Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral? [asy] size(4cm); fill((-5,0)--(0,12)--(0,6)--(8,0)--cycle, gray+opacity(0.3)); draw((0,0)--(0,12)--(-5,0)--cycle); draw((0,0)--(8,0)--(0,6)); label("5", (-2.5,0), S); label("13", (-2.5,6), dir(140)); label("6", (0,3), E); label("8", (4,0), S); [/asy] [i]Proposed by Nathan Xiong[/i]

Let $P$ be a point on the circumcircle of square $ABCD$ such that $PA \cdot PC = 56$ and $PB \cdot PD = 90.$ What is the area of square $ABCD?$

Let $ABC$ be a triangle with $AB = 20$ and $AC = 22$. Suppose its incircle touches $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at $D$, $E$, and $F$ respectively, and $P$ is the foot of the perpendicular from $D$ to $\overline{EF}$. If $\angle BPC = 90^{\circ}$, then compute $BC^2$. [i]Proposed by Ankan Bhattacharya[/i]

In the accompanying figure, the outer square has side length 40. A second square S' of side length 15 is constructed inside S with the same center as S and with sides parallel to those of S. From each midpoint of a side of S, segments are drawn to the two closest vertices of S'. The result is a four-pointed starlike figure inscribed in S. The star figure is cut out and then folded to form a pyramid with base S'. Find the volume of this pyramid. [asy] draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0)); draw((2.5,2.5)--(4,0)--(5.5,2.5)--(8,4)--(5.5,5.5)--(4,8)--(2.5,5.5)--(0,4)--(2.5,2.5)--(5.5,2.5)--(5.5,5.5)--(2.5,5.5)--(2.5,2.5)); [/asy]

[b]p1.[/b] Will is distributing his money to three friends. Since he likes some friends more than others, the amount of money he gives each is in the ratio of $5 : 3 : 2$. If the person who received neither the least nor greatest amount of money was given $42$ dollars, how many dollars did Will distribute in all? [b]p2.[/b] Fan, Zhu, and Ming are driving around a circular track. Fan drives $24$ times as fast as Ming and Zhu drives $9$ times as fast as Ming. All three drivers start at the same point on the track and keep driving until Fan and Zhu pass Ming at the same time. During this interval, how many laps have Fan and Zhu driven together? [b]p3.[/b] Mr. DoBa is playing a game with Gunga the Gorilla. They both agree to think of a positive integer from $1$ to $120$, inclusive. Let the sum of their numbers be $n$. Let the remainder of the operation $\frac{n^2}{4}$ be $r$. If $r$ is $0$ or $1$, Mr. DoBa wins. Otherwise, Gunga wins. Let the probability that Mr. DoBa wins a given round of this game be $p$. What is $120p$? [b]p4.[/b] Let S be the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. How many subsets of $S$ are there such that if $a$ is the number of even numbers in the subset and $b$ is the number of odd numbers in the subset, then $a$ and $b$ are either both odd or both even? By definition, subsets of $S$ are unordered and only contain distinct elements that belong to $S$. [b]p5.[/b] Phillips Academy has five clusters, $WQN$, $WQS$, $PKN$, $FLG$ and $ABB$. The Blue Key heads are going to visit all five clusters in some order, except $WQS$ must be visited before $WQN$. How many total ways can they visit the five clusters? [b]p6.[/b] An astronaut is in a spaceship which is a cube of side length $6$. He can go outside but has to be within a distance of $3$ from the spaceship, as that is the length of the rope that tethers him to the ship. Out of all the possible points he can reach, the surface area of the outer surface can be expressed as $m+n\pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? [b]p7.[/b] Let $ABCD$ be a square and $E$ be a point in its interior such that $CDE$ is an equilateral triangle. The circumcircle of $CDE$ intersects sides $AD$ and $BC$ at $D$, $F$ and $C$, $G$, respectively. If $AB = 30$, the area of $AFGB$ can be expressed as $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and c is not divisible by the square of any prime. Find $a + b + c$. [b]p8.[/b] Suppose that $x, y, z$ satisfy the equations $$x + y + z = 3$$ $$x^2 + y^2 + z^2 = 3$$ $$x^3 + y^3 + z^3 = 3$$ Let the sum of all possible values of $x$ be $N$. What is $12000N$? [b]p9.[/b] In circle $O$ inscribe triangle $\vartriangle ABC$ so that $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be the midpoint of arc $BC$, and let $AD$ intersect $BC$ at $E$. Determine the value of $DE \cdot DA$. [b]p10.[/b] How many ways are there to color the vertices of a regular octagon in $3$ colors such that no two adjacent vertices have the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The points $A_1,B_1,C_1$ lie on the sides sides $BC,AC$ and $AB$ of the triangle $ABC$ respectively. Suppose that $AB_1-AC_1=CA_1-CB_1=BC_1-BA_1$. Let $I_A, I_B, I_C$ be the incentres of triangles $AB_1C_1,A_1BC_1$ and $A_1B_1C$ respectively. Prove that the circumcentre of triangle $I_AI_BI_C$ is the incentre of triangle $ABC$.

Two identical balls of radius $\sqrt{15}$ and two identical balls of a smaller radius are located on a plane so that each ball touches the other three. Find the area of the surface $S$ of the ball with the smaller radius.

1. There are $100$ participants. Out of every group of $12$ participants, there is one pair of familiar participants. Each participant is given a number (not necessarily $1$ through $100$). Prove that there is a pair of familiar participants whose number has the same starting digit. 2. $\sqrt{x + \sqrt{x + \sqrt{x + \dots + \sqrt{x}}}} = y$. If the left side is finite, find all integer solutions. 3. Is there a geometric sequence such that $a_0 > 0, b > 1$, and so that $a_l$ is an integer for $0 \le l \le 9$, but $a_l$ is not an integer for $l>9$? If so, find it. 4. Suppose r and s are positive integers and that $2^r$ is a permutation of the decimal representation of $2^s$. Prove that $r=s$. 5. Find the minimum area of a right triangle with an inscribed circle that has a radius of $1$ cm. [hide = Note]This isn't exactly verbatim, just paraphrased. I will update the questions when the official problems/solutions are released. In the meanwhile, feel free to post your solutions below![/hide]

A circle of radius $ r$ is concentric with and outside a regular hexagon of side length 2. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is 1/2. What is $ r$? $ \textbf{(A) } 2\sqrt {2} \plus{} 2\sqrt {3} \qquad \textbf{(B) } 3\sqrt {3} \plus{} \sqrt {2} \qquad \textbf{(C) } 2\sqrt {6} \plus{} \sqrt {3} \qquad \textbf{(D) } 3\sqrt {2} \plus{} \sqrt {6}\\ \textbf{(E) } 6\sqrt {2} \minus{} \sqrt {3}$

A swimmer is standing at a corner of a square swimming pool. She swims at a fixed speed and runs at a fixed speed (possibly different). No time is taken entering or leaving the pool. What path should she follow to reach the opposite corner of the pool in the shortest possible time?

Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$

The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?

Point $B$ is on $\overline{AC}$ with $AB = 9$ and $BC = 21$. Point $D$ is not on $\overline{AC}$ so that $AD = CD$, and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\triangle ACD$. Find $s$.

In parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}.$ Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB.$ Find the greatest integer that does not exceed $1000r.$