Four circles are situated in the plane so that each is tangent to the other three. If three of the radii are $5$, $5$, and $8$, the largest possible radius of the fourth circle is $a/b$, where $a$ and $b$ are positive integers and gcd$(a, b) = 1$. Find $a + b$.

In the $ x \minus{} y$ plane, let $ l$ be the tangent line at the point $ A\left(\frac {a}{2},\ \frac {\sqrt {3}}{2}b\right)$ on the ellipse $ \frac {x^2}{a^2} \plus{} \frac {y^2}{b^2}\equal{}1\ (0 < b < 1 < a)$. Let denote $ S$ be the area of the figure bounded by $ l,$ the $ x$ axis and the ellipse. (1) Find the equation of $ l$. (2) Express $ S$ in terms of $ a,\ b$. (3) Find the maximum value of $ S$ with the constraint $ a^2 \plus{} 3b^2 \equal{} 4$.

[hide=R stands for Ramanujan, P stands for Pascal]they had two problem sets under those two names[/hide] [b]R1.[/b] What is the distance between the points $(6, 0)$ and $(-2, 0)$? [b]R2 / P1.[/b] Angle $X$ has a degree measure of $35$ degrees. What is the supplement of the complement of angle $X$? [i]The complement of an angle is $90$ degrees minus the angle measure. The supplement of an angle is $180$ degrees minus the angle measure. [/i] [b]R3.[/b] A cube has a volume of $729$. What is the side length of the cube? [b]R4 / P2.[/b] A car that always travels in a straight line starts at the origin and goes towards the point $(8, 12)$. The car stops halfway on its path, turns around, and returns back towards the origin. The car again stops halfway on its return. What are the car’s final coordinates? [b]R5.[/b] A full, cylindrical soup can has a height of $16$ and a circular base of radius $3$. All the soup in the can is used to fill a hemispherical bowl to its brim. What is the radius of the bowl? [b]R6.[/b] In square $ABCD$, the numerical value of the length of the diagonal is three times the numerical value of the area of the square. What is the side length of the square? [b]R7.[/b] Consider triangle $ABC$ with $AB = 3$, $BC = 4$, and $AC = 5$. The altitude from $B$ to $AC$ intersects $AC$ at $H$. Compute $BH$. [b]R8.[/b] Mary shoots $5$ darts at a square with side length $2$. Let $x$ be equal to the shortest distance between any pair of her darts. What is the maximum possible value of $x$? [b]P3.[/b] Let $ABC$ be an isosceles triangle such that $AB = BC$ and all of its angles have integer degree measures. Two lines, $\ell_1$ and $\ell_2$, trisect $\angle ABC$. $\ell_1$ and $\ell_2$ intersect $AC$ at points $D$ and $E$ respectively, such that $D$ is between $A$ and $E$. What is the smallest possible integer degree measure of $\angle BDC$? [b]P4.[/b] In rectangle $ABCD$, $AB = 9$ and $BC = 8$. $W$, $X$, $Y$ , and $Z$ are on sides $AB$, $BC$, $CD$, and $DA$, respectively, such that $AW = 2WB$, $CX = 3BX$, $CY = 2DY$ , and $AZ = DZ$. If $WY$ and $XZ$ intersect at $O$, find the area of $OWBX$. [b]P5.[/b] Consider a regular $n$-gon with vertices $A_1A_2...A_n$. Find the smallest value of $n$ so that there exist positive integers $i, j, k \le n$ with $\angle A_iA_jA_k = \frac{34^o}{5}$. [b]P6.[/b] In right triangle $ABC$ with $\angle A = 90^o$ and $AB < AC$, $D$ is the foot of the altitude from $A$ to $BC$, and $M$ is the midpoint of $BC$. Given that $AM = 13$ and $AD = 5$, what is $\frac{AB}{AC}$ ? [b]P7.[/b] An ant is on the circumference of the base of a cone with radius $2$ and slant height $6$. It crawls to the vertex of the cone $X$ in an infinite series of steps. In each step, if the ant is at a point $P$, it crawls along the shortest path on the exterior of the cone to a point $Q$ on the opposite side of the cone such that $2QX = PX$. What is the total distance that the ant travels along the exterior of the cone? [b]P8.[/b] There is an infinite checkerboard with each square having side length $2$. If a circle with radius $1$ is dropped randomly on the checkerboard, what is the probability that the circle lies inside of exactly $3$ squares? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A triangle $ ABC $ is given, and a segment $ PQ=t $ on $ BC $ such that $ P $ is between $ B $ and $ Q $ and $ Q $ is between $ P $ and $ C $. Let $ PP_1 || AB $, $ P_1 $ is on $ AC $, and $ PP_2 || AC $, $ P_2 $ is on $ AB $. Points $ Q_1 $ and $ Q_2 $ аrе defined similar. Prove that the sum of the areas of $ PQQ_1P_1 $ and $ PQQ_2P_2 $ does not depend from the position of $ PQ $ on $ BC $.

[u]Round 5[/u] [b]5.1.[/b] Quadrilateral $ABCD$ is such that $\angle ABC = \angle ADC = 90^o$ , $\angle BAD = 150^o$ , $AD = 3$, and $AB = \sqrt3$. The area of $ABCD$ can be expressed as $p\sqrt{q}$ for positive integers $p, q$ where $q$ is not divisible by the square of any prime. Find $p + q$. [b]5.2.[/b] Neetin wants to gamble, so his friend Akshay describes a game to him. The game will consist of three dice: a $100$-sided one with the numbers $1$ to $100$, a tetrahedral one with the numbers $1$ to $4$, and a normal $6$-sided die. If Neetin rolls numbers with a product that is divisible by $21$, he wins. Otherwise, he pays Akshay $100$ dollars. The number of dollars that Akshay must pay Neetin for a win in order to make this game fair is $a/b$ for relatively prime positive integers $a, b$. Find $a + b$. (Fair means the expected net gain is $0$. ) [b]5.3.[/b] What is the sum of the fourth powers of the roots of the polynomial $P(x) = x^2 + 2x + 3$? [u]Round 6[/u] [b]6.1.[/b] Consider the set $S = \{1, 2, 3, 4,..., 25\}$. How many ordered $n$-tuples $S_1 = (a_1, a_2, a_3,..., a_n)$ of pairwise distinct ai exist such that $a_i \in S$ and $i^2 | a_i$ for all $1 \le i \le n$? [b]6.2.[/b] How many ways are there to place $2$ identical rooks and $ 1$ queen on a $ 4 \times 4$ chessboard such that no piece attacks another piece? (A queen can move diagonally, vertically or horizontally and a rook can move vertically or horizontally) [b]6.3.[/b] Let $L$ be an ordered list $\ell_1$, $\ell_2$, $...$, $\ell_{36}$ of consecutive positive integers who all have the sum of their digits not divisible by $11$. It is given that $\ell_1$ is the least element of $L$. Find the least possible value of $\ell_1$. [u]Round 7[/u] [b]7.1.[/b] Spencer, Candice, and Heather love to play cards, but they especially love the highest cards in the deck - the face cards (jacks, queens, and kings). They also each have a unique favorite suit: Spencer’s favorite suit is spades, Candice’s favorite suit is clubs, and Heather’s favorite suit is hearts. A dealer pulls out the $9$ face cards from every suit except the diamonds and wants to deal them out to the $3$ friends. How many ways can he do this so that none of the $3$ friends will see a single card that is part of their favorite suit? [b]7.2.[/b] Suppose a sequence of integers satisfies the recurrence $a_{n+3} = 7a_{n+2} - 14a_{n+1} + 8a_n$. If $a_0 = 4$, $a_1 = 9$, and $a_2 = 25$, find $a_{16}$. Your answer will be in the form $2^a + 2^b + c$, where $2^a < a_{16} < 2^{a+1}$ and $b$ is as large as possible. Find $a + b + c$. [b]7.3.[/b] Parallel lines $\ell_1$ and $\ell_2$ are $1$ unit apart. Unit square $WXYZ$ lies in the same plane with vertex $W$ on $\ell_1$. Line $\ell_2$ intersects segments $YX$ and $YZ$ at points $U$ and $O$, respectively. Given $UO =\frac{9}{10}$, the inradius of $\vartriangle YOU$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of contestants who participated in at least one of the three rounds of the 2020 ABMC April contest. Let $B$ be the number of times the letter b appears in the Accuracy Round. Let $M$ be the number of people who submitted both the speed and accuracy rounds before 2:00 PM EST. Further, let $C$ be the number of times the letter c appears in the Speed Round. Estimate $$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766239p24226402]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$. Let $K$ be a point inside $ABC$ which is not $O$ nor $H$. Point $L$ and $M$ are located outside the triangle $ABC$ such that $AKCL$ and $AKBM$ are parallelogram. At last, let $BL$ and $CM$ intersects at $N$, and let $J$ be the midpoint of $HK$. Show that $KONJ$ is also a parallelogram. [i]Raja Oktovin, Pekanbaru[/i]

Write out the positive integers consisting of only $1$s, $6$s, and $9$s in ascending order as in: $1,6,9,11,16,\dots$. a. Find the order of $1996$ in the sequence. b. Find the $1996$th term in the sequence.

Three equal circles $k_1, k_2, k_3$ intersect non-tangentially at a point $P$. Let $A$ and $B$ be the centers of circles $k_1$ and $k_2$. Let $D$ and $C$ be the intersection of $k_3$ with $k_1$ and $k_2$ respectively, which is different from $P$. Show that $ABCD$ is a parallelogram.

Let $ \mathcal{M} $ a set of $ 3n\ge 3 $ planar points such that the maximum distance between two of these points is $ 1 $. Prove that: [b]a)[/b] among any four points,there are two aparted by a distance at most $ \frac{1}{\sqrt{2}} . $ [b]b)[/b] for $ n=2 $ and any $ \epsilon >0, $ it is possible that $ 12 $ or $ 15 $ of the distances between points from $ \mathcal{M} $ lie in the interval $ (1-\epsilon , 1]; $ but any $ 13 $ of the distances can´t be found all in the interval $ \left(\frac{1}{\sqrt 2} ,1\right]. $ [b]c)[/b] there exists a circle of diameter $ \sqrt{6} $ that contains $ \mathcal{M} . $ [b]d)[/b] some two points of $ \mathcal{M} $ are on a distance not exceeding $ \frac{4}{3\sqrt n-\sqrt 3} . $

Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? $ \textbf{(A) }60 \qquad \textbf{(B) }65 \qquad \textbf{(C) }70 \qquad \textbf{(D) }75 \qquad \textbf{(E) }80 \qquad $

Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space. Let $V$ be the cone obtained by rotating the triangle around the $x$-axis. Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis.

In the parallelogram $ABCD$, rays are released from its vertices towards its interior. The rays coming out of the vertices $A{}$ and $D{}$ intersect at $E{}$ and the rays coming out of the vertices $B{}$ and $C{}$ at point $F{}$. It is known that $\angle BAE=\angle BCF$ and $\angle CDE = \angle CBF$. Prove that $AB \parallel EF$. [i]Proposed by V. Eisenstadt[/i]

Let $ABC$ be an acuted-angle triangle and let $H$ be it's orthocenter. Let $PQ$ be a segment through $H$ such that $P$ lies on $AB$ and $Q$ lies on $AC$ and such that $ \angle PHB= \angle CHQ$. Finally, in the circumcircle of $\triangle ABC$, consider $M$ such that $M$ is the mid point of the arc $BC$ that doesn't contain $A$. Prove that $MP=MQ$ Proposed by Eduardo Velasco/Marco Figueroa

We have $1998$ polygon such that sum of the areas is $1997,5$ $cm^2$. These polygons placing inside a square with side lenght $1$ $cm$. (Polygons no overflow). Prove that we can find a point such that, all polygons have this point.

Let $ABCD$ be a quadtrilateral with no parallel sides. The diagonals intersect at $E$, and $P, Q$ are points on sides $AB, CD$ respectively such that $\frac{AP}{PB} = \frac{CQ}{QD}$. $PQ$ meet $AC$ and $BD$ at $R,S$. Prove that $(EAB),(ECD),(ERS)$ all meet a point other than $E$.

Let $ABC$ be a triangle with $\angle C=90^\circ$, and $D$ be a point outside $ABC$, such that $\angle ADC=\angle BAC$. The segments $CD$ and $AB$ meet at point $E$. It is known that the distance from $E$ to $AC$ is equal to the circumradius of triangle $ADE$. Find the angles of triangle $ABC$.

The points $ A,B,C$ are on a circle $ O$. The tangent line at $ A$ and the secant $ BC$ intersect at $ P$, $ B$ lying between $ C$ and $ P$. If $ \overline{BC} \equal{} 20$ and $ \overline{PA} \equal{} 10\sqrt {3}$, then $ \overline{PB}$ equals: $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 10\sqrt {3} \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 30$

In $\Delta ABC$, $AD \perp BC$ at $D$. $E,F$ lie on line $AB$, such that $BD=BE=BF$. Let $I,J$ be the incenter and $A$-excenter. Prove that there exist two points $P,Q$ on the circumcircle of $\Delta ABC$ , such that $PB=QC$, and $\Delta PEI \sim \Delta QFJ$ .

In the triangle $ABC$, points $D, E, F$ are on the sides $BC, CA$ and $AB$ respectively such that $FE$ is parallel to $BC$ and $DF$ is parallel to $CA$, Let P be the intersection of $BE$ and $DF$, and $Q$ the intersection of $FE$ and $AD$. Prove that $PQ$ is parallel to $AB$.

In an acute triangle $ABC$ with $|AB| \not= |AC|$, let $I$ be the incenter and $O$ the circumcenter. The incircle is tangent to $\overline{BC}, \overline{CA}$ and $\overline{AB}$ in $D,E$ and $F$ respectively. Prove that if the line parallel to $EF$ passing through $I$, the line parallel to $AO$ passing through $D$ and the altitude from $A$ are concurrent, then the point of concurrence is the orthocenter of the triangle $ABC$. [i]Proposed by Petar Nizié-Nikolac[/i]

Prove that if the sides $a, b, c$ of a non-equilateral triangle satisfy $a + b = 2c$, then the line passing through the incenter and centroid is parallel to one of the sides of the triangle.

[u]Round 1[/u] [b]p1.[/b] Nineteen witches, all of different heights, stand in a circle around a campfire. Each witch says whether she is taller than both of her neighbors, shorter than both, or in-between. Exactly three said “I am taller.” How many said “I am in-between”? [b]p2.[/b] Alex is writing a sequence of $A$’s and $B$’s on a chalkboard. Any $20$ consecutive letters must have an equal number of $A$’s and $B$’s, but any 22 consecutive letters must have a different number of $A$’s and $B$’s. What is the length of the longest sequence Alex can write?. [b]p3.[/b] A police officer patrols a town whose map is shown. The officer must walk down every street segment at least once and return to the starting point, only changing direction at intersections and corners. It takes the officer one minute to walk each segment. What is the fastest the officer can complete a patrol? [img]https://cdn.artofproblemsolving.com/attachments/a/3/78814b37318adb116466ede7066b0d99d6c64d.png[/img] [b]p4.[/b] A zebra is a new chess piece that jumps in the shape of an “L” to a location three squares away in one direction and two squares away in a perpendicular direction. The picture shows all the moves a zebra can make from its given position. Is it possible for a zebra to make a sequence of $64$ moves on an $8\times 8$ chessboard so that it visits each square exactly once and returns to its starting position? [img]https://cdn.artofproblemsolving.com/attachments/2/d/01a8af0214a2400b279816fc5f6c039320e816.png[/img] [b]p5.[/b] Ann places the integers $1, 2,..., 100$ in a $10 \times 10$ grid, however she wants. In each round, Bob picks a row or column, and Ann sorts it from lowest to highest (left-to-right for rows; top-to-bottom for columns). However, Bob never sees the grid and gets no information from Ann. After eleven rounds, Bob must name a single cell that is guaranteed to contain a number that is at least $30$ and no more than $71$. Can he find a strategy to do this, no matter how Ann originally arranged the numbers? [u]Round 2[/u] [b]p6.[/b] Evelyn and Odette are playing a game with a deck of $101$ cards numbered $1$ through $101$. At the start of the game the deck is split, with Evelyn taking all the even cards and Odette taking all the odd cards. Each shuffles her cards. On every move, each player takes the top card from her deck and places it on a table. The player whose number is higher takes both cards from the table and adds them to the bottom of her deck, first the opponent’s card, then her own. The first player to run out of cards loses. Card $101$ was played against card $2$ on the $10$th move. Prove that this game will never end. [img]https://cdn.artofproblemsolving.com/attachments/8/1/aa16fe1fb4a30d5b9e89ac53bdae0d1bdf20b0.png[/img] [b]p7.[/b] The Vogon spaceship Tempest is descending on planet Earth. It will land on five adjacent buildings within a $10 \times 10$ grid, crushing any teacups on roofs of buildings within a $5 \times 1$ length of blocks (vertically or horizontally). As Commander of the Space Force, you can place any number of teacups on rooftops in advance. When the ship lands, you will hear how many teacups the spaceship breaks, but not where they were. (In the figure, you would hear $4$ cups break.) What is the smallest number of teacups you need to place to ensure you can identify at least one building the spaceship landed on? [img]https://cdn.artofproblemsolving.com/attachments/8/7/2a48592b371bba282303e60b4ff38f42de3551.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a quadrilateral with sides $\left|\overline{AB}\right|=2$, $\left|\overline{BC}\right|=\left|\overline{CD}\right|=4$ and $\left|\overline{DA}\right|=5$. The opposite angles, $\angle A$ and $\angle C$ are equal. The length of diagonal $BD$ equals $\textbf{(A)}~2\sqrt6$ $\textbf{(B)}~3\sqrt3$ $\textbf{(C)}~3\sqrt6$ $\textbf{(D)}~2\sqrt3$

A $2^{2014} + 1$ by $2^{2014} + 1$ grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer $n$ greater than $2$, there do not exist pairwise distinct black squares $s_1$, $s_2$, \dots, $s_n$ such that $s_i$ and $s_{i+1}$ share an edge for $i=1,2, \dots, n$ (here $s_{n+1}=s_1$). What is the maximum possible number of filled black squares? [i]Proposed by David Yang[/i]

Let $ABC$ be a triangle and let the points $D, E$ be on the rays $AB$, $AC$ such that $BCED$ is cyclic. Prove that the following two statements are equivalent: $\bullet$ There is a point $X$ on the circumcircle of $ABC$ such that $BDX$, $CEX$ are tangent to each other. $\bullet$ $AB \cdot AD \le 4R^2$, where $R$ is the radius of the circumcircle of $ABC$.