$P$ and $Q$ are points on the equal sides $AB$ and $AC$ respectively of an isosceles triangle $ABC$ such that $AP=CQ$. Moreover, neither $P$ nor $Q$ is a vertex of $ABC$. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of the triangle $ABC$.

Consider a pyramid $P-ABCD$ whose base $ABCD$ is a square and whose vertex $P$ is equidistant from $A$, $B$, $C$, and $D$. If $AB=1$ and $\angle APD=2\theta$ then the volume of the pyramid is $\text{(A)} \ \frac{\sin \theta}{6} \qquad \text{(B)} \ \frac{\cot \theta}{6} \qquad \text{(C)} \ \frac1{6\sin \theta} \qquad \text{(D)} \ \frac{1-\sin 2\theta}{6} \qquad \text{(E)} \ \frac{\sqrt{\cos 2\theta}}{6\sin \theta}$

A square in the [i]xy[/i]-plane has area [i]A[/i], and three of its vertices have [i]x[/i]-coordinates $2,0,$ and $18$ in some order. Find the sum of all possible values of [i]A[/i].

[u]Round 5[/u] [i]Each of the three problems in this round depends on the answer to two of the other problems. There is only one set of correct answers to these problems; however, each problem will be scored independently, regardless of whether the answers to the other problems are correct. [/i] [b]p13.[/b] Let $B$ be the answer to problem $14$, and let $C$ be the answer to problem $15$. A quadratic function $f(x)$ has two real roots that sum to $2^{10} + 4$. After translating the graph of $f(x)$ left by $B$ units and down by $C$ units, the new quadratic function also has two real roots. Find the sum of the two real roots of the new quadratic function. [b]p14.[/b] Let $A$ be the answer to problem $13$, and let $C$ be the answer to problem $15$. In the interior of angle $\angle NOM = 45^o$, there is a point $P$ such that $\angle MOP = A^o$ and $OP = C$. Let $X$ and $Y$ be the reflections of $P$ over $MO$ and $NO$, respectively. Find $(XY)^2$. [b]p15.[/b] Let $A$ be the answer to problem $13$, and let $B$ be the answer to problem $14$. Totoro hides a guava at point $X$ in a flat field and a mango at point $Y$ different from $X$ such that the length $XY$ is $B$. He wants to hide a papaya at point $Z$ such that $Y Z$ has length $A$ and the distance $ZX$ is a nonnegative integer. In how many different locations can he hide the papaya? [u]Round 6[/u] [b]p16.[/b] Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, $AB = 4$, $CD = 8$, $BC = 5$, and $AD = 6$. Given that point $E$ is on segment $CD$ and that $AE$ is parallel to $BC$, find the ratio between the area of trapezoid $ABCD$ and the area of triangle $ABE$. [b]p17.[/b] Find the maximum possible value of the greatest common divisor of $\overline{MOO}$ and $\overline{MOOSE}$, given that $S$, $O$, $M$, and $E$ are some nonzero digits. (The digits $S$, $O$, $M$, and $E$ are not necessarily pairwise distinct.) [b]p18.[/b] Suppose that $125$ politicians sit around a conference table. Each politician either always tells the truth or always lies. (Statements of a liar are never completely true, but can be partially true.) Each politician now claims that the two people beside them are both liars. Suppose that the greatest possible number of liars is $M$ and that the least possible number of liars is $N$. Determine the ordered pair $(M,N)$. [u]Round 7[/u] [b]p19.[/b] Define a [i]lucky [/i] number as a number that only contains $4$s and $7$s in its decimal representation. Find the sum of all three-digit lucky numbers. [b]p20.[/b] Let line segment $AB$ have length $25$ and let points $C$ and $D$ lie on the same side of line $AB$ such that $AC = 15$, $AD = 24$, $BC = 20$, and $BD = 7$. Given that rays $AC$ and $BD$ intersect at point $E$, compute $EA + EB$. [b]p21.[/b] A $3\times 3$ grid is filled with positive integers and has the property that each integer divides both the integer directly above it and directly to the right of it. Given that the number in the top-right corner is $30$, how many distinct grids are possible? [u]Round 8[/u] [b]p22.[/b] Define a sequence of positive integers $s_1, s_2, ... , s_{10}$ to be [i]terrible [/i] if the following conditions are satisfied for any pair of positive integers $i$ and $j$ satisfying $1 \le i < j \le 10$: $\bullet$ $s_i > s_j $ $\bullet$ $j - i + 1$ divides the quantity $s_i + s_{i+1} + ... + s_j$ Determine the minimum possible value of $s_1 + s_2 + ...+ s_{10}$ over all terrible sequences. [b]p23.[/b] The four points $(x, y)$ that satisfy $x = y^2 - 37$ and $y = x^2 - 37$ form a convex quadrilateral in the coordinate plane. Given that the diagonals of this quadrilateral intersect at point $P$, find the coordinates of $P$ as an ordered pair. [b]p24.[/b] Consider a non-empty set of segments of length $1$ in the plane which do not intersect except at their endpoints. (In other words, if point $P$ lies on distinct segments $a$ and $b$, then $P$ is an endpoint of both $a$ and $b$.) This set is called $3$-[i]amazing [/i] if each endpoint of a segment is the endpoint of exactly three segments in the set. Find the smallest possible size of a $3$-amazing set of segments. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2934024p26255963]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In circle $ O$, the midpoint of radius $ OX$ is $ Q$; at $ Q$, $ \overline{AB} \perp \overline{XY}$. The semi-circle with $ \overline{AB}$ as diameter intersects $ \overline{XY}$ in $ M$. Line $ \overline{AM}$ intersects circle $ O$ in $ C$, and line $ \overline{BM}$ intersects circle $ O$ in $ D$. Line $ \overline{AD}$ is drawn. Then, if the radius of circle $ O$ is $ r$, $ AD$ is: [asy]defaultpen(linewidth(.8pt)); unitsize(2.5cm); real m = 0; real b = 0; pair O = origin; pair X = (-1,0); pair Y = (1,0); pair Q = midpoint(O--X); pair A = (Q.x, -1*sqrt(3)/2); pair B = (Q.x, -1*A.y); pair M = (Q.x + sqrt(3)/2,0); m = (B.y - M.y)/(B.x - M.x); b = (B.y - m*B.x); pair D = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b)); m = (A.y - M.y)/(A.x - M.x); b = (A.y - m*A.x); pair C = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b)); draw(Circle(O,1)); draw(Arc(Q,sqrt(3)/2,-90,90)); draw(A--B); draw(X--Y); draw(B--D); draw(A--C); draw(A--D); dot(O);dot(M); label("$B$",B,NW); label("$C$",C,NE); label("$Y$",Y,E); label("$D$",D,SE); label("$A$",A,SW); label("$X$",X,W); label("$Q$",Q,SW); label("$O$",O,SW); label("$M$",M,NE+2N);[/asy]$ \textbf{(A)}\ r\sqrt {2} \qquad \textbf{(B)}\ r\qquad \textbf{(C)}\ \text{not a side of an inscribed regular polygon}\qquad \textbf{(D)}\ \frac {r\sqrt {3}}{2}\qquad \textbf{(E)}\ r\sqrt {3}$

During the van ride from the Grand Canyon to the beach, Michael asks his dad about the costs of renewable energy resources. "How much more does it really cost for a family like ours to switch entirely to renewable energy?" Jerry explains, "Part of that depends on where the family lives. In the Western states, solar energy pays off more than it does where we live in the Southeast. But as technology gets better, costs of producing more photovoltaic power go down, so in just a few years more people will have reasonably inexpensive options for switching to clearner power sources. Even now most families could switch to biomass for between $\$200$ and $\$1000$ per year. The energy comes from sawdust, switchgrass, and even landfill gas. We pay that premium ourselves, but some families operate on a tighter budget, or don't understand the alternatives yet." "Ew, landfill gas!" Alexis complains mockingly. Wanting to save her own energy, Alexis decides to take a nap. She falls asleep and dreams of walking around a $2-\text{D}$ coordinate grid, looking for a wormhole that she believes will transport her to the beach (bypassing the time spent in the family van). In her dream, Alexis finds herself holding a device that she recognizes as a $\textit{tricorder}$ from one of the old $\textit{Star Trek}$ t.v. series. The tricorder has a button labeled "wormhole" and when Alexis presses the button, a computerized voice from the tricorder announces, "You are at the origin. Distance to the wormhole is $2400$ units. Your wormhole distance allotment is $\textit{two}$."' Unsure as to how to reach, Alexis begins walking forward. As she walks, the tricorder displays at all times her distance from her starting point at the origin. When Alexis is $2400$ units from the origin, she again presses the "wormhole" buttom. The same computerized voice as before begins, "Distance to the origin is $2400$ units. Distance to the wormhole is $3840$ units. Your wormhole distance allotment is $\textit{two}$." Alexis begins to feel disoriented. She wonders what is means that her $\textit{wormhole distance allotment is two}$, and why that number didn't change as she pushed the button. She puts her hat down to mark her position, then wanders aroud a bit. The tricorder shows her two readings as she walks. The first she recognizes as her distance to the origin. The second reading clearly indicates her distance from the point where her hat lies - where she last pressed the button that gave her distance to the wormhole. Alexis picks up her hat and begins walking around. Eventually Alexis finds herself at a spot $2400$ units from the origin and $3840$ units from where she last pressed the button. Feeling hopeful, Alexis presses the tricorder's wormhole button again. Nothing happens. She presses it again, and again nothing happens. "Oh," she thinks, "my wormhole allotment was $\textit{two}$, and I used it up already!" Despair fills poor Alexis who isn't sure what a wormhole looks like or how she's supposed to find it. Then she takes matters into her own hands. Alexis sits down and scribbles some notes and realizes where the wormhole must be. Alexis gets up and runs straight from her "third position" to the wormhole. As she gets closer, she sees the wormhole, which looks oddly like a huge scoop of icecream. Alexis runs into the wormhole, then wakes up. How many units did Alexis run from her third position to the wormhole?

The points $A,B,$ and $C$ lies on the circumference of the unit circle. Furthermore, it is known that $AB$ is a diameter of the circle and \[\frac{|AC|}{|CB|}=\frac{3}{4}.\] The bisector of $ABC$ intersects the circumference at the point $D$. Determine the length of the $AD$.

[b]p1.[/b] What is the sum of the first $12$ positive integers? [b]p2.[/b] How many positive integers less than or equal to $100$ are multiples of both $2$ and $5$? [b]p3. [/b]Alex has a bag with $4$ white marbles and $4$ black marbles. She takes $2$ marbles from the bag without replacement. What is the probability that both marbles she took are black? Express your answer as a decimal or a fraction in lowest terms. [b]p4.[/b] How many $5$-digit numbers are there where each digit is either $1$ or $2$? [b]p5.[/b] An integer $a$ with $1\le a \le 10$ is randomly selected. What is the probability that $\frac{100}{a}$ is an integer? Express your answer as decimal or a fraction in lowest terms. [b]p6.[/b] Two distinct non-tangent circles are drawn so that they intersect each other. A third circle, distinct from the previous two, is drawn. Let $P$ be the number of points of intersection between any two circles. How many possible values of $P$ are there? [b]p7.[/b] Let $x, y, z$ be nonzero real numbers such that $x + y + z = xyz$. Compute $$\frac{1 + yz}{yz}+\frac{1 + xz}{xz}+\frac{1 + xy}{xy}.$$ [b]p8.[/b] How many positive integers less than $106$ are simultaneously perfect squares, cubes, and fourth powers? [b]p9.[/b] Let $C_1$ and $C_2$ be two circles centered at point $O$ of radii $1$ and $2$, respectively. Let $A$ be a point on $C_2$. We draw the two lines tangent to $C_1$ that pass through $A$, and label their other intersections with $C_2$ as $B$ and $C$. Let x be the length of minor arc $BC$, as shown. Compute $x$. [img]https://cdn.artofproblemsolving.com/attachments/7/5/915216d4b7eba0650d63b26715113e79daa176.png[/img] [b]p10.[/b] A circle of area $\pi$ is inscribed in an equilateral triangle. Find the area of the triangle. [b]p11.[/b] Julie runs a $2$ mile route every morning. She notices that if she jogs the route $2$ miles per hour faster than normal, then she will finish the route $5$ minutes faster. How fast (in miles per hour) does she normally jog? [b]p12.[/b] Let $ABCD$ be a square of side length $10$. Let $EFGH$ be a square of side length $15$ such that $E$ is the center of $ABCD$, $EF$ intersects $BC$ at $X$, and $EH$ intersects $CD$ at $Y$ (shown below). If $BX = 7$, what is the area of quadrilateral $EXCY$ ? [img]https://cdn.artofproblemsolving.com/attachments/d/b/2b2d6de789310036bc42d1e8bcf3931316c922.png[/img] [b]p13.[/b] How many solutions are there to the system of equations $$a^2 + b^2 = c^2$$ $$(a + 1)^2 + (b + 1)^2 = (c + 1)^2$$ if $a, b$, and $c$ are positive integers? [b]p14.[/b] A square of side length $ s$ is inscribed in a semicircle of radius $ r$ as shown. Compute $\frac{s}{r}$. [img]https://cdn.artofproblemsolving.com/attachments/5/f/22d7516efa240d00d6a9743a4dc204d23d190d.png[/img] [b]p15.[/b] $S$ is a collection of integers n with $1 \le n \le 50$ so that each integer in $S$ is composite and relatively prime to every other integer in $S$. What is the largest possible number of integers in $S$? [b]p16.[/b] Let $ABCD$ be a regular tetrahedron and let $W, X, Y, Z$ denote the centers of faces $ABC$, $BCD$, $CDA$, and $DAB$, respectively. What is the ratio of the volumes of tetrahedrons $WXYZ$ and $WAYZ$? Express your answer as a decimal or a fraction in lowest terms. [b]p17.[/b] Consider a random permutation $\{s_1, s_2, ... , s_8\}$ of $\{1, 1, 1, 1, -1, -1, -1, -1\}$. Let $S$ be the largest of the numbers $s_1$, $s_1 + s_2$, $s_1 + s_2 + s_3$, $...$ , $s_1 + s_2 + ... + s_8$. What is the probability that $S$ is exactly $3$? Express your answer as a decimal or a fraction in lowest terms. [b]p18.[/b] A positive integer is called [i]almost-kinda-semi-prime[/i] if it has a prime number of positive integer divisors. Given that there $are 168$ primes less than $1000$, how many almost-kinda-semi-prime numbers are there less than $1000$? [b]p19.[/b] Let $ABCD$ be a unit square and let $X, Y, Z$ be points on sides $AB$, $BC$, $CD$, respectively, such that $AX = BY = CZ$. If the area of triangle $XYZ$ is $\frac13$ , what is the maximum value of the ratio $XB/AX$? [img]https://cdn.artofproblemsolving.com/attachments/5/6/cf77e40f8e9bb03dea8e7e728b21e7fb899d3e.png[/img] [b]p20.[/b] Positive integers $a \le b \le c$ have the property that each of $a + b$, $b + c$, and $c + a$ are prime. If $a + b + c$ has exactly $4$ positive divisors, find the fourth smallest possible value of the product $c(c + b)(c + b + a)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two parallel chords in a circle have lengths $10$ and $14$, and the distance between them is $6$. The chord parallel to these chords and midway between them is of length $\sqrt{a}$ where $a$ is [asy] // note: diagram deliberately not to scale -- azjps void htick(pair A, pair B, real r){ D(A--B); D(A-(r,0)--A+(r,0)); D(B-(r,0)--B+(r,0)); } size(120); pathpen = linewidth(0.7); pointpen = black+linewidth(3); real min = -0.6, step = 0.5; pair[] A, B; D(unitcircle); for(int i = 0; i < 3; ++i) { A.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[0]); B.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[1]); D(D(A[i])--D(B[i])); } MP("10",(A[0]+B[0])/2,N); MP("\sqrt{a}",(A[1]+B[1])/2,N); MP("14",(A[2]+B[2])/2,N); htick((B[1].x+0.1,B[0].y),(B[1].x+0.1,B[2].y),0.06); MP("6",(B[1].x+0.1,B[0].y/2+B[2].y/2),E);[/asy] $\textbf{(A)}\ 144 \qquad \textbf{(B)}\ 156 \qquad \textbf{(C)}\ 168 \qquad \textbf{(D)}\ 176 \qquad \textbf{(E)}\ 184$

A semicircle with diameter length $16$ contains a circle radius $3$ tangent both to the inside of the semicircle and its diameter as shown. A second larger circle is tangent to the inside of the semicircle, the outside of the circle, and the diameter of the semicircle. The diameter of the second circle can be written as $\frac{n + k\sqrt{2}}{m}$ where $m$, $n$, and $k$ are positive integers and $m$ and $n$ have no factors in common. Find $m + n + k$. [asy] size(200); pair O=(0,0); real R=10, r=4.7; draw(arc(O,R,0,180)--cycle); pair P=(sqrt((R-r)^2-r^2),r),Q; draw(circle(P,r)); real a=0,b=r,c; for(int k=0;k<20;++k) { c=(a+b)/2; Q=(-sqrt((R-c)^2-c^2),c); if(abs(P-Q)>c+r) a=c; else b=c; } draw(circle(Q,c));[/asy]

[u]Round 5[/u] [b]p13.[/b] Find all ordered pairs of real numbers $(x, y)$ satisfying the following equations: $$\begin{cases} \dfrac{1}{xy} + \dfrac{y}{x}= 2 \\ \dfrac{1}{xy^2} + \dfrac{y^2}{x} = 7 \end{cases}$$ [b]p14.[/b] An egg plant is a hollow prism of negligible thickness, with height $2$ and an equilateral triangle base. Inside the egg plant, there is enough space for four spherical eggs of radius $1$. What is the minimum possible volume of the egg plant? [b]p15.[/b] How many ways are there for Farmer James to color each square of a $2\times 6$ grid with one of the three colors eggshell, cream, and cornsilk, so that no two adjacent squares are the same color? [u]Round 6[/u] [b]p16.[/b] In a triangle $ABC$, $\angle A = 45^o$, and let $D$ be the foot of the perpendicular from $A$ to segment $BC$. $BD = 2$ and $DC = 4$. Let $E$ be the intersection of the line $AD$ and the perpendicular line from $B$ to line $AC$. Find the length of $AE$. [b]p17.[/b] Find the largest positive integer $n$ such that there exists a unique positive integer $m$ satisfying $$\frac{1}{10} \le \frac{m}{n} \le \frac19$$ [b]p18.[/b] How many ordered pairs $(A,B)$ of positive integers are there such that $A+B = 10000$ and the number $A^2 + AB + B$ has all distinct digits in base $10$? [u]Round 7[/u] [b]p19.[/b] Pentagon $JAMES$ satisfies $JA = AM = ME = ES = 2$. Find the maximum possible area of $JAMES$. [b]p20.[/b] $P(x)$ is a monic polynomial (a polynomial with leading coecient $1$) of degree $4$, such that $P(2^n+1) =8^n + 1$ when $n = 1, 2, 3, 4$. Find the value of $P(1)$. [b]p21[/b]. PEAcock and Zombie Hen Hao are at the starting point of a circular track, and start running in the same direction at the same time. PEAcock runs at a constant speed that is $2018$ times faster than Zombie Hen Hao's constant speed. At some point in time, Farmer James takes a photograph of his two favorite chickens, and he notes that they are at different points along the track. Later on, Farmer James takes a second photograph, and to his amazement, PEAcock and Zombie Hen Hao have now swapped locations from the first photograph! How many distinct possibilities are there for PEAcock and Zombie Hen Hao's positions in Farmer James's first photograph? (Assume PEAcock and Zombie Hen Hao have negligible size.) [u]Round 8[/u] [b]p22.[/b] How many ways are there to scramble the letters in $EGGSEATER$ such that no two consecutive letters are the same? [b]p23.[/b] Let $JAMES$ be a regular pentagon. Let $X$ be on segment $JA$ such that $\frac{JX}{XA} = \frac{XA}{JA}$ . There exists a unique point $P$ on segment $AE$ such that $XM = XP$. Find the ratio $\frac{AE}{PE}$ . [b]p24.[/b] Find the minimum value of the function $$f(x) = \left|x - \frac{1}{x} \right|+ \left|x - \frac{2}{x} \right| + \left|x - \frac{3}{x} \right|+... + \left|x - \frac{9}{x} \right|+ \left|x - \frac{10}{x} \right|$$ over all nonzero real numbers $x$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949191p26406082]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is $\tfrac{m+\sqrt{n}}{k}$ where $m$, $n$, and $k$ are integers, and $k$ is not divisible by the square of any prime. Find $m + n + k$.

Don't have original wording: In $\triangle{ABC}$ medians $\overline{AD}$ and $\overline{BE}$ intersect at $G$ and $\triangle{AGE}$ is equilateral. Then $\cos(C)$ can be written as $\frac{m\sqrt p}n$, where $m$ and $n$ are relatively prime positive integers and $p$ is a positive integer not divisible by the square of any prime. What is $m+n+p?$ [asy] import geometry; unitsize(2cm); real arg(pair p) { return atan2(p.y, p.x) * 180/pi; } pair G=(0,0),E=(1,0),A=(1/2,sqrt(3)/2),D=1.5*G-0.5*A,C=2*E-A,B=2*D-C; pair t(pair p) { return rotate(-arg(dir(B--C)))*p; } path t(path p) { return rotate(-arg(dir(B--C)))*p; } void d(path p, pen q = black+linewidth(1.5)) { draw(t(p),q); } void o(pair p, pen q = 5+black) { dot(t(p),q); } void l(string s, pair p, pair d) { label(s, t(p),d); } d(A--B--C--cycle); d(A--D); d(B--E); o(A); o(B); o(C); o(D); o(E); o(G); l("$A$",A,N); l("$B$",B,SW); l("$C$",C,SE); l("$D$",D,S); l("$E$",E,NE); l("$G$",G,NW); [/asy] $\textbf{(A)}44~\textbf{(B)}48~\textbf{(C)}52~\textbf{(D)}56~\textbf{(E)}60$

For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible? $\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

Tetrahedron $ABCD$ has $AD=BC=28$, $AC=BD=44$, and $AB=CD=52$. For any point $X$ in space, define $f(X)=AX+BX+CX+DX$. The least possible value of $f(X)$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.