Aurick has a cup, a right cone with a circular base of radius $\frac12$, filled with milk tea. The slant height of the cup is $1$, and the tea fills the cup $\frac12$ of the way up the cup’s side. Suppose that Aurick tips the cup just to the point of spilling, as shown in the diagram. The new slant height EA and the tilted tea surface’s major axis $ET$ form $\angle T EA$. Compute $\cos(\angle T EA)$. [img]https://cdn.artofproblemsolving.com/attachments/e/e/76e12ee31ce4ba8a5daaf0f5538b98726a0d37.png[/img]

Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a right octagon with center $O$ and $\lambda_1$,$\lambda_2$, $\lambda_3$, $\lambda_4$ be some rational numbers for which: $\lambda_1 \overrightarrow{OA_1}+\lambda_2 \overrightarrow{OA_2}+\lambda_3 \overrightarrow{OA_3}+\lambda_4 \overrightarrow{OA_4} =\overrightarrow{o}$. Prove that $\lambda_1=\lambda_2=\lambda_3=\lambda_4=0$.

Let $ K$ be the circumscribed circle of the trapezoid $ ABCD$ . In this trapezoid the diagonals $ AC$ and $ BD$ are perpendicular. The parallel sides $ AB\equal{}a$ and $ CD\equal{}c$ are diameters of the circles $ K_{a}$ and $ K_{b}$ respectively. Find the perimeter and the area of the part inside the circle $ K$, that is outside circles $ K_{a}$ and $ K_{b}$.

[u]Round 9[/u] [b]p25.[/b] Maisy the Bear is at the origin of the Cartesian Plane. WhenMaisy is on the point $(m,n)$ then it can jump to either $(m,n +1)$ or $(m+1,n)$. Let $L(x, y)$ be the number of pathsMaisy can take to reach the point $(x, y)$. The sum of $L(x, y)$ over all lattice points $(x, y)$ with both coordinates between $0$ and $2020$, inclusive, can be written as ${2k \choose k} - j$ for a minimum positive integer k and corresponding positive integer $j$ . Find $k + j$ . [b]p26.[/b] A circle with center $O$ and radius $2$ and a circle with center $P$ and radius $3$ are externally tangent at $A$. Points $B$ and $C$ are on the circle with center $O$ such that $\vartriangle ABC$ is equilateral. Segment $AB$ extends past B to point $D$ and $AC$ extends past $C$ to point $E$ such that $BD = CE =\sqrt3$. A line through $D$ is tangent to circle $P$ at $F$. The value of $EF^2$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are integers, c is squarefree, and $gcd(a,b,d) = 1$. Find $a +b +c +d$. [b]p27.[/b] Find the number of trailing zeroes at the end of $$\sum^{2021}_{i=1}(2021^i -1) = (2021^1 -1)...(2021^{2021}-1).$$ [u]Round 10[/u] [b]p28.[/b] Points $A, B, C, P$, and $D$ lie on circle ω in that order. Let $AC$ and $BD$ intersect at $I$ . Given that $PI = PC = PD$, $\angle DAB = 137^o$, and $\angle ABC = 109^o$, find the measure of $\angle BIC$ in degrees. [b]p29.[/b] Find the sum of all positive integers $n < 2021$ such that when ${d_1,d_2,... ,d_k}$ are the positive integer factors of $n$, then $$\left( \sum^{k}_{i=1}d_i \right) \left( \sum^{k}_{i=1} \frac{1}{d_i} \right)= r^2$$ for some rational number $r$ . [b]p30.[/b] Let $a, b, c, d$ and $e$ be positive real numbers. Define the function $f (x, y) = \frac{x}{y}+\frac{y}{x}$ for all positive real numbers. Given that $f (a,b) = 7$, $f (b,c) = 5$, $f (c,d) = 3$, and $f (d,e) = 2$, find the sum of all possible values of $f (e,a)$. [u]Round 11[/u] [b]p31.[/b] There exist $100$ (not necessarily distinct) complex numbers $r_1, r_2,..., r_{100}$ such that for any positive integer $1 \le k \le 100$, we have that $P(r_k ) = 0$ where the polynomial $P$ is defined as $$P(x) = \sum^{101}_{i=1}i \cdot x^{101-i} = x^{100} +2x^{99} +3x^{98} +...+99x^2 +100x +101.$$ Find the value of $$\prod^{100}_{j=1} (r^2_j+1) = (r^2_1 +1)(r^2_2 +1)...(r^2_{100} +1).$$ [b]p32.[/b] Let $BT$ be the diameter of a circle $\omega_1$, and $AT$ be a tangent of $\omega_1$. Line $AB$ intersects $\omega_1$ at $C$, and $\vartriangle ACT$ has circumcircle $\omega_2$. Points $P$ and $S$ exist such that $PA$ and $PC$ are tangent to $\omega_2$ and $SB = BT = 20$. Given that $AT = 15$, the length of $PS$ can be written as $\frac{a\sqrt{b}}{c}$ , where $a$, $b$, and $c$ are positive integers, $b$ is squarefree, and $gcd(a,b) = 1$. Find $a +b +c$. [b]p33.[/b] There are a hundred students in math team. Each pair of students are either mutually friends or mutually enemies. It is given that if any three students are chosen, then they are not all mutually friends. The maximum possible number of ways to choose four students such that it is possible to label them $A, B, C$, and $D$ such that $A$ and $B$ are friends, $B$ and $C$ are friends, $C$ and $D$ are friends, and D and A are friends can be expressed as $n^4$. Find $n$. [u]Round 12[/u] [b]p34.[/b] Let $\{p_i\}$ be the prime numbers, such that $p_1 = 2, p_2 = 3, p_3 = 5, ...$ For each $i$ , let $q_i$ be the nearest perfect square to $p_i$ . Estimate $\sum^{2021}_{i=1}|p_i=q_i |$. If the correct answer is $A$ and your answer is $E$, your score will be $\left \lfloor 30 \cdot \max - \left(0,1-5 \cdot \left| \log_{10} \frac{A}{E} \right| \right)\right \rfloor.$ [b]p35.[/b] Estimate the number of digits of $(2021!)^{2021}$. If the correct answer is $A$ and your answer is $E$, your score will be $\left \lfloor 15 \cdot \max \left(0,2- \cdot \left| \log_{10} \frac{A}{E} \right| \right) \right \rfloor.$ [b]p36.[/b] Pick a positive integer between$ 1$ and $1000$, inclusive. If your answer is $E$ and a quarter of the mean of all the responses to this problem is $A$, your score will be $$ \lfloor \max \left(0,30- |A-E|, 2-|E-1000| \right) \rfloor.$$ Note that if you pick $1000$, you will automatically get $2$ points. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166489p28814241]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166494p28814284]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a rectangle and denote by $M$ and $N$ the midpoints of $AD$ and $BC$ respectively. The point $P$ is on $(CD$ such that $D\in (CP)$, and $PM$ intersects $AC$ in $Q$. Prove that $m(\angle{MNQ})=m(\angle{MNP})$.

Find the smallest real number $A$, such that there are two different triangles, with integer sidelengths and so that the area of each be $A$.

[b]p1.[/b] Consider a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$ such that the perimeter of the right triangle is numerically (ignoring units) equal to its area. Prove that there is only one possible value of $a + b - c$, and determine that value. [b]p2.[/b] Last August, Jennifer McLoud-Mann, along with her husband Casey Mann and an undergraduate David Von Derau at the University of Washington, Bothell, discovered a new tiling pattern of the plane with a pentagon. This is the fifteenth pattern of using a pentagon to cover the plane with no gaps or overlaps. It is unknown whether other pentagons tile the plane, or even if the number of patterns is finite. Below is a portion of this new tiling pattern. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOS8xLzM4M2RjZDEzZTliYTlhYTJkZDU4YTA4ZGMwMTA0MzA5ODk1NjI0LnBuZw==&rn=bW1wYyAyMDE1LnBuZw==[/img] Determine the five angles (in degrees) of the pentagon $ABCDE$ used in this tiling. Explain your reasoning, and give the values you determine for the angles at the bottom. [b]p3.[/b] Let $f(x) =\sqrt{2019 + 4\sqrt{2015}} +\sqrt{2015} x$. Find all rational numbers $x$ such that $f(x)$ is a rational number. [b]p4.[/b] Alice has a whiteboard and a blackboard. The whiteboard has two positive integers on it, and the blackboard is initially blank. Alice repeats the following process. $\bullet$ Let the numbers on the whiteboard be $a$ and $b$, with $a \le b$. $\bullet$ Write $a^2$ on the blackboard. $\bullet$ Erase $b$ from the whiteboard and replace it with $b - a$. For example, if the whiteboard began with 5 and 8, Alice first writes $25$ on the blackboard and changes the whiteboard to $5$ and $3$. Her next move is to write $9$ on the blackboard and change the whiteboard to $2$ and $3$. Alice stops when one of the numbers on the whiteboard is 0. At this point the sum of the numbers on the blackboard is $2015$. a. If one of the starting numbers is $1$, what is the other? b. What are all possible starting pairs of numbers? [b]p5.[/b] Professor Beatrix Quirky has many multi-volume sets of books on her shelves. When she places a numbered set of $n$ books on her shelves, she doesn’t necessarily place them in order with book $1$ on the left and book $n$ on the right. Any volume can be placed at the far left. The only rule is that, except the leftmost volume, each volume must have a volume somewhere to its left numbered either one more or one less. For example, with a series of six volumes, Professor Quirky could place them in the order $123456$, or $324561$, or $564321$, but not $321564$ (because neither $4$ nor $6$ is to the left of $5$). Let’s call a sequence of numbers a [i]quirky [/i] sequence of length $n$ if: 1. the sequence contains each of the numbers from $1$ to $n$, once each, and 2. if $k$ is not the first term of the sequence, then either $k + 1$ or $k - 1$ occurs somewhere before $k$ in the sequence. Let $q_n$ be the number of quirky sequences of length $n$. For example, $q_3 = 4$ since the quirky sequences of length $3$ are $123$, $213$, $231$, and $321$. a. List all quirky sequences of length $4$. b. Find an explicit formula for $q_n$. Prove that your formula is correct. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the trapezoid $ABCD$ we know that $CD \perp BC, $ and $CD \perp AD .$ Circle $w$ with diameter $AB$ intersects $AD$ in points $A$ and $P,$ tangent from $P$ to $w$ intersects $CD$ at $M.$ The second tangent from $M$ to $w$ touches $w$ at $Q.$ Prove that midpoint of $CD$ lies on $BQ.$

Prove that if the angles $ A $, $ B $, $ C $ of a triangle satisfy the equation $$\cos 3A + \cos 3B + \cos 3C = 1,$$ then one of these angles equals $120^\circ $.

Let be two fixed points $ B,C. $ Find the locus of the spatial points $ A $ such that $ ABC $ is a nondegenerate triangle and the expression $$ R^2 (A)\cdot\sin \left( 2\angle ABC\right)\cdot\sin \left( 2\angle BCA\right) $$ has the greatest value possible, where $ R(A) $ denotes the radius of the excirlce of $ ABC. $

In a right circular cone, $A$ is the vertex, $B$ is the center of the base, and $C$ is a point on the circumference of the base with $BC = 1$ and $AB = 4$. There is a trapezoid $ABCD$ with $\overline{AB} \parallel \overline{CD}$. A right circular cylinder whose surface contains the points $A, C$, and $D$ intersects the cone such that its axis of symmetry is perpendicular to the plane of the trapezoid, and $\overline{CD}$ is a diameter of the cylinder. A sphere radius $r$ lies inside the cone and inside the cylinder. The greatest possible value of $r$ is $\frac{a\sqrt{b}-c}{d}$ , where $a, b, c$, and $d$ are positive integers, $a$ and $d$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a + b + c + d$.

In triangle $ABC$, $AB > BC > CA$, $AB=6$, $\angle{B}-\angle{C}=90^o$. The incircle touches $BC$ at $E$ and $EF$ is a diameter of the incircle. Radical $AF$ intersect $BC$ at $D$. $DE$ equals to the circumradius of $\triangle{ABC}$. Find $BC$ and $AC$.

In a right-angled triangle $ABC$ ($\angle A = 90^o$), the perpendicular bisector of $BC$ intersects the line $AC$ in $K$ and the perpendicular bisector of $BK$ intersects the line $AB$ in $L$. If the line $CL$ be the internal bisector of angle $C$, find all possible values for angles $B$ and $C$. by Mahdi Etesami Fard

A right triangle $ABC$ with hypotenuse $AB$ has side $AC = 15$. Altitude $CH$ divides $AB$ into segments $AH$ And $HB$, with $HB = 16$. The area of $\triangle ABC$ is: [asy] size(200); defaultpen(linewidth(0.8)+fontsize(11pt)); pair A = origin, H = (5,0), B = (13,0), C = (5,6.5); draw(C--A--B--C--H^^rightanglemark(C,H,B,16)); label("$A$",A,W); label("$B$",B,E); label("$C$",C,N); label("$H$",H,S); label("$15$",C/2,NW); label("$16$",(H+B)/2,S); [/asy] $\textbf{(A) }120\qquad \textbf{(B) }144\qquad \textbf{(C) }150\qquad \textbf{(D) }216\qquad \textbf{(E) }144\sqrt5$

Let $ABCDE$ be a convex pentagon with five equal sides and right angles at $C$ and $D$. Let $P$ denote the intersection point of the diagonals $AC$ and $BD$. Prove that the segments $PA$ and $PD$ have the same length. (Gottfried Perz)

Let $ABCD$ be a convex quadrilateral with shortest side $AB$ and longest side $CD$, and suppose that $AB < CD$. Show that there is a point $E \ne C, D$ on segment $CD$ with the following property: For all points $P \ne E$ on side $CD$, if we define $O_1$ and $O_2$ to be the circumcenters of $\vartriangle APD$ and $\vartriangle BPE$ respectively, then the length of $O_1O_2$ does not depend on $P$.

For square $ABCD$, $M$ is a midpoint of segment $CD$ and $E$ is a point on $AD$ satisfying $\angle BEM = \angle MED$. $P$ is an intersection of $AM$, $BE$. Find the value of $\frac{PE}{BP}$

it is given a polyhedral constructed from two regular pyramids with bases heptagons (a polygon with $7$ vertices) with common base $A_1A_2A_3A_4A_5A_6A_7$ and vertices respectively the points $B$ and $C$. The edges $BA_i , CA_i$ $(i = 1,...,7$), diagonals of the common base are painted in blue or red. Prove that there exists three vertices of the polyhedral given which forms a triangle with all sizes in the same color.

There are four houses, located on the vertices of a square. You want to draw a road network, so that you can go from any house to any other. Prove that the network formed by the diagonals is not the shortest. Find a shorter network.

Circles $O_1, O_2$ intersects at $A, B$. The circumcircle of $O_1BO_2$ intersects $O_1, O_2$ and line $AB$ at $R, S, T$ respectively. Prove that $TR = TS$

Eight $1\times 1$ square tiles are arranged as shown so their outside edges form a polygon with a perimeter of $14$ units. Two additional tiles of the same size are added to the figure so that at least one side of each tile is shared with a side of one of the squares in the original figure. Which of the following could be the perimeter of the new figure? [asy] for (int a=1; a <= 4; ++a) { draw((a,0)--(a,2)); } draw((0,0)--(4,0)); draw((0,1)--(5,1)); draw((1,2)--(5,2)); draw((0,0)--(0,1)); draw((5,1)--(5,2)); [/asy] $\text{(A)}\ 15 \qquad \text{(B)}\ 17 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qquad \text{(E)}\ 20$

A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, $A_0,$ and the hunter's starting point, $B_0$ are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}.$ In the $n^{\text{th}}$ round of the game, three things occur in order: [list=i] [*]The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1.$ [*]A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most $1.$ [*]The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly $1.$ [/list] Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds, she can ensure that the distance between her and the rabbit is at most $100?$ [i]Proposed by Gerhard Woeginger, Austria[/i]

A right-triangular wooden block of mass $M$ is at rest on a table, as shown in figure. Two smaller wooden cubes, both with mass $m$, initially rest on the two sides of the larger block. As all contact surfaces are frictionless, the smaller cubes start sliding down the larger block while the block remains at rest. What is the normal force from the system to the table? $\textbf{(A) } 2mg\\ \textbf{(B) } 2mg + Mg\\ \textbf{(C) } mg + Mg\\ \textbf{(D) } Mg + mg( \sin \alpha + \sin \beta)\\ \textbf{(E) } Mg + mg( \cos \alpha + \cos \beta)$

Let $ABCDE$ be a regular pentagon. The star $ACEBD$ has area 1. $AC$ and $BE$ meet at $P$, while $BD$ and $CE$ meet at $Q$. Find the area of $APQD$.

In the figure below, points $A$, $B$, and $C$ are distance 6 from each other. Say that a point $X$ is [i]reachable[/i] if there is a path (not necessarily straight) connecting $A$ and $X$ of length at most 8 that does not intersect the interior of $\overline{BC}$. (Both $X$ and the path must lie on the plane containing $A$, $B$, and $C$.) Let $R$ be the set of reachable points. What is the area of $R$? [asy] unitsize(40); pair A = dir(90); pair B = dir(210); pair C = dir(330); dot(A); dot(B); dot(C); draw(B -- C); label("$A$", A, N); label("$B$", B, W); label("$C$", C, E); [/asy]