Let $\triangle BC$ be a triangle with side lengths $AB = 9, BC = 10, CA = 11$. Let $O$ be the circumcenter of $\triangle ABC$. Denote $D = AO \cap BC, E = BO \cap CA, F = CO \cap AB$. If $\frac{1}{AD} + \frac{1}{BE} + \frac{1}{FC}$ can be written in simplest form as $\frac{a \sqrt{b}}{c}$, find $a + b + c$.

The sides of triangle $BAC$ are in the ratio $2: 3: 4$. $BD$ is the angle-bisector drawn to the shortest side $AC$, dividing it into segments $AD$ and $CD$. If the length of $AC$ is $10$, then the length of the longer segment of $AC$ is: $\text{(A)} \ 3\frac12 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 5\frac57 \qquad \text{(D)} \ 6 \qquad \text{(E)} \ 7\frac12$

A cube is divided into $27$ equal smaller cubes. A plane intersects the cube. Find the maximum possible number of smaller cubes the plane can intersect.

Let $AKL$ be a triangle such that $\angle ALK > 90^o +\angle LAK$. Construct an isosceles trapezoid $ABCD$ with $AB \parallel CD$ such that $K$ lies on the side $BC, L$ on the diagonal $AC$ and the lines $AK$ and $BL$ intersect at the circumcenter of the trapezoid.

A circle meets the three sides $BC,CA,AB$ of a triangle $ABC$ at points $D_1,D_2;E_1,E_2; F_1,F_2$ respectively. Furthermore, line segments $D_1E_1$ and $D_2F_2$ intersect at point $L$, line segments $E_1F_1$ and $E_2D_2$ intersect at point $M$, line segments $F_1D_1$ and $F_2E_2$ intersect at point $N$. Prove that the lines $AL,BM,CN$ are concurrent.

A parallelogram $ABCD$ is given. Two circles with centers at the vertices $A$ and $C$ pass through $B$. The straight line $\ell$ that passes through $B$ and crosses the circles at second time at points $X, Y$ respectively. Prove that $DX = DY$.

$I_b$ is the $B$-excenter of the triangle $ABC$ and $\omega$ is the circumcircle of this triangle. $M$ is the middle of arc $BC$ of $\omega$ which doesn't contain $A$. $MI_b$ meets $\omega$ at $T\not =M$. Prove that $$ TB\cdot TC=TI_b^2.$$

The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$ so that $CD = BC$. The side $CA$ is extended beyond $A$ to $E$ so that $AE = 2CA$. Prove that, if $AD=BE$, then the triangle $ABC$ is right-angled.

Let $ABC$ be a triangle with $\angle{BAC} = 40^{\circ}$ and $\angle{ABC}=60^{\circ}$. Let $D$ and $E$ be the points lying on the sides $AC$ and $AB$, respectively, such that $\angle{CBD} = 40^{\circ}$ and $\angle{BCE} = 70^{\circ}$. Let $F$ be the point of intersection of the lines $BD$ and $CE$. Show that the line $AF$ is perpendicular to the line $BC$.

Two circles $ \omega_1$ and $ \omega_2$ tangents internally in point $ P$. On their common tangent points $ A$, $ B$ are chosen such that $ P$ lies between $ A$ and $ B$. Let $ C$ and $ D$ be the intersection points of tangent from $ A$ to $ \omega_1$, tangent from $ B$ to $ \omega_2$ and tangent from $ A$ to $ \omega_2$, tangent from $ B$ to $ \omega_1$, respectively. Prove that $ CA \plus{} CB \equal{} DA \plus{} DB$.

Given is a convex pentagon $ABCDE$ in which $\angle A = 60^o$, $\angle B = 100^o$, $\angle C = 140^o$. Show that this pentagon can be placed in a circle with a radius of $\frac23 AD$.

$ ABC$ and $ A'B'C'$ are two triangles in the same plane such that the lines $ AA',BB',CC'$ are mutually parallel. Let $ [ABC]$ denotes the area of triangle $ ABC$ with an appropriate $ \pm$ sign, etc.; prove that \[ 3([ABC] \plus{} [A'B'C']) \equal{} [AB'C'] \plus{} [BC'A'] \plus{} [CA'B'] \plus{} [A'BC] \plus{} [B'CA] \plus{} [C'AB].\]

Let $ABCD$ be a square. Choose points $E$ on $BC$ and $F$ on $CD$ so that $\angle EAF=45^\circ$ and so that neither $E$ nor $F$ is a vertex of the square. The lines $AE$ and $AF$ intersect the circumcircle of the square in the points $G$ and $H$ distinct from $A$, respectively. Show that the lines $EF$ and $GH$ are parallel.

The medians of the sides $AC$ and $BC$ in the triangle $ABC$ are perpendicular to each other. Prove that $\frac12 <\frac{|AC|}{|BC|}<2$.

A point $P$ lies in the interior of the triangle $ABC$. The lines $AP, BP$, and $CP$ intersect $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Prove that if two of the quadrilaterals $ABDE, BCEF, CAFD, AEPF, BFPD$, and $CDPE$ are concyclic, then all six are concyclic.

[b]p1.[/b] Fluffy the Dog is an extremely fluffy dog. Because of his extreme fluffiness, children always love petting Fluffy anywhere. Given that Fluffy likes being petted $1/4$ of the time, out of $120$ random people who each pet Fluffy once, what is the expected number of times Fluffy will enjoy being petted? [b]p2.[/b] Andy thinks of four numbers $27$, $81$, $36$, and $41$ and whispers the numbers to his classmate Cynthia. For each number she hears, Cynthia writes down every factor of that number on the whiteboard. What is the sum of all the different numbers that are on the whiteboard? (Don't include the same number in your sum more than once) [b]p3.[/b] Charles wants to increase the area his square garden in his backyard. He increases the length of his garden by $2$ and increases the width of his garden by $3$. If the new area of his garden is $182$, then what was the original area of his garden? [b]p4.[/b] Antonio is trying to arrange his flute ensemble into an array. However, when he arranges his players into rows of $6$, there are $2$ flute players left over. When he arranges his players into rows of $13$, there are $10$ flute players left over. What is the smallest possible number of flute players in his ensemble such that this number has three prime factors? [b]p5.[/b] On the AMC $9$ (Acton Math Competition $9$), $5$ points are given for a correct answer, $2$ points are given for a blank answer and $0$ points are given for an incorrect answer. How many possible scores are there on the AMC $9$, a $15$ problem contest? [b]p6.[/b] Charlie Puth produced three albums this year in the form of CD's. One CD was circular, the second CD was in the shape of a square, and the final one was in the shape of a regular hexagon. When his producer circumscribed a circle around each shape, he noticed that each time, the circumscribed circle had a radius of $10$. The total area occupied by $1$ of each of the different types of CDs can be expressed in the form $a + b\pi + c\sqrt{d}$ where $d$ is not divisible by the square of any prime. Find $a + b + c + d$. [b]p7.[/b] You are picking blueberries and strawberries to bring home. Each bushel of blueberries earns you $10$ dollars and each bushel of strawberries earns you $8$ dollars. However your cart can only fit $24$ bushels total and has a weight limit of $100$ lbs. If a bushel of blueberries weighs $8$ lbs and each bushel of strawberries weighs $6$ lbs, what is your maximum profit. (You can only pick an integer number of bushels) [b]p8.[/b] The number $$\sqrt{2218 + 144\sqrt{35} + 176\sqrt{55} + 198\sqrt{77}}$$ can be expressed in the form $a\sqrt5 + b\sqrt7 + c\sqrt{11}$ for positive integers $a, b, c$. Find $abc$. [b]p9.[/b] Let $(x, y)$ be a point such that no circle passes through the three points $(9,15)$, $(12, 20)$, $(x, y)$, and no circle passes through the points $(0, 17)$, $(16, 19)$, $(x, y)$. Given that $x - y = -\frac{p}{q}$ for relatively prime positive integers $p$, $q$, Find $p + q$. [b]p10.[/b] How many ways can Alfred, Betty, Catherine, David, Emily and Fred sit around a $6$ person table if no more than three consecutive people can be in alphabetical order (clockwise)? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A rope of length 10 [i]m[/i] is tied tautly from the top of a flagpole to the ground 6 [i]m[/i] away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 [i]cm/min[/i]. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of Cancer so that the sun is directly overhead.)

Given a sheet of tin $6\times 6$. It is allowed to bend it and to cut it but in such a way that it doesn’t fall to pieces. How to make a cube with edge $2$, divided by partitions into unit cubes?

Let $ ABCD$ be a rectangle with center $ O$, $ AB\neq BC$. The perpendicular from $ O$ to $ BD$ cuts the lines $ AB$ and $ BC$ in $ E$ and $ F$ respectively. Let $ M,N$ be the midpoints of the segments $ CD,AD$ respectively. Prove that $ FM \perp EN$.

Let \( ABC \) be a scalene triangle. Let \( E \) and \( F \) be the midpoints of sides \( AC \) and \( AB \), respectively, and let \( D \) be any point on segment \( BC \). The circumcircles of triangles \( BDF \) and \( CDE \) intersect line \( EF \) at points \( K \neq F \), and \( L \neq E \), respectively, and intersect at points \( X \neq D \). The point \( Y \) is on line \( DX \) such that \( AY \) is parallel to \( BC \). Prove that points \( K \), \( L \), \( X \), and \( Y \) lie on the same circle.

A convex quadrilateral $ABCD$ inscribed in a circle $k$ has the property that the rays $DA$ and $CB$ meet at a point $E$ for which $CD^2=AD\cdot
ED.$ The perpendicular to $ED$ at $A$ intersects $k$ again at point $F.$ Prove that the segments $AD$ and $CF$ are congruent if and only if the circumcenter of $\triangle ABE$ lies on $ED.$

Is there a convex polygon with each side equal to some diagonal, and each diagonal equal to some side?

Let $ABC$ be a triangle with $\angle B = 30^{\circ }$. We consider the closed disks of radius $\frac{AC}3$, centered in $A$, $B$, $C$. Does there exist an equilateral triangle with one vertex in each of the 3 disks? [i]Radu Gologan, Dan Schwarz[/i]

On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $K, L$ and $M$ are selected, respectively, such that $AK = AM$ and $BK = BL$. If $\angle{MLB} = \angle{CAB}$, Prove that $ML = KI$, where $I$ is the incenter of triangle $CML$.

Let $O = (0,0), A = (0,a), and B = (0,b)$, where $0<b<a$ are reals. Let $\Gamma$ be a circle with diameter $\overline{AB}$ and let $P$ be any other point on $\Gamma$. Line $PA$ meets the x-axis again at $Q$. Prove that angle $\angle BQP = \angle BOP$.