Let $\Gamma_1$ and $\Gamma_2$ be two circles touching each other externally at $R.$ Let $O_1$ and $O_2$ be the centres of $\Gamma_1$ and $\Gamma_2,$ respectively. Let $\ell_1$ be a line which is tangent to $\Gamma_2$ at $P$ and passing through $O_1,$ and let $\ell_2$ be the line tangent to $\Gamma_1$ at $Q$ and passing through $O_2.$ Let $K=\ell_1\cap \ell_2.$ If $KP=KQ$ then prove that the triangle $PQR$ is equilateral.

a) Given two squares $ABCD$ and $DEFG$, with point $E$ lying on the segment $CD$, and points$ F,G$ outside the square $ABCD$. Find the angle between lines $AE$ and $BF$. b) Two regular pentagons $OKLMN$ and $OPRST$ are given, and the point $P$ lies on the segment $ON$, and the points $R, S, T$ are outside the pentagon $OKLMN$. Find the angle between straight lines $KP$ and $MS$.

Let U be the union of a finite number (not necessarily connected and not necessarily disjoint) of closed unit squares lying in the plane. Can the quotient of the perimeter and area of U be arbitrarily large? @below: i think "single" means "connected".

Circles $S_1$ and $S_2$ meet at points $A$ and $B$. A line through $A$ is parallel to the line through the centers of $S_1$ and $S_2$ and meets $S_1$ and $S_2$ again $C$ and $D$ respectively. Circle $S_3$ having $CD$ as its diameter meets $S_1$ and $S_2$ again at $P$ and $Q$ respectively. Prove that lines $CP$, $DQ$, and $AB$ are concurent.

A block $Z$ is formed by gluing one face of a solid cube with side length 6 onto one of the circular faces of a right circular cylinder with radius $10$ and height $3$ so that the centers of the square and circle coincide. If $V$ is the smallest convex region that contains Z, calculate $\lfloor\operatorname{vol}V\rfloor$ (the greatest integer less than or equal to the volume of $V$).

A quadrilateral $ABCD$ is inscribed in a circle. On each of the sides $AB,BC,CD,DA$ one erects a rectangle towards the interior of the quadrilateral, the other side of the rectangle being equal to $CD,DA,AB,BC,$ respectively. Prove that the centers of these four rectangles are vertices of a rectangle.

The diagonals of the cyclic quadrilateral $ABCD$ intersect at the point $E$. Let $P$ and $Q$ are the centers of the circles circumscribed around the triangles $BCE$ and $DCE$, respectively. A straight line passing through the point $P$ parallel to $AB$, and a straight line passing through the point $Q$ parallel to $AD$, intersect at the point $R$. Prove that the point $R$ lies on segment $AC$.

Let $ABC$ be a triangle with incircle $\omega$, incenter $I$ and circumcircle $\Gamma$. Let $D$ be the tangency point of $\omega$ with $BC$, let $M$ be the midpoint of $ID$, and let $A'$ be the diametral opposite of $A$ with respect to $\Gamma$. If we denote $X=A'M\cap \Gamma$, then prove that the circumcircle of triangle $AXD$ is tangent to $BC$.

We have “bricks” made in the following way: we take a unit cube and glue to three of its faces which have a common vertex three more cubes in such a way that the faces glued together coincide. Is it possible to construct from these bricks an $11 \times 12 \times 13$ box? (A Andjans, Riga )

Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.

Let $ABC$ be a triangle with $BA=BC$ and $\angle ABC=90^{\circ}$. Let $D$ and $E$ be the midpoints of $CA$ and $BA$ respectively. The point $F$ is inside of $\triangle ABC$ such that $\triangle DEF$ is equilateral. Let $X=BF\cap AC$ and $Y=AF\cap DB$. Prove that $DX=YD$.

Let $M$ be a point on the side $AB$ of triangle $ABC$. The length $AB = c$ and $\angle CMA=\phi$ are given. Find the distance between the orthocentres (intersection points of altitudes) of the triangles $AMC$ and $BMC$. (IF Sharygin)

Circles of radius 3 and 6 are externally tangent to each other and are internally tangent to a circle of radius 9. The circle of radius 9 has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.

Let \( ABC \) be a triangle with \( AB < AC \). Let \( M \) be the midpoint of \( AC \), and let \( D \) be a point on segment \( AC \) such that \( DB = DC \). Let \( E \) be the point of intersection, different from \( B \), of the circumcircle of triangle \( ABM \) and line \( BD \). Define \( P \) and \( Q \) as the points of intersection of line \( BC \) with \( EM \) and \( AE \), respectively. Prove that \( P \) is the midpoint of \( BQ \).

A point moves on the sides of the triangle $ABC$, defined by the vertices $A(-1.8, 0)$, $B(3.2, 0)$, $C(0, 2.4)$ . Determine the positions of said point, in which the sum of their distance to the three vertices is absolute maximum or minimum. [img]https://cdn.artofproblemsolving.com/attachments/2/5/9e5bb48cbeefaa5f4c069532bf5605b9c1f5ea.png[/img]

A triangle $ABC$ is inscribed in a circle $\omega$. A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$). Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$. Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$. [i](Russia) Vasily Mokin and Fedor Ivlev[/i]

Let $K$ be a point in the interior of an acute triangle $ABC$ and $ARBPCQ$ be a convex hexagon whose vertices lie on the circumcircle $\Gamma$ of the triangle $ABC.$ Let $A_1$ be the second point where the circle passing through $K$ and tangent to $\Gamma$ at $A$ intersects the line $AP.$ The points $B_1$ and $C_1$ are defined similarly. Prove that \[ \min\left\{\frac{PA_1}{AA_1}, \: \frac{QB_1}{BB_1}, \: \frac{RC_1}{CC_1}\right\} \leq 1.\]

Does there exist a triangle, whose side is equal to some of its altitudes, another side is equal to some of its bisectors, and the third is equal to some of its medians?

In the quadrilateral $ABCD$ is $AD \parallel BC$ and the angles $\angle A$ and $\angle D$ are acute. The diagonals intersect in $P$. The circumscribed circles of $\vartriangle ABP$ and $\vartriangle CDP$ intersect the line $AD$ again at $S$ and $T$ respectively. Call $M$ the midpoint of $[ST]$. Prove that $\vartriangle BCM$ is isosceles. [img]https://1.bp.blogspot.com/-C5MqC0RTqwY/Xy1fAavi_aI/AAAAAAAAMSM/2MXMlwb13McCYTrOHm1ZzWc0nkaR1J6zQCLcBGAsYHQ/s0/flanders%2B2016%2Bp1.png[/img]

Let $ABCD$ be a cyclic quadrilateral, and angle bisectors of $\angle BAD$ and $\angle BCD$ meet at point $I$. Show that if $\angle BIC = \angle IDC$, then $I$ is the incenter of triangle $ABD$.

[b]p1.[/b] The total cost of $1$ football, $3$ tennis balls and $7$ golf balls is $\$14$ , while that of $1$ football, $4$ tennis balls and $10$ golf balls is $\$17$.If one has $\$20$ to spend, is this sufficient to buy a) $3$ footballs and $2$ tennis balls? b) $2$ footballs and $3$ tennis balls? [b]p2.[/b] Let $\overline{AB}$ and $\overline{CD}$ be two chords in a circle intersecting at a point $P$ (inside the circle). a) Prove that $AP \cdot PB = CP\cdot PD$. b) If $\overline{AB}$ is perpendicular to $\overline{CD}$ and the length of $\overline{AP}$ is $2$, the length of $\overline{PB}$ is $6$, and the length of $\overline{PD}$ is $3$, find the radius of the circle. [b]p3.[/b] A polynomial $P(x)$ of degree greater than one has the remainder $2$ when divided by $x-2$ and the remainder $3$ when divided by $x-3$. Find the remainder when $P(x)$ is divided by $x^2-5x+6$. [b]p4.[/b] Let $x_1= 2$ and $x_{n+1}=x_n+ (3n+2)$ for all $n$ greater than or equal to one. a) Find a formula expressing $x_n$ as a function of$ n$. b) Prove your result. [b]p5.[/b] The point $M$ is the midpoint of side $\overline{BC}$ of a triangle $ABC$. a) Prove that $AM \le \frac12 AB + \frac12 AC$. b) A fly takes off from a certain point and flies a total distance of $4$ meters, returning to the starting point. Explain why the fly never gets outside of some sphere with a radius of one meter. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] If $x + z = v$, $w + z = 2v$, $z - w = 2y$, and $y \ne 0$, compute the value of $$\left(x + y +\frac{x}{y} \right)^{101}.$$ [b]p2. [/b]Every minute, a snail picks one cardinal direction (either north, south, east, or west) with equal probability and moves one inch in that direction. What is the probability that after four minutes the snail is more than three inches away from where it started? [b]p3.[/b] What is the probability that a point chosen randomly from the interior of a cube is closer to the cube’s center than it is to any of the cube’s eight vertices? [b]p4.[/b] Let $ABCD$ be a rectangle where $AB = 4$ and $BC = 3$. Inscribe circles within triangles $ABC$ and $ACD$. What is the distance between the centers of these two circles? [b]p5.[/b] $C$ is a circle centered at the origin that is tangent to the line $x - y\sqrt3 = 4$. Find the radius of $C$. [b]p6.[/b] I have a fair $100$-sided die that has the numbers $ 1$ through $100$ on its sides. What is the probability that if I roll this die three times that the number on the first roll will be greater than or equal to the sum of the two numbers on the second and third rolls? [b]p7. [/b] List all solutions $(x, y, z)$ of the following system of equations with x, y, and z positive real numbers: $$x^2 + y^2 = 16$$ $$x^2 + z^2 = 4 + xz$$ $$y^2 + z^2 = 4 + yz\sqrt3$$ [b]p8.[/b] $A_1A_2A_3A_4A_5A_6A_7$ is a regular heptagon ($7$ sided-figure) centered at the origin where $A_1 = (\sqrt[91]{6}, 0)$. $B_1B_2B_3... B_{13}$ is a regular triskaidecagon ($13$ sided-figure) centered at the origin where $B_1 =(0,\sqrt[91]{41})$. Compute the product of all lengths $A_iB_j$ , where $i$ ranges between $1$ and $7$, inclusive, and $j$ ranges between $1$ and $13$, inclusive. [b]p9.[/b] How many three-digit integers are there such that one digit of the integer is exactly two times a digit of the integer that is in a different place than the first? (For example, $100$, $122$, and $124$ should be included in the count, but $42$ and $130$ should not.) [b]p10.[/b] Let $\alpha$ and $\beta$ be the solutions of the quadratic equation $$x^2 - 1154x + 1 = 0.$$ Find $\sqrt[4]{\alpha}+\sqrt[4]{\beta}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Triangle $ABC$ is said to be perpendicular to triangle $DEF$ if the perpendiculars from $A$ to $EF$,from $B$ to $FD$ and from $C$ to $DE$ are concurrent.Prove that if $ABC$ is perpendicular to $DEF$,then $DEF$ is perpendicular to $ABC$

Let $A,B,C,D,E$ be consecutive points on a circle with center $O$ such that $AC=BD=CE=DO$. Let $H_1,H_2,H_3$ be the orthocenters triangles $ACD,BCD,BCE$ respectively. Prove that the triangle $H_1H_2H_3$ is right.

[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].