A cyclic quadrilateral $ABCD$ has diameter $AC$ with circumcircle $\omega$. Let $K$ be the foot of the perpendicular from $C$ to $BD$, and the tangent to $\omega$ at $A$ meets $BD$ at $T$. Let the line $AK$ meets $\omega$ at $X$ and choose a point $Y$ on line $AK$ such that $\angle TYA=90^{\circ}$. Prove that $AY=KX$. [i]Proposed by Anzo Teh Zhao Yang[/i]

A particle is located on the coordinate plane at $ (5,0)$. Define a [i]move[/i] for the particle as a counterclockwise rotation of $ \pi/4$ radians about the origin followed by a translation of $ 10$ units in the positive $ x$-direction. Given that the particle's position after $ 150$ moves is $ (p,q)$, find the greatest integer less than or equal to $ |p|\plus{}|q|$.

Equilateral triangle $ABC$ has side length $6$. Circles with centers at $A$, $B$, and $C$ are drawn such that their respective radii $r_A$, $r_B$, and $r_C$ form an arithmetic sequence with $r_A<r_B<r_C$. If the shortest distance between circles $A$ and $B$ is $3.5$, and the shortest distance between circles $A$ and $C$ is $3$, then what is the area of the shaded region? Express your answer in terms of pi. [asy] size(8cm); draw((0,0)--(6,0)--6*dir(60)--cycle); draw(circle((0,0),1)); draw(circle(6*dir(60),1.5)); draw(circle((6,0),2)); filldraw((0,0)--arc((0,0),1,0,60)--cycle, grey); filldraw(6*dir(60)--arc(6*dir(60),1.5,240,300)--cycle, grey); filldraw((6,0)--arc((6,0),2,120,180)--cycle, grey); label("$A$",(0,0),SW); label("$B$",6*dir(60),N); label("$C$",(6,0),SE); [/asy]

An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is $2$, then the area of the hexagon is $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad \textbf{(E) }12$

Let $ABCD$ be a trapezium (trapezoid) with $AD$ parallel to $BC$ and $J$ be the intersection of the diagonals $AC$ and $BD$. Point $P$ a chosen on the side $BC$ such that the distance from $C$ to the line $AP$ is equal to the distance from $B$ to the line $DP$. [i]The following three questions 1, 2 and 3 are independent, so that a condition in one question does not apply in another question.[/i] 1.Suppose that $Area( \vartriangle AJB) =6$ and that $Area(\vartriangle BJC) = 9$. Determine $Area(\vartriangle APD)$. 2. Find all points $Q$ on the plane of the trapezium such that $Area(\vartriangle AQB) = Area(\vartriangle DQC)$. 3. Prove that $PJ$ is the angle bisector of $\angle APD$.

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [b]L.10[/b] Given the following system of equations where $x, y, z$ are nonzero, find $x^2 + y^2 + z^2$. $$x + 2y = xy$$ $$3y + z = yz$$ $$3x + 2z = xz$$ [u]Set 4[/u] [b]L.16 / D.23[/b] Anson, Billiam, and Connor are looking at a $3D$ figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a $5 \times 5$ square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure. [b]L.17[/b] The repeating decimal $0.\overline{MBMT}$ is equal to $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, and $M, B, T$ are distinct digits. Find the minimum value of $q$. [b]L.18[/b] Annie, Bob, and Claire have a bag containing the numbers $1, 2, 3, . . . , 9$. Annie randomly chooses three numbers without replacement, then Bob chooses, then Claire gets the remaining three numbers. Find the probability that everyone is holding an arithmetic sequence. (Order does not matter, so $123$, $213$, and $321$ all count as arithmetic sequences.) [b]L.19[/b] Consider a set $S$ of positive integers. Define the operation $f(S)$ to be the smallest integer $n > 1$ such that the base $2^k$ representation of $n$ consists only of ones and zeros for all $k \in S$. Find the size of the largest set $S$ such that $f(S) < 2^{2019}$. [b]L.20 / D.25[/b] Find the largest solution to the equation $$2019(x^{2019x^{2019}-2019^2+2019})^{2019} = 2019^{x^{2019}+1}.$$ [u]Set 5[/u] [b]L.21[/b] Steven is concerned about his artistic abilities. To make himself feel better, he creates a $100 \times 100$ square grid and randomly paints each square either white or black, each with probability $\frac12$. Then, he divides the white squares into connected components, groups of white squares that are connected to each other, possibly using corners. (For example, there are three connected components in the following diagram.) What is the expected number of connected components with 1 square, to the nearest integer? [img]https://cdn.artofproblemsolving.com/attachments/e/d/c76e81cd44c3e1e818f6cf89877e56da2fc42f.png[/img] [b]L.22[/b] Let x be chosen uniformly at random from $[0, 1]$. Let n be the smallest positive integer such that $3^n x$ is at most $\frac14$ away from an integer. What is the expected value of $n$? [b]L.23[/b] Let $A$ and $B$ be two points in the plane with $AB = 1$. Let $\ell$ be a variable line through $A$. Let $\ell'$ be a line through $B$ perpendicular to $\ell$. Let X be on $\ell$ and $Y$ be on $\ell'$ with $AX = BY = 1$. Find the length of the locus of the midpoint of $XY$ . [b]L.24[/b] Each of the numbers $a_i$, where $1 \le i \le n$, is either $-1$ or $1$. Also, $$a_1a_2a_3a_4+a_2a_3a_4a_5+...+a_{n-3}a_{n-2}a_{n-1}a_n+a_{n-2}a_{n-1}a_na_1+a_{n-1}a_na_1a_2+a_na_1a_2a_3 = 0.$$ Find the number of possible values for $n$ between $4$ and $100$, inclusive. [b]L.25[/b] Let $S$ be the set of positive integers less than $3^{2019}$ that have only zeros and ones in their base $3$ representation. Find the sum of the squares of the elements of $S$. Express your answer in the form $a^b(c^d - 1)(e^f - 1)$, where $a, b, c, d, e, f$ are positive integers and $a, c, e$ are not perfect powers. PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and D.16-30/ L10-15 [url=https://artofproblemsolving.com/community/c3h2790818p24541688]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] What is the units digit of the product of the first seven primes? [b]p2. [/b]In triangle $ABC$, $\angle BAC$ is a right angle and $\angle ACB$ measures $34$ degrees. Let $D$ be a point on segment $ BC$ for which $AC = CD$, and let the angle bisector of $\angle CBA$ intersect line $AD$ at $E$. What is the measure of $\angle BED$? [b]p3.[/b] Chad numbers five paper cards on one side with each of the numbers from $ 1$ through $5$. The cards are then turned over and placed in a box. Jordan takes the five cards out in random order and again numbers them from $ 1$ through $5$ on the other side. When Chad returns to look at the cards, he deduces with great difficulty that the probability that exactly two of the cards have the same number on both sides is $p$. What is $p$? [b]p4.[/b] Only one real value of $x$ satisfies the equation $kx^2 + (k + 5)x + 5 = 0$. What is the product of all possible values of $k$? [b]p5.[/b] On the Exeter Space Station, where there is no effective gravity, Chad has a geometric model consisting of $125$ wood cubes measuring $ 1$ centimeter on each edge arranged in a $5$ by $5$ by $5$ cube. An aspiring carpenter, he practices his trade by drawing the projection of the model from three views: front, top, and side. Then, he removes some of the original $125$ cubes and redraws the three projections of the model. He observes that his three drawings after removing some cubes are identical to the initial three. What is the maximum number of cubes that he could have removed? (Keep in mind that the cubes could be suspended without support.) [b]p6.[/b] Eric, Meena, and Cameron are studying the famous equation $E = mc^2$. To memorize this formula, they decide to play a game. Eric and Meena each randomly think of an integer between $1$ and $50$, inclusively, and substitute their numbers for $E$ and $m$ in the equation. Then, Cameron solves for the absolute value of $c$. What is the probability that Cameron’s result is a rational number? [b]p7.[/b] Let $CDE$ be a triangle with side lengths $EC = 3$, $CD = 4$, and $DE = 5$. Suppose that points $ A$ and $B$ are on the perimeter of the triangle such that line $AB$ divides the triangle into two polygons of equal area and perimeter. What are all the possible values of the length of segment $AB$? [b]p8.[/b] Chad and Jordan are raising bacteria as pets. They start out with one bacterium in a Petri dish. Every minute, each existing bacterium turns into $0, 1, 2$ or $3$ bacteria, with equal probability for each of the four outcomes. What is the probability that the colony of bacteria will eventually die out? [b]p9.[/b] Let $a = w + x$, $b = w + y$, $c = x + y$, $d = w + z$, $e = x + z$, and $f = y + z$. Given that $af = be = cd$ and $$(x - y)(x - z)(x - w) + (y - x)(y - z)(y - w) + (z - x)(z - y)(z - w) + (w - x)(w - y)(w - z) = 1,$$ what is $$2(a^2 + b^2 + c^2 + d^2 + e^2 + f^2) - ab - ac - ad - ae - bc - bd - bf - ce - cf - de - df - ef ?$$ [b]p10.[/b] If $a$ and $b$ are integers at least $2$ for which $a^b - 1$ strictly divides $b^a - 1$, what is the minimum possible value of $ab$? Note: If $x$ and $y$ are integers, we say that $x$ strictly divides $y$ if $x$ divides $y$ and $|x| \ne |y|$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

Let $ABCDEF$ be a convex hexagon with area $S$ such that $AB \parallel DE$, $BC \parallel EF$, $CD \parallel FA$ holds, and whose all angles are obtuse and opposite sides are not the same length. Prove that the following inequality holds: $$A_{ABC} + A_{BCD} + A_{CDE} + A_{DEF} + A_{EFA} + A_{FAB} < S$$ , where $A_{XYZ}$ is the area of triangle $XYZ$

In a given disc, construct a subset such that its area equals the half of the disc area and its intersection with its reflection over an arbitrary diameter has the area equal to the quarter of the disc area.

Let $ABCD$ be a square and let $X$ be any point on side $BC$ between $B$ and $C$. Let $Y$ be the point on line $CD$ such that $BX = YD$ and $D$ is between $C$ and $Y$ . Prove that the midpoint of $XY$ lies on diagonal $BD$.

Prove that every multiple of $3$ can be written as a sum of four cubes (positive or negatives).

A circle $c$ with center $A$ passes through the vertices $B$ and $E$ of a regular pentagon $ABCDE$. The line $BC$ intersects the circle $c$ for second time at point $F$. Prove that the lines $DE$ and $EF$ are perpendicular.

The letter T is formed by placing two $ 2\times 4$ inch rectangles next to each other, as shown. What is the perimeter of the T, in inches? [asy]size(150); draw((0,6)--(4,6)--(4,4)--(3,4)--(3,0)--(1,0)--(1,4)--(0,4)--cycle, linewidth(1));[/asy] $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 24$

Let $a, b, c$ be fìxed natural numbers. Suppose that, for every positive integer n, there is a triangle whose sides have lengths $a^n$, $b^n$, and $c^n$ respectively. Prove that these triangles are isosceles.

Given three non-collinear points $A,B,C$, construct a circle with centre $C$ such that the tangents from $A$ and $B$ are parallel.

Let $P$ be a point inside a triangle $ABC$ such that $\angle PAB = \angle PCA$ and $\angle PAC = \angle PBA$. If $O \ne P$ is the circumcenter of $\triangle ABC$, prove that $\angle APO$ is right.

Circle $ O$ has diameters $ AB$ and $ CD$ perpendicular to each other. $ AM$ is any chord intersecting $ CD$ at $ P$. Then $ AP\cdot AM$ is equal to: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm); pair O = origin; pair A = (-1,0); pair B = (1,0); pair C = (0,1); pair D = (0,-1); pair M = dir(45); pair P = intersectionpoint(O--C,A--M); draw(Circle(O,1)); draw(A--B); draw(C--D); draw(A--M); label("$A$",A,W); label("$B$",B,E); label("$C$",C,N); label("$D$",D,S); label("$M$",M,NE); label("$O$",O,NE); label("$P$",P,NW);[/asy]$ \textbf{(A)}\ AO\cdot OB \qquad \textbf{(B)}\ AO\cdot AB\qquad \textbf{(C)}\ CP\cdot CD \qquad \textbf{(D)}\ CP\cdot PD\qquad$ $ \textbf{(E)}\ CO\cdot OP$

[b]p1.[/b] What is the largest number of five dollar footlongs Jimmy can buy with 88 dollars? [b]p2.[/b] Austin, Derwin, and Sylvia are deciding on roles for BMT $2021$. There must be a single Tournament Director and a single Head Problem Writer, but one person cannot take on both roles. In how many ways can the roles be assigned to Austin, Derwin, and Sylvia? [b]p3.[/b] Sofia has$ 7$ unique shirts. How many ways can she place $2$ shirts into a suitcase, where the order in which Sofia places the shirts into the suitcase does not matter? [b]p4.[/b] Compute the sum of the prime factors of $2021$. [b]p5.[/b] A sphere has volume $36\pi$ cubic feet. If its radius increases by $100\%$, then its volume increases by $a\pi$ cubic feet. Compute $a$. [b]p6.[/b] The full price of a movie ticket is $\$10$, but a matinee ticket to the same movie costs only $70\%$ of the full price. If $30\%$ of the tickets sold for the movie are matinee tickets, and the total revenue from movie tickets is $\$1001$, compute the total number of tickets sold. [b]p7.[/b] Anisa rolls a fair six-sided die twice. The probability that the value Anisa rolls the second time is greater than or equal to the value Anisa rolls the first time can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p8.[/b] Square $ABCD$ has side length $AB = 6$. Let point $E$ be the midpoint of $\overline{BC}$. Line segments $\overline{AC}$ and $\overline{DE}$ intersect at point $F$. Compute the area of quadrilateral ABEF. [b]p9.[/b] Justine has a large bag of candy. She splits the candy equally between herself and her $4$ friends, but she needs to discard three candies before dividing so that everyone gets an equal number of candies. Justine then splits her share of the candy between herself and her two siblings, but she needs to discard one candy before dividing so that she and her siblings get an equal number of candies. If Justine had instead split all of the candy that was originally in the large bag between herself and $14$ of her classmates, what is the fewest number of candies that she would need to discard before dividing so that Justine and her $14$ classmates get an equal number of candies? [b]p10.[/b] For some positive integers $a$ and $b$, $a^2 - b^2 = 400$. If $a$ is even, compute $a$. [b]p11.[/b] Let $ABCDEFGHIJKL$ be the equilateral dodecagon shown below, and each angle is either $90^o$ or $270^o$. Let $M$ be the midpoint of $\overline{CD}$, and suppose $\overline{HM}$ splits the dodecagon into two regions. The ratio of the area of the larger region to the area of the smaller region can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/3/e/387bcdf2a6c39fcada4f21f24ceebd18a7f887.png[/img] [b]p12.[/b] Nelson, who never studies for tests, takes several tests in his math class. Each test has a passing score of $60/100$. Since Nelson's test average is at least $60/100$, he manages to pass the class. If only nonnegative integer scores are attainable on each test, and Nelson gets a dierent score on every test, compute the largest possible ratio of tests failed to tests passed. Assume that for each test, Nelson either passes it or fails it, and the maximum possible score for each test is 100. [b]p13.[/b] For each positive integer $n$, let $f(n) = \frac{n}{n+1} + \frac{n+1}{n}$ . Then $f(1)+f(2)+f(3)+...+f(10)$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p14.[/b] Triangle $\vartriangle ABC$ has point $D$ lying on line segment $\overline{BC}$ between $B$ and $C$ such that triangle $\vartriangle ABD$ is equilateral. If the area of triangle $\vartriangle ADC$ is $\frac14$ the area of triangle $\vartriangle ABC$, then $\left( \frac{AC}{AB}\right)^2$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p15.[/b] In hexagon $ABCDEF$, $AB = 60$, $AF = 40$, $EF = 20$, $DE = 20$, and each pair of adjacent edges are perpendicular to each other, as shown in the below diagram. The probability that a random point inside hexagon $ABCDEF$ is at least $20\sqrt2$ units away from point $D$ can be expressed in the form $\frac{a-b\pi}{c}$ , where $a$, $b$, $c$ are positive integers such that gcd$(a, b, c) = 1$. Compute $a + b + c$. [img]https://cdn.artofproblemsolving.com/attachments/3/c/1b45470265d10a73de7b83eff1d3e3087d6456.png[/img] [b]p16.[/b] The equation $\sqrt{x} +\sqrt{20-x} =\sqrt{20 + 20x - x^2}$ has $4$ distinct real solutions, $x_1$, $x_2$, $x_3$, and $x_4$. Compute $x_1 + x_2 + x_3 + x_4$. [b]p17.[/b] How many distinct words with letters chosen from $B, M, T$ have exactly $12$ distinct permutations, given that the words can be of any length, and not all the letters need to be used? For example, the word $BMMT$ has $12$ permutations. Two words are still distinct even if one is a permutation of the other. For example, $BMMT$ is distinct from $TMMB$. [b]p18.[/b] We call a positive integer binary-okay if at least half of the digits in its binary (base $2$) representation are $1$'s, but no two $1$s are consecutive. For example, $10_{10} = 1010_2$ and $5_{10} = 101_2$ are both binary-okay, but $16_{10} = 10000_2$ and $11_{10} = 1011_2$ are not. Compute the number of binary-okay positive integers less than or equal to $2020$ (in base $10$). [b]p19.[/b] A regular octahedron (a polyhedron with $8$ equilateral triangles) has side length $2$. An ant starts on the center of one face, and walks on the surface of the octahedron to the center of the opposite face in as short a path as possible. The square of the distance the ant travels can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/f/8/3aa6abe02e813095e6991f63fbcf22f2e0431a.png[/img] [b]p20.[/b] The sum of $\frac{1}{a}$ over all positive factors $a$ of the number $360$ can be expressed in the form $\frac{m}{n}$ ,where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Triangle $ABC$ has circumcenter $O$. $D$ is the foot from $A$ to $BC$, and $P$ is apoint on $AD$. The feet from $P$ to $CA, AB$ are $E, F$, respectively, and the foot from $D$ to $EF$ is $T$. $AO$ meets $(ABC)$ again at $A'$. $A'D$ meets $(ABC)$ again at $R$. If $Q$ is a point on $AO$ satisfying $\angle ABP = \angle QBC$, prove that $D, P, T, R$ lie on acircle and $DQ$ is tangent to it.

Let $ n > 4$ be a non-negative integer. Given is the in a circle inscribed convex $ n$-gon $ A_0A_1A_2\dots A_{n \minus{} 1}A_n$ $ (A_n \equal{} A_0)$ where the side $ A_{i \minus{} 1}A_i \equal{} i$ (for $ 1 \le i \le n$). Moreover, let $ \phi_i$ be the angle between the line $ A_iA_{i \plus{} 1}$ and the tangent to the circle in the point $ A_i$ (where the angle $ \phi_i$ is less than or equal $ 90^o$, i.e. $ \phi_i$ is always the smaller angle of the two angles between the two lines). Determine the sum $ \Phi \equal{} \sum_{i \equal{} 0}^{n \minus{} 1} \phi_i$ of these $ n$ angles.

In the plane a point $O$ is and a sequence of points $P_1, P_2, P_3, \ldots$ are given. The distances $OP_1, OP_2, OP_3, \ldots$ are $r_1, r_2, r_3, \ldots$ Let $\alpha$ satisfies $0 < \alpha < 1.$ Suppose that for every $n$ the distance from the point $P_n$ to any other point of the sequence is $\geq r^{\alpha}_n.$ Determine the exponent $\beta$, as large as possible such that for some $C$ independent of $n$ \[r_n \geq Cn^{\beta}, n = 1,2, \ldots\]

Let points $ A = (0,0) , \ B = (1,2), \ C = (3,3), $ and $ D = (4,0) $. Quadrilateral $ ABCD $ is cut into equal area pieces by a line passing through $ A $. This line intersects $ \overline{CD} $ at point $ \left (\frac{p}{q}, \frac{r}{s} \right ) $, where these fractions are in lowest terms. What is $ p + q + r + s $? $ \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D)} \ 70 \qquad \textbf{(E)} \ 75 $

Can we draw $\triangle ABC$ and points $X,Y$, such that $AX=BY=AB$, $ BX = CY = BC$, $CX = AY = CA$?