2 +\sqrt 3 \qquad\textbf{(B)}\ 5-\sqrt 6 \qquad\textbf{(C)}\ \sqrt 6 -\sqrt 2 \qquad\textbf{(D)}\ \sqrt 6 + 1 \qquad\textbf{(E)}\ \text{None} $

The sphere inscribed in a tetrahedron $ABCD$ touches face $ABC$ at point $H$. Another sphere touches face $ABC$ at $O$ and the planes containing the other three faces at points exterior to the faces. Prove that if $O$ is the circumcenter of triangle $ABC$, then $H$ is the orthocenter of that triangle.

The total area of all the faces of a rectangular solid is $22 \text{cm}^2$, and the total length of all its edges is $24 \text{cm}$. Then the length in $\text{cm}$ of any one of its internal diagonal is A. $\sqrt{11}$ B. $\sqrt{12}$ C. $\sqrt{13}$ D. $\sqrt{14}$ E. Not uniquely determined

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

A sphere is inscribed in tetrahedron $ ABCD$ and is tangent to faces $ BCD,CAD,ABD,ABC$ at points $ P,Q,R,S$ respectively. Segment $ PT$ is the sphere's diameter, and lines $ TA,TQ,TR,TS$ meet the plane $ BCD$ at points $ A',Q',R',S'$. respectively. Show that $ A$ is the center of a circumcircle on the triangle $ S'Q'R'$.

$ a,b,c$ are sides of triangle $ ABC$. For any choosen triple from $ (a \plus{} 1,b,c),(a,b \plus{} 1,c),(a,b,c \plus{} 1)$ there exist a triangle which sides are choosen triple. Find all possible values of area which triangle $ ABC$ can take.

Find the largest cube which can be placed inside a regular tetrahedron with side $1$ so that one of its faces lies on the base of the tetrahedron.

Let $ABCD$ be a convex quadrilateral such that $AC = BD$ and the sides $AB$ and $CD$ are not parallel. Let $P$ be the intersection point of the diagonals $AC$ and $BD$. Points $E$ and $F$ lie, respectively, on segments $BP$ and $AP$ such that $PC=PE$ and $PD=PF$. Prove that the circumcircle of the triangle determined by the lines $AB, CD, EF$ is tangent to the circumcircle of the triangle $ABP$.

$ABCD$ is a cyclic quadrilateral, with diagonals $AC,BD$ perpendicular to each other. Let point $F$ be on side $BC$, the parallel line $EF$ to $AC$ intersect $AB$ at point $E$, line $FG$ parallel to $BD$ intersect $CD$ at $G$. Let the projection of $E$ onto $CD$ be $P$, projection of $F$ onto $DA$ be $Q$, projection of $G$ onto $AB$ be $R$. Prove that $QF$ bisects $\angle PQR$.

In a right-angled triangle $ABC$ ($\angle A = 90^o$), the perpendicular bisector of $BC$ intersects the line $AC$ in $K$ and the perpendicular bisector of $BK$ intersects the line $AB$ in $L$. If the line $CL$ be the internal bisector of angle $C$, find all possible values for angles $B$ and $C$. by Mahdi Etesami Fard

Let $AA_1$ and $CC_1$ be the altitudes of the acute-angled triangle $ABC$. Let $K,L$ and $M$ be the midpoints of the sides $AB,BC$ and $CA$ respectively. Prove that if $\angle C_1MA_1 =\angle ABC$, then $C_1 K = A_1L$.

Given is a convex quadrilateral $ ABCD$ with $ \angle ADC\equal{}\angle BCD>90^{\circ}$. Let $ E$ be the point of intersection of the line $ AC$ with the parallel line to $ AD$ through $ B$ and $ F$ be the point of intersection of the line $ BD$ with the parallel line to $ BC$ through $ A$. Show that $ EF$ is parallel to $ CD$