Consider a cyclic quadrilateral $ABCD$, and let $S$ be the intersection of $AC$ and $BD$. Let $E$ and $F$ the orthogonal projections of $S$ on $AB$ and $CD$ respectively. Prove that the perpendicular bisector of segment $EF$ meets the segments $AD$ and $BC$ at their midpoints.

Let $a$, $b$, $c$, $d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.

Let $G=(S^1,\cdot)$ be a group. Then its nontrivial subgroups $\textbf{(A)}~\text{are necessarily finite}$ $\textbf{(B)}~\text{can be infinite}$ $\textbf{(C)}~\text{can be dense in }S^1$ $\textbf{(D)}~\text{None of the above}$

Let $ABC$ be a triangle. Construct points $B'$ and $ C'$ such that $ACB'$ and $ABC'$ are equilateral triangles that have no overlap with $ \vartriangle ABC$. Let $BB'$ and $CC'$ intersect at X. If $AX = 3$, $BC = 4$, and $CX = 5$, find the area of quadrilateral $BCB'C'$. .

Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.

Let $A$ and $B$ be two subsets of $S = \{1, 2, . . . , 2000\}$ with $|A| \cdot |B| \geq 3999$. For a set $X$ , let $X-X$ denotes the set $\{s-t | s, t \in X, s \not = t\}$. Prove that $(A-A) \cap (B-B)$ is nonempty.

Suppose $a$ is a complex number such that \[a^2+a+\frac{1}{a}+\frac{1}{a^2}+1=0\] If $m$ is a positive integer, find the value of \[a^{2m}+a^m+\frac{1}{a^m}+\frac{1}{a^{2m}}\]

[b]p1.[/b] An animal farm has geese and pigs with a total of $30$ heads and $84$ legs. Find the number of pigs and geese on this farm. [b]p2.[/b] What is the maximum number of $1 \times 1$ squares of a $7 \times 7$ board that can be colored black in such a way that the black squares don’t touch each other even at their corners? Show your answer on the figure below and explain why it is not possible to get more black squares satisfying the given conditions. [img]https://cdn.artofproblemsolving.com/attachments/d/5/2a0528428f4a5811565b94061486699df0577c.png[/img] [b]p3.[/b] Decide whether it is possible to divide a regular hexagon into three equal not necessarily regular hexagons? A regular hexagon is a hexagon with equal sides and equal angles. [img]https://cdn.artofproblemsolving.com/attachments/3/7/5d941b599a90e13a2e8ada635e1f1f3f234703.png[/img] [b]p4.[/b] A rectangle is subdivided into a number of smaller rectangles. One observes that perimeters of all smaller rectangles are whole numbers. Is it possible that the perimeter of the original rectangle is not a whole number? [b]p5.[/b] Place parentheses on the left hand side of the following equality to make it correct. $$ 4 \times 12 + 18 : 6 + 3 = 50$$ [b]p6.[/b] Is it possible to cut a $16\times 9$ rectangle into two equal parts which can be assembled into a square? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve, for integers $x$ and $y$ : $$2x^2y = (x+2)^2(y + 1), $$ provided that $(x+2)^2(y + 1)> 1000$.

A positive integer $n$ is called [i]Olympic[/i], if there exists a quadratic trinomial with integer coeffecients $f(x)$ satisfying $f(f(\sqrt{n}))=0$. Determine, with proof, the largest Olympic number not exceeding $2015$. [i]A. Khrabrov[/i]

Let a city be in the form of a square grid which has $n \times n$ cells, each of which contain a skyscraper . At first the $m$ skyscrapers burn, but the fire spreads: everyone skyscraper that has at least two burning neighboring houses (by neighboring houses we mean only houses that border the house along a street, not just at a corner) immediately gets fire. Prove that when in the end the whole city burns down, of must have been $m \ge n$. [hide=original wording in German] Eine Stadt habe die Form eines quadratischen Gitters, welches n × n Zellen habe, von denen jede ein Hochhaus enthalte. Anfangs brennen m der Hochh¨auser, doch der Brand pflanzt sich fort: Jedes Hochhaus, das mindestens zwei brennende Nachbarh¨auser hat (unter Nachbarh¨ausern verstehen wir dabei nur H¨auser, die entlang einer Straße an das Haus angrenzen, nicht nur an einer Ecke), f¨angt sofort Feuer. Man beweise: Wenn am Ende die gesamte Stadt abgebrannt ist,muss m ≥ n gewesen sein.[/hide]

Let $ABC$ be an acute-angled triangle with $\angle BAC = 60^{\circ}$ and with incenter $I$ and circumcenter $O$. Let $H$ be the point diametrically opposite(antipode) to $O$ in the circumcircle of $\triangle BOC$. Prove that $IH=BI+IC$.

A container is shaped like a right circular cone open at the top surmounted by a frustum which is open at the top and bottom as shown below. The lower cone has a base with radius 2 centimeters and height 6 centimeters while the frustum has bases with radii 2 and 8 centimeters and height 6 centimeters. If there is a rainfall measuring 2 centimeter of rain, the rain falling into the container will fill the container to a height of $m + 3\sqrt{n}$ cm, where m and n are positive integers. Find m + n.

In $\triangle ABC, \angle A = 100^\circ, \angle B = 50^\circ, \angle C = 30^\circ, \overline{AH}$ is an altitude, and $\overline{BM}$ is a median. Then $\angle MHC =$ [asy] draw((0,0)--(16,0)--(6,6)--cycle); draw((6,6)--(6,0)--(11,3)--(0,0)); dot((6,6)); dot((0,0)); dot((11,3)); dot((6,0)); dot((16,0)); label("A", (6,6), N); label("B", (0,0), W); label("C", (16,0), E); label("H", (6,0), S); label("M", (11,3), NE);[/asy] $\text{(A)} \ 15^\circ \qquad \text{(B)} \ 22.5^\circ \qquad \text{(C)} \ 30^\circ \qquad \text{(D)} \ 40^\circ \qquad \text{(E)} \ 45^\circ$

Let $\omega_1$ and $\omega_2$ be two non-intersecting circles. Suppose the following three conditions hold: [list] [*]The smallest of a common internal tangent of $\omega_1$ and $\omega_2$ is equal to $19.$ [*]The length of a common external tangent of $\omega_1$ and $\omega_2$ is equal to $37.$ [*]If two points $X$ and $Y$ are selected on $\omega_1$ and $\omega_2,$ respectively, uniformly at random, then the expected value of $XY^2$ is $2023.$ [/list] Compute the distance between the centers of $\omega_1$ and $\omega_2.$

Prove that every integer $n \ge 12$ is the sum of two composite numbers.

Vincent is playing a game with Evil Bill. The game uses an infinite number of red balls, an infinite number of green balls, and a very large bag. Vincent first picks two nonnegative integers $g$ and $k$ such that $g < k \le 2016$, and Evil Bill places $g$ green balls and $2016 - g$ red balls in the bag, so that there is a total of $2016$ balls in the bag. Vincent then picks a ball of either color and places it in the bag. Evil Bill then inspects the bag. If the ratio of green balls to total balls in the bag is ever exactly $\frac{k}{2016}$ , then Evil Bill wins. If the ratio of green balls to total balls is greater than $\frac{k}{2016}$ , then Vincent wins. Otherwise, Vincent and Evil Bill repeat the previous two actions (Vincent picks a ball and Evil Bill inspects the bag). If $S$ is the sum of all possible values of $k$ that Vincent could choose and be able to win, determine the largest prime factor of $S$.

Find $ \sum_{k \in A} \frac{1}{k-1}$ where $A= \{ m^n : m,n \in \mathbb{Z} m,n \geq 2 \} $. Problem was post earlier [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=67&t=29456&hilit=silk+road]here[/url] , but solution not gives and olympiad doesn't indicate, so I post it again :blush: Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

Let $ABC$ be a triangle with $AB<AC$ and let $M $ be the midpoint of $BC$. $MI$ ($I$ incenter) intersects $AB$ at $D$ and $CI$ intersects the circumcircle of $ABC$ at $E$. Prove that $\frac{ED }{ EI} = \frac{IB }{IC}$ [img]https://cdn.artofproblemsolving.com/attachments/0/5/4639d82d183247b875128a842a013ed7415fba.jpg[/img] [hide=.][url=http://artofproblemsolving.com/community/c6h602657p10667541]source[/url], translated by Antreas Hatzipolakis in fb, corrected by me in order to be compatible with it's figure[/hide]

Find the least positive integer $n$ satisfying the following statement: for eash pair of positive integers $a$ and $b$ such that $36$ divides $a+b$ and $n$ divides $ab$ it follows that $36$ divides both $a$ and $b$.

Say that a positive integer is [i]red[/i] if it is of the form $n^{2020}$, where $n$ is a positive integer. Say that a positive integer is [i]blue[/i] if it is not red and is of the form $n^{2019}$, where $n$ is a positive integer. True or false: Between every two different red positive integers greater than $10^{100{,}000{,}000}$, there are at least 2019 blue positive integers. Prove that your answer is correct.

A polygon is decomposed into triangles by drawing some non-intersecting interior diagonals in such a way that for every pair of triangles of the triangulation sharing a common side, the sum of the angles opposite to this common side is greater than $180^o$. a) Prove that this polygon is convex. b) Prove that the circumcircle of every triangle used in the decomposition contains the entire polygon. [i]Proposed by Morteza Saghafian - Iran[/i]

Positive real numbers are arranged in the form: $ 1 \ \ \ 3 \ \ \ 6 \ \ \ 10 \ \ \ 15 ...$ $ 2 \ \ \ 5 \ \ \ 9 \ \ \ 14 ...$ $ 4 \ \ \ 8 \ \ \ 13 ...$ $ 7 \ \ \ 12 ...$ $ 11 ...$ Find the number of the line and column where the number 2002 stays.

Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $I$ be the incentre of triangle $ABC$, and let $D,E,F$ be the points where the incircle touches the sides $BC,CA,AB,$ respectively. Let $BI,CI$ meet the line $EF$ at $Y,X$ respectively. Further assume that both $X$ and $Y$ are outside the triangle $ABC$. Prove that $\text{(i)}$ $B,C,Y,X$ are concyclic. $\text{(ii)}$ $I$ is also the incentre of triangle $DYX$.

Let $n\ge 2$ be an integer, and let $a_1, a_2, \cdots , a_n$ be positive integers. Show that there exist positive integers $b_1, b_2, \cdots, b_n$ satisfying the following three conditions: $\text{(A)} \ a_i\le b_i$ for $i=1, 2, \cdots , n;$ $\text{(B)} \ $ the remainders of $b_1, b_2, \cdots, b_n$ on division by $n$ are pairwise different; and $\text{(C)} \ $ $b_1+b_2+\cdots b_n \le n\left(\frac{n-1}{2}+\left\lfloor \frac{a_1+a_2+\cdots a_n}{n}\right \rfloor \right)$ (Here, $\lfloor x \rfloor$ denotes the integer part of real number $x$, that is, the largest integer that does not exceed $x$.)