On the side $CD$ of the square $ABCD$, the point $F$ is chosen and the equal squares $DGFE$ and $AKEH$ are constructed ($E$ and $H$ lie inside the square). Let $M$ be the midpoint of $DF$, $J$ is the incenter of the triangle $CFH$. Prove that: a) the points $D, K, H, J, F$ lie on the same circle; b) the circles inscribed in triangles $CFH$ and $GMF$ have the same radii.

A circle $\omega$ is strictly inside triangle $ABC$. The tangents from $A$ to $\omega$ intersect $BC$ in $A_1,A_2$ define $B_1,B_2,C_1,C_2$ similarly. Prove that if five of six points $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a circle the sixth one lie on the circle too.

[u]Round 5[/u] [b]p13.[/b] The expression $\sqrt2 \times \sqrt[3]{3} \times \sqrt[6]{6}$ can be expressed as a single radical in the form $\sqrt[n]{m}$, where $m$ and $n$ are integers, and $n$ is as small as possible. What is the value of $m + n$? [b]p14.[/b] Bertie Bott also produces Bertie Bott’s Every Flavor Pez. Each package contains $6$ peppermint-, $2$ kumquat-, $3$ chili pepper-, and $5$ garlic-flavored candies in a random order. Harold opens a package and slips it into his Dumbledore-shaped Pez dispenser. What is the probability that of the first four candies, at least three are garlic-flavored? [b]p15.[/b] Quadrilateral $ABCD$ with $AB = BC = 1$ and $CD = DA = 2$ is circumscribed around and inscribed in two different circles. What is the area of the region between these circles? [u] Round 6[/u] [b]p16.[/b] Find all values of x that satisfy $\sqrt[3]{x^7} + \sqrt[3]{x^4} = \sqrt[3]{x}$. [b]p17.[/b] An octagon has vertices at $(2, 1)$, $(1, 2)$, $(-1, 2)$, $(-2, 1)$, $(-2, -1)$, $(-1, -2)$, $(1, -2)$, and $(2, -1)$. What is the minimum total area that must be cut off of the octagon so that the remaining figure is a regular octagon? [b]p18.[/b] Ron writes a $4$ digit number with no zeros. He tells Ronny that when he sums up all the two-digit numbers that are made by taking 2 consecutive digits of the number, he gets 99. He also reveals that his number is divisible by 8. What is the smallest possible number Ron could have written? [u]Round 7[/u] [b]p19.[/b] In a certain summer school, 30 kids enjoy geometry, 40 kids enjoy number theory, and 50 kids enjoy algebra. Also, the number of kids who only enjoy geometry is equal to the number of kids who only enjoy number theory and also equal to the number of kids who only enjoy algebra. What is the difference between the maximum and minimum possible numbers of kids who only enjoy geometry and algebra? [b]p20.[/b] A mouse is trying to run from the origin to a piece of cheese, located at $(4, 6)$, by traveling the shortest path possible along the lattice grid. However, on a lattice point within the region $\{0 \le x \le 4, 0 \le y \le 6$, $(x, y) \ne (0, 0),(4, 6)\}$ lies a rock through which the mouse cannot travel. The number of paths from which the mouse can choose depends on where the rock is placed. What is the difference between the maximum possible number of paths and the minimum possible number of paths available to the mouse? [b]p21.[/b] The nine points $(x, y)$ with $x, y \in \{-1, 0, 1\}$ are connected with horizontal and vertical segments to their nearest neighbors. Vikas starts at $(0, 1)$ and must travel to $(1, 0)$, $(0, -1)$, and $(-1, 0)$ in any order before returning to $(0, 1)$. However, he cannot travel to the origin $4$ times. If he wishes to travel the least distance possible throughout his journey, then how many possible paths can he take? [u]Round 8[/u] [b]p22.[/b] Let $g(x) = x^3 - x^2- 5x + 2$. If a, b, and c are the roots of g(x), then find the value of $((a + b)(b + c)(c + a))^3$. [b]p23.[/b] A regular octahedron composed of equilateral triangles of side length $1$ is contained within a larger tetrahedron such that the four faces of the tetrahedron coincide with four of the octahedron’s faces, none of which share an edge. What is the ratio of the volume of the octahedron to the volume of the tetrahedron? [b]p24.[/b] You are the lone soul at the south-west corner of a square within Elysium. Every turn, you have a $\frac13$ chance of remaining at your corner and a $\frac13$ chance of moving to each of the two closest corners. What is the probability that after four turns, you will have visited every corner at least once? PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134177p28401527]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134489p28406583]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let two circles be externally tangent at point $C$, with parallel diameters $A_1A_2, B_1B_2$ (i.e. the quadrilateral $A_1B_1B_2A_2$ is a trapezoid with bases $A_1A_2$ and $B_1B_2$ or parallelogram). Circle with the center on the common internal tangent to these two circles, passes through the intersection point of lines $A_1B_2$ and $A_2B_1$ as well intersects those lines at points $M, N$. Prove that the line $MN$ is perpendicular to the parallel diameters $A_1A_2, B_1B_2$. (Yuri Biletsky)

On a circle of center $O$, let $A$ and $B$ be points on the circle such that $\angle AOB = 120^o$. Point $C$ lies on the small arc $AB$ and point $D$ lies on the segment $AB$. Let also $AD = 2, BD = 1$ and $CD = \sqrt2$. Calculate the area of triangle $ABC$.

On the legs of $\angle AOB$, the segments $OA$ and $OB$ lie, $OA > OB$. Points $M$ and $N$ on lines $OA$ and $OB$, respectively, are such that $AM = BN = x$. Find $x$ for which the length of $MN$ is minimal.

$I,\Omega$ are the incenter and the circumcircle of triangle $ABC$, respectively, and the tangents of $B,C$ to $\Omega$ intersect at $L$. Assume that $P\neq C$ is a point on $\Omega$ such that $CI,AP$, and the circle with center $L$ and radius $LC$ are concurrent. Let the foot from $I$ to $AB$ be $F$, the midpoint of $BC$ be $M$, $X$ is a point on $\Omega$ s.t. $AI,BC,PX$ are concurrent. Prove that the lines $AI,AX,MF$ form an isosceles triangle. [i]Proposed by ckliao914[/i]

Let $ABCDEFGH$ be a parallelepiped with $AE \parallel BF \parallel CG \parallel DH$. Prove the inequality \[AF + AH + AC \leq AB + AD + AE + AG.\] In what cases does equality hold? [i]Proposed by France.[/i]

There are $n > 1$ distinct points marked in the plane. Prove that there exists a set of circles $\mathcal C$ such that [color=#FFFFFF]___[/color]$\bullet$ Each circle in $\mathcal C$ has unit radius. [color=#FFFFFF]___[/color]$\bullet$ Every marked point lies in the (strict) interior of some circle in $\mathcal C$. [color=#FFFFFF]___[/color]$\bullet$ There are less than $0.3n$ pairs of circles in $\mathcal C$ that intersect in exactly $2$ points. [i]Note: Weaker results with $\it{0.3n}$ replaced by $\it{cn}$ may be awarded points depending on the value of the constant $\it{c > 0.3}$.[/i] [i]Proposed by Siddharth Choppara, Archit Manas, Ananda Bhaduri, Manu Param[/i]

Let $ABCD$ be a regular triangular pyramid with base $ABC$ (this means that $ABC$ is a regular triangle, and edges $AD$, $BD$ and $CD$ are equal) and plane angles at the opposite vertex equal to $a$. A plane parallel to $ABC$ intersects $AD$, $BD$ and $CD$, respectively, at points $A_1$, $B_1$ and $C_1$. The surface of the polyhedron $ABCA_1B_1C_1$ is cut along five edges: $A_1B_1$, $B_1C_1$, $C_1C$, $CA$ and $AB$, after which this surface is turned onto a plane. At what values of $a$ will the resulting scan necessarily cover itself?

Nine points, no three of which lie on the same straight line, are located inside an equilateral triangle of side $4$. Prove that some three of these points are vertices of a triangle whose area is not greater than $\sqrt3$.

Consider a convex pentagon circumscribed about a circle. We name the lines that connect vertices of the pentagon with the opposite points of tangency with the circle [i]gergonnians[/i]. (a) Prove that if four gergonnians are conncurrent, the all five of them are concurrent. (b) Prove that if there is a triple of gergonnians that are concurrent, then there is another triple of gergonnians that are concurrent.

Let $C$ and $D$ be points on the semicircle with center $O$ and diameter $AB$ such that $ABCD$ is a convex quadrilateral. Let $Q$ be the intersection of the diagonals $[AC]$ and $[BD]$, and $P$ be the intersection of the lines tangent to the semicircle at $C$ and $D$. If $m(\widehat{AQB})=2m(\widehat{COD})$ and $|AB|=2$, then what is $|PO|$? $ \textbf{(A)}\ \sqrt 2 \qquad\textbf{(B)}\ \sqrt 3 \qquad\textbf{(C)}\ \frac{1+\sqrt 3} 2 \qquad\textbf{(D)}\ \frac{1+\sqrt 3}{2\sqrt 2} \qquad\textbf{(E)}\ \frac{2\sqrt 3} 3 $

Let the intersection of the diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$ be point $E$. Let $M$ be the midpoint of $AE$ and $N$ be the midpoint of $CD$. It's known that $BD$ bisects $\angle ABC$. Prove that $ABCD$ is cyclic if and only if $MBCN$ is cyclic.

Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.

Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $ 4 : 3$. The horizontal length of a “$ 27$-inch” television screen is closest, in inches, to which of the following? [asy]import math; unitsize(7mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((0,0)--(4,0)--(4,3)--(0,3)--(0,0)--(4,3)); fill((0,0)--(4,0)--(4,3)--cycle,mediumgray); label(rotate(aTan(3.0/4.0))*"Diagonal",(2,1.5),NW); label(rotate(90)*"Height",(4,1.5),E); label("Length",(2,0),S);[/asy]$ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 20.5 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 21.5 \qquad \textbf{(E)}\ 22$

Two circles $c$ and $c'$ with centers $O$ and $O'$ lie completely outside each other. Points $A, B$, and $C$ lie on the circle $c$ and points $A', B'$, and $C$ lie on the circle $c'$ so that segment $AB\parallel A'B'$, $BC \parallel B'C'$, and $\angle ABC = \angle A'B'C'$. The lines $AA', BB$', and $CC'$ are all different and intersect in one point $P$, which does not coincide with any of the vertices of the triangles $ABC$ or $A'B'C'$. Prove that $\angle AOB = \angle A'O'B'$.

The diagram shows two congruent isosceles triangles in a $20\times20$ square which has been partitioned into four $10\times10$ squares. Find the area of the shaded region. [asy] import graph; size(4.4cm); real labelscalefactor = 0.5; pen dotstyle = black; fill((-2,5)--(0,1)--(1,3)--(1,5)--cycle,gray); draw((-3,5)--(1,5), linewidth(2.2)); draw((1,5)--(1,1), linewidth(2.2)); draw((1,1)--(-3,1), linewidth(2.2)); draw((-3,1)--(-3,5), linewidth(2.2)); draw((-1,5)--(-1,1), linewidth(2.2)); draw((-3,3)--(1,3), linewidth(2.2)); draw((-2,5)--(-3,3), linewidth(1.4)); draw((-2,5)--(0,1), linewidth(1.4)); draw((0,1)--(1,3), linewidth(1.4)); draw((-2,5)--(0,1)); draw((0,1)--(1,3)); draw((1,3)--(1,5)); draw((1,5)--(-2,5));[/asy]

Let $ABC$ be an equilateral triangle with circumcircle $k$. A circle $q$ touches $k$ from outside in a point $D$, where the point $D$ on $k$ is chosen so that $D$ and $C$ lie on different sides of the line $AB$. We now draw tangent lines from $A,B$ and $C$ to the circle $q$ and denote the lengths of the respective tangent line segments by $a,b$ and $c$. Prove that $a+b=c$.

Suppose $A_1 A_2 \ldots A_n$ is an $n$-sided polygon with $n \geq 3$ and $\angle A_j \leq 180^{\circ}$ for each $j$ (in other words, the polygon is convex or has fewer than $n$ distinct sides). For each $i \leq n$, suppose $\alpha_i$ is the smallest possible value of $\angle{A_i A_j A_{i+1}}$ where $j$ is neither $i$ nor $i+1$. (Here, we define $A_{n+1} = A_1$.) Prove that \[ \alpha_1 + \alpha_2 + \cdots + \alpha_n \leq 180^{\circ} \] and determine all equality cases.

As shown in the figure, let point $E$ be the intersection of the diagonals $AC$ and $BD$ of the cyclic quadrilateral $ABCD$. The circumcenter of triangle $ABE$ is denoted as $K$. Point $X$ is the reflection of point $B$ with respect to the line $CD$, and point $Y$ is the point on the plane such that quadrilateral $DKEY$ is a parallelogram. Prove that the points $D,E,X,Y$ are concyclic. [img]https://cdn.artofproblemsolving.com/attachments/3/4/df852f90028df6f09b4ec1342f5330fc531d12.jpg[/img]

A point $C$ lies on a line segment $AB$ between $A$ and $B$ and circles are drawn having $AC$ and $CB$ as diameters. A common tangent to both circles touches the circle with $AC$ as diameter at $P \ne C$ and the circle with $CB$ as diameter at $Q \ne C$. Prove that $AP, BQ$ and the common tangent to both circles at $C$ all meet at a single point which lies on the circumference of the circle with $AB$ as diameter.

In the plane there are six different points $A, B, C, D, E, F$ such that $ABCD$ and $CDEF$ are parallelograms. What is the maximum number of those points that can be located on one circle?

Circle $\odot O$ with diameter $\overline{AB}$ has chord $\overline{CD}$ drawn such that $\overline{AB}$ is perpendicular to $\overline{CD}$ at $P$. Another circle $\odot A$ is drawn, sharing chord $\overline{CD}$. A point $Q$ on minor arc $\overline{CD}$ of $\odot A$ is chosen so that $\text{m} \angle AQP + \text{m} \angle QPB = 60^\circ$. Line $l$ is tangent to $\odot A$ through $Q$ and a point $X$ on $l$ is chosen such that $PX=BX$. If $PQ = 13$ and $BQ = 35$, find $QX$. [i]Proposed by Aaron Lin[/i]

Let $n\ge3$ be an integer. Two regular $n$-gons $\mathcal{A}$ and $\mathcal{B}$ are given in the plane. Prove that the vertices of $\mathcal{A}$ that lie inside $\mathcal{B}$ or on its boundary are consecutive. (That is, prove that there exists a line separating those vertices of $\mathcal{A}$ that lie inside $\mathcal{B}$ or on its boundary from the other vertices of $\mathcal{A}$.)